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Cosmic ray acceleration of gas in active galactic nuclei Eilek, Jean Anne 1975

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COSMIC BAY ACCELERATION OF GAS IN ACTIVE GALACTIC NUCLEI by JEAN ANNE EILEK A., U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , 1968 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS AND ASTRONOMY We accept t h i s t h e s i s as comforming t o the re g u i r e d ^ s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Co lumb ia , I ag ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date i ABSTRACT Dynamical models of S e y f e r t n u c l e i and q u a s i - s t e l l a r ' o b j e c t s are presented. The c e n t r a l energy source o f t e n p o s t u l a t e d f o r these a c t i v e o b j e c t s p r o v i d e s a means of h e a t i n g and i o n i z i n g the n u c l e a r gas, and a l s o e x e r t s an/outward f o r c e on the gas. Si n c e the gas w i l l be f u l l y ionize/d, i t w i l l be n e a r l y t r a n s p a r e n t t o X-rays, while cosmic rays w i l l i n t e r a c t s t r o n g l y with i t . P r e l i m i n a r y c a l c u l a t i o n s of t h i s " i o n i z a t i o n " p ressure on d i s c r e t e c l o u d s show that photons are u n l i k e l y to produce the high gas v e l o c i t i e s r e l a t i v e to the nucleus which are i n d i c a t e d by the emission l i n e p r o f i l e s i n S e y f e r t n u c l e i and the b l u e s h i f t e d quasar a b s o r p t i o n l i n e s , but that cosmic r a y s can a c c e l e r a t e the c l o u d s up to these v e l o c i t i e s . A more d e t a i l e d c a l c u l a t i o n t a k i n g i n t o account the dynamics of the gas i s c a l l e d f o r . A computer code was w r i t t e n to s o l v e the s p h e r i c a l l y symmetric hydrodynaraic equations n u m e r i c a l l y . I t uses a f i n i t e d i f f e r e n c e , i m p l i c i t E u l e r i a n scheme to s o l v e the time dependent equ a t i o n s . As w e l l as the mass c o n s e r v a t i o n and momentum t r a n s f e r e q u a t i o n s , the numerical system i n c l u d e s an energy equation which a l l o w s f o r i o n i z a t i o n and Coulomb hea t i n g , and r a d i a t i v e c o o l i n g . The code was used t o o b t a i n a set of n u c l e a r e v o l u t i o n a r y models. These models i n v o l v e a s t a t i c gas surrounding a q u i e s c e n t energy source which t u r n s on suddenly. A range of i n p u t p h y s i c a l parameters i s re p r e s e n t e d : f o r s i z e s 0.1 to 1 pc, a t o t a l cosmic ray f l u x from 10*^ ergs s ~ l to 1 0 4 8 ergs s _ 1 , a gas d e n s i t y of 10* to 10 8 cm - 3, a lowest p a r t i c l e energy i n a power law spectrum of 0.1 to 10.0 MeV, and a c e n t r a l mass of 10 8 or 10 ' f l 0 . i i Such s o f t cosmic rays have a very s h o r t a b s o r p t i o n l e n g t h i n the n u c l e a r gas. T h i s means a narrow r e g i o n i n r a d i a l extent w i l l g ain the momentum of the cosmic ray beam, and an outward moving s h e l l w i l l form. I t snowplows the c o o l e r gas ahead of i t and l e a v e s a l e s s dense, hot c a v i t y behind. T h i s t h i n c a v i t y reaches temperatures of 10 8 K, and the dense s h e l l reaches an e q u i l i b r i u m temperature i n the range 10 4-10 5 K. The s h e l l v e l o c i t i e s i n c r e a s e d as the cosmic ray f l u x was i n c r e a s e d , ranging from 500 to 8000 km s - 1 . The l i f e t i m e of t h i s phenomenon i s the time f o r the s h e l l t o escape the n u c l e a r r e g i o n , which i s only a few parsecs a c r o s s . At these v e l o c i t i e s , the t i m e s c a l e i s only 10 J to 10 4 y e a r s . T h i s suggests r e p e t i t i v e r a t h e r than continuous a c t i v i t y of the c e n t r a l source. A q u i e s c e n t phase would allow replenishment of the gas from e x t r a - n u c l e a r s t e l l a r s o u r c e s . The i n t e r f a c e between the hot c a v i t y and the s h e l l i s R a y l e i g h - T a y l o r u n s t a b l e with a fragmentation time approximately equal to the s h e l l escape time. T h i s may e x p l a i n the c l o u d s t r u c t u r e observed i n these o b j e c t s . Thermal i n s t a b i l i t i e s may a l s o a r i s e i f the c e n t r a l source t u r n s o f f . P r e d i c t i o n of the sources of the p e r m i t t e d and f o r b i d d e n emission l i n e s i s dependent on the behavior of the i n s t a b i l i t i e s . The very dense s h e l l suggests a p h y s i c a l d i s t i n c t i o n between the r e g i o n s producing the two types of spectra,- which may e x p l a i n the wider p e r m i t t e d l i n e s i n some sour c e s . The hot gas near the energy source w i l l produce thermal X-rays. The l u m i n o s i t y and temperature p r e d i c t e d f o r the X-rays i s c o n s i s t e n t with o b s e r v a t i o n s . i i i TAELE OF CONTENTS page An ABSTRACT precades t h i s i TABLE OF CONTENTS. There f o l l o w s a i i i LIST OF TABLES, a V LIST OF FIGURES, and v i ACKNOWLEDGEMENTS. v i i i CHAPTER I i s an INTRODUCTION t o t h i s t h e s i s , 1 wherein some i n t r i g u i n g t a c t s about S e y f e r t g a l a x i e s are presented, and a b r i e f d i s c u s s i o n of the plan of t h i s t h e s i s i s l a i d out. In CHAPTER I I , SIMPLE MODELS of the gas are 8 di s c u s s e d i n terms of what p h y s i c a l q u a n t i t i e s can be i n f e r r e d from o b s e r v a t i o n s . An attempt to estimate c l o u d a c c e l e r a t i o n f o l l o w s , and t h i s f u r t h e r encourages CHAPTER I I I , which presents the CONTINUOUS FLOW 33 CALCULATIONS. The model to be c a l c u l a t e d i s d e s c r i b e d , and the m i c r o s c o p i c p h y s i c s are c o n s i d e r e d . CHAPTER IV d e s c r i b e s the NUMERICAL CODE w r i t t e n 48 to s o l v e the hydrodynamic e q u a t i o n s . These c a l c u l a t i o n s form the b a s i s of CHAPTER V, where the EVOLUTION OF THE GAS i s 70 presented i n terms of s e v e r a l numerical models. . These are f u r t h e r d i s c u s s e d i n CHAPTER VI, wherein the IMPLICATIONS of these 155 c a l c u l a t i o n s are i n v e s t i g a t e d , i n c l u d i n g the s t a b i l i t y of the models to f r a g a i e n t a t i o n , and t h e i r r e l a t i o n to o b s e r v a t i o n s . F i n a l l y , i n CHAPTER VII, the work i s summarized, CONCLUSIONS drawn and s u g g e s t i o n s f o r f u t u r e work are given. BIBLIOGRAPHY LIST OF SYMBOLS APPENDIX I pr e s e n t s some r a t h e r lengthy c a l c u l a t i o n s of the ATTENUATION OF THE IONIZING FLUX, whether cosmic r a y s or photons. APPENDIX I I d i s c u s s e s the NUMERICAL STABILITY of the computer program and pr e s e n t s v a r i o u s t e s t s of same. APPENDIX I I I i s a l i s t i n g of the COMPUTER CODE. LIST OF TABLES I . V e l o c i t y S t r u c t u r e Of Line P r o f i l e s 11 I I . Parameters Of A Standard A c t i v e Nucleus 17 I I I . Cosmic Ray A c c e l e r a t i o n Of F r e e l y Expanding Clouds 31 / IV. Computational Parameters / 52 V. The Hydrodynamic Equations / 54 VI. The C a l c u l a t i o n C y c l e 68 V I I . C o o l Mode Models, Computational Parameters 93 V I I I . S h a l l Models, P h y s i c a l Parameters 153 i X . Comparison Of Models With O b s e r v a t i o n s 167 v i LIST OF FIGURES 0 . S e y f e r t Nuclear S p e c t r a 18 1 . R a d i a t i v e C o o l i n g Curve 72 2 . I o n i z a t i o n S t r u c t u r e 76 3 . X Ray Model - Densi t y 81 4 . X Ray Model - V e l o c i t y 82 5 . Hot Mode C a l c u l a t i o n . 90 6 . S h e l l Model 1 (40 point) - D e n s i t y 100 7 . S h e l l Model 1 (40 point) - V e l o c i t y 101 8a. S h e l l Model 1 (40 Point) - Temperature 103 8b. S h e l l Model 1 (40 Point) - Cosmic Ray A b s o r p t i o n 105 9 . S h e l l Model 1 (40 point) - Density Contour 107 10 . S h e l l Model 1 (40 point) - V e l o c i t y Contour 108 11 . S h e l l Model 1 - B (t) And E (t) 110 12 . S h e l l Model 1 (60 point) - Density 112 13 . S h e l l Model 1 (60 point) - V e l o c i t y 113 14 . S h e l l Model 1 (60 point) - Density Contour 115 15 . S h e l l Model 1 (60 point) - V e l o c i t y Contour 116 16 . S h e l l Model 1 ( a d i a b a t i c ) - Density 118 17 . S h e l l Model 1 ( a d i a b a t i c ) - V e l o c i t y 119 18 . S h e l l V e l o c i t y v (t) - Models 1A, 1B, 2, 3 124 19 . S h e l l V e l o c i t y v (t) - Models 4, 5, 6, 7 131 20 . S h e l l Model 5 - Density 133 21 . S h e l l Model 5 - V e l o c i t y 134 2 2 . S h e l l Model 5 - Density Contour 136 23 . S h e l l Model 5 - V e l o c i t y Contour 137 23a- S h e l l Model 5 - Temperature 139 24 - S h e l l Model 6 - Density 14 1 v i i 25 . S h e l l Model 6 - V e l o c i t y 142 26 . S h e l l Model 6 - Density Contour 144 27 . S h e l l Model 6 - V e l o c i t y Contour 145 28 . S h e l l Model 7 - Density 147 29 . S h e l l Model 7 - V e l o c i t y 148 30 - S h e l l Model 7 - Density Contour 150 31 . S h e l l Model 7 - V e l o c i t y Contour 151 A. 1 S i m i l a r i t y S o l u t i o n - Density 196 A.2 S i m i l a r i t y S o l u t i o n - Temperature 197 A. 3 S i m i l a r i t y S o l u t i o n - V e l o c i t y 198 A.4 Homogeneous Sphere - Density 203 A.5 Homogeneous Sphere - Temperature 204 A. 6 Homogeneous Sphere - V e l o c i t y 205 A.7 Ha l f G r i d Te St 208 v i i i ACKNOWLEDGEMENTS The o r i g i n a l m o t i v a t i o n f o r t h i s problem arose through work with the DBC i s o c o n o b s e r v i n g group, which i n c l u d e d the e f f o r t s and i n s p i r a t i o n of Ann Gower, John Glaspey and Gordon Walker. My a d v i s o r , Jason Auman, provided much h e l p f u l d i s c u s s i o n and s p e c i f i c c r i t i c i s m of the p h y s i c a l concepts and numerical models, as d i d Greg Pahlman. The c a l c u l a t i o n s of Chapter I I owe much to s u g g e s t i o n s from L a r r y C a r o f f . Ian Easson provided v a l u a b l e a d v i c e on numerical methods. Ian Thompson a s s i s t e d i n producing t h i s t h e s i s , and Ingemar Olson o f f e r e d a f i n e c r i t i c a l edge. I am g r a t e f u l f o r support from UBC graduate f e l l o w s h i p s and from the N a t i o n a l Research C o u n c i l of Canada. F i n a l l y , I am indebted to many members of the department — e s p e c i a l l y Drs. Auman, Fahlman and Glaspey — f o r t h e i r moral support and undeserved p a t i e n c e over the y e a r s . a l l s t r o n g women 1 CHAPTER I INTRODUCTION Within the l a s t decade or so, a great deal of e f f o r t w i t h i n the a s t r o n o m i c a l community has turned toward i n v e s t i g a t i o n of many i n t e r e s t i n g e x t r a g a l a c t i c o b j e c t s , some of which have been newly d i s c o v e r e d , o t h e r s which are more f a m i l i a r but have r e c e n t l y seemed e s p e c i a l l y i n t r i g u i n g . Our p i c t u r e of the " e x t r a g a l a c t i c nebulae" has been extended from a v i s i o n of s t a t i c " i s l a n d u n i v e r s e s " of s t a r s to i n c l u d e some r a p i d l y changing, h i g h l y e n e r g e t i c o b j e c t s , with p o s s i b l y n o n - s t e l l a r components. Some of these e x t r a g a l a c t i c o b j e c t s vary over months or years, yet seem to c o n t a i n and produce as much energy as a normal s t e l l a r g a l a x y . Many of these o b j e c t s have been d i s c o v e r e d w i t h i n the l a s t three decades, but some have been f a m i l i a r much lon g e r than t h a t . The newly-discovered r a d i o s o u r c e s were i d e n t i f i e d with e x t e r n a l g a l a x i e s i n the 1950's (f o r Cyg A, e.g.., c f . B o l t o n and S t a n l e y , 1948, and Baade and Minkowski, -1954). These r a d i o g a l a x i e s o f t e n show a double s t r u c t u r e , l a r g e r than the o p t i c a l g a l a x y , which h i n t s a t e j e c t i o n from the nuc l e u s . Q u a s i - s t e l l a r o b j e c t s gained wide i n t e r e s t when str o n g r a d i o sources were i d e n t i f i e d with o p t i c a l s t a r l i k e o b j e c t s o f hig h r e d s h i f t ( for 3C 273, e.g., c f . Hazard, Mackey and Shimming, 1963, and Schmidt,. 1963). More r e c e n t l y , o b j e c t s of the BL Lac e r t a e c l a s s have been observed widely ( S t r i t t m a t t e r et a l . , 1972, f o r OJ 287; MacLeod et a l . , 1971, f o r BL L a c ) . On t h e oth e r hand, some of the g a l a x i e s now c l a s s i f i e d as S e y f e r t g a l a x i e s have been known as g a l a x i e s with broad n u c l e a r 2 e m i s s i o n l i n e s s i n c e almost the t u r n of the c e n t u r y . (Fath, 1908, on NGC 1068; Campbell and Moore, 1918, on NGC 4151.) S e y f e r t (1943) suggested that these g a l a x i e s c o n s t i t u t e a d i s t i n c t c l a s s . T h i s c l a s s i s c u r r e n t l y d e f i n e d as a l l g a l a x i e s / r e c o g n i z a b l e on Sky Survey p r i n t s with broad emission l i n e s and /' a b r i g h t s e m i - s t e l l a r nucleus (Khachikian and Weedman, 1974). These emission l i n e s are c h a r a c t e r i s t i c of a t h i n , e x c i t e d hydrogen gas with m u l t i p l e i o n i z a t i o n s t a t e s of helium and hea v i e r elements present. Besides the emission l i n e s , S e y f e r t n u c l e i o f t e n d i s p l a y a nonthermal o p t i c a l continuum, s t r o n g (probably thermal) i n f r a r e d e m i s s i o n , and i n two cases show X-ray e m i s s i o n . Most of the g a l a x i e s are a s s o c i a t e d with r a d i o s o u r c e s , although these are o f t e n r e l a t i v e l y weak. Rapid v a r i a b i l i t y has been observed i n the emission l i n e s or the continuum i n some cases. ( D e t a i l e d d e s c r i p t i o n s and r e f e r e n c e s w i l l be given below.) The s p e c t r a of the other e n e r g e t i c e x t r a g a l a c t i c o b j e c t s mentioned above show i n t e r e s t i n g s i m i l a r i t i e s t o S e y f a r t n u c l e a r s p e c t r a . The emi s s i o n l i n e s c h a r a c t e r i s t i c of S e y f e r t n u c l e i appear i n quasars (Burbidge 1967), and sometimes i n e x t r a g a l a c t i c r a d i o sources and other types of g a l a x i e s (Osterbrock and M i l l e r , 1975; Burbidge, 1970 and r e f e r e n c e s t h e r e i n ) . The emission l i n e s do vary from o b j e c t to o b j e c t i n t h e i r r e l a t i v e i n t e n s i t i e s and i o n i z a t i o n l e v e l s present. The quasar 3C 273 has been observed i n the X-ray band and s e v e r a l quasars have been observed i n the i n f r a r e d (Margon et ajL. , 1975; Rieke and Low, 1972). * O p t i c a l and r a d i o continuum v a r i a b i l i t y has been r e p o r t e d i n at l e a s t twelve quasars on a time s c a l e of 3 years or l e s s ( r e f e r e n c e s i n Burbidge, 1967, Schmidt, 1969, and Kellerman and P a u l i n y - T o t h , 1968). The VLBI s t r u c t u r e a l s o v a r i e s on t h i s time s c a l e i n some quasars (Gubbay et a l . , 1969, Moff e t et a l . , 1971} . / BL Lacertae o b j e c t s a l s o e x h i b i t non-therma'l c o n t i n u o u s r a d i a t i o n which v a r i e s on a time s c a l e of years' or l e s s ( E p s t e i n et a l . , 1972; MacLeod e t a l . , 197 1; Stannard e t a l . , 1975). The VLBI s t r u c t u r e of BL Lac has been observed to change over a year (Clark e t a l . , 1973). No s p e c t r a l l i n e s have been confirmed i n i t s spectrum (Oke and Gunn, 1974; Baldwin et a l - , 1975). I t may be dangerous to g e n e r a l i z e about these v a r i e d o b j e c t s as a s i n g l e group, based on these minimal o b s e r v a t i o n s . Nonetheless, the f a c t stands out t h a t a l l show evidence of v i o l e n t a c t i v i t y . The presence of a hot nebular gas, nonthermal s p e c t r a and X-ray s p e c t r a which may be due to r e l a t i v i s t i c p a r t i c l e s , the s t r u c t u r a l h i n t s at e j e c t i o n of matter, the high gas v e l o c i t y i m p l i e d from the l i n e p r o f i l e s , and the r a p i d v a r i a b i l i t y a l l i n d i c a t e a c t i v i t y . These o b j e c t s have t h e r e f o r e been c o n s i d e r e d i n the l i t e r a t u r e as examples of " a c t i v e o b j e c t s " or "compact nonthermal s o u r c e s " . T h i s t h e s i s i s mainly concerned with S e y f e r t n u c l e i . The models to be d i s c u s s e d , however, seem t o o v e r l a p p o s s i b l e models of these other a c t i v e o b j e c t s , and f o r t h i s reason I c o n s i d e r the d i s c u s s i o n here to apply to a " t y p i c a l " a c t i v e g a l a c t i c nucleus. S e y f e r t n u c l e i and r a d i o g a l a x i e s put out l a r g e amounts of energy. I f guasars and BL Lac o b j e c t s are at c o s m o l o g i c a l d i s t a n c e s , they too are very e n e r g e t i c s o u r c e s . These high t o t a l e n e r g i e s and high l u m i n o s i t i e s have not been 4 s a t i s f a c t o r i l y e x p l a i n e d , although there has been much d i s c u s s i o n of the energy source. Suggestions f o r the energy source i n c l u d e dense s t a r c l u s t e r s ( S p i t z e r and Saslaw, 1966, S p i t z e r and Stone, 1967, Shara and Shaviv, 1974); c l u s t e r s of / supernovae (Colgate, 1967) ; massive r o t a t i n g condensed o b j e c t s (Morrison 1969); b l a c k holes (Wolfe and Burbidge, 1970, H i l l s , 1975); and more w h i m s i c a l l y perhaps, white h o l e s (Jeans, 1929, N a r l i k a r , Appa Rao and Dadhich, 1974) and f i r e b r e a t h i n g dragons ( S h i e l d s , Oke and Sargent, 1972). 1 A more pragmatic approach to these o b j e c t s has been to i n v e s t i g a t e only the secondary e f f e c t s of the energy source. T h i s approach t r e a t s i n d e t a i l the s t r u c t u r e of the matter producing the observed r a d i a t i o n , about which we have o b s e r v a t i o n a l evidence, without s p e c i f y i n g the energy source. In t h i s t h e s i s , T w i l l f o l l o w the l a t t e r , w e l l - e s t a b l i s h e d t r a d i t i o n . I w i l l make use of a l a r g e energy i n p u t , about that of an e n t i r e s t e l l a r galaxy, while t o t a l l y i g n o r i n g i t s o r i g i n . T h i s i s j u s t i f i e d , because whatever the o r i g i n a l source of the energy, i t must couple to the matter producing the observed spectrum by means of a more p r o s a i c mechanism, such as e n e r g e t i c p a r t i c l e s or photons. My main focus here w i l l be the dynamics of the matter r e s u l t i n g from t h i s energy i n p u t , s p e c i f i c a l l y of the gas producing the s p e c t r a l l i n e s , s i n c e the re c a n t high r e s o l u t i o n o b s e r v a t i o n s of these l i n e s allow d e t a i l e d i n v e s t i g a t i o n of t h i s n u c l e a r component. Where r e l e v a n t , * S i n c e some of these i d e a s occur throughout the l i t e r a t u r e on S e y f e r t g a l a x i e s and quasars, the r e f e r e n c e s c i t e d here can only be r e p r e s e n t a t i v e , r a t h e r than e x h a u s t i v e . 5 p a s s i n g remarks w i l l be made on the other s p e c t r a l components. Most of t h e p r e v i o u s work on these o b j e c t s has i n v e s t i g a t e d the i o n i z a t i o n s t r u c t u r e and r a d i a t i o n mechanisms, with the assumption of a non-evolving nucleus. T h i s work w i l l a t t e n d to the dynamical e v o l u t i o n of the gas, r a t h e r than attempting to p r e d i c t the observed l i n e s i n d e t a i l . The s p e c t r a l l i n e s , both a b s o r p t i o n and e m i s s i o n , show d e t a i l e d s t r u c t u r e which i s probably due to high v e l o c i t y motions. These v e l o c i t i e s are u s u a l l y w e l l above the escape v e l o c i t y , which i n d i c a t e s mass i s being l o s t . Models of t h i s mass l o s s can be found i n the l i t e r a t u r e . Y a h i l and O s t r i k e r (1973), Wolfe (1974) and I p a v i c h (1975) present v a r i o u s time independent " g a l a c t i c wind" models. Matthews and Baker (1971) c a l c u l a t e d non-steady winds from normal e l l i p t i c a l g a l a x i e s . Matthews (1974), and Kippenhahn, Perry and Roser (1974) among o t h e r s , c o n s i d e r e d r a d i a t i v e a c c e l e r a t i o n of d i s c r e t e gas c l o u d s near the nucleus. Models of the i o n i z a t i o n or l i n e f o r m a t i o n mechanisms (Osterbrock and Parker, 1965, MacAlpina, 1972, Davidson, 1973, Ptak and Stoner, 1973) g e n e r a l l y p o s t u l a t e a c e n t r a l p o i n t source of i o n i z a t i o n i n s i d e the o p t i c a l gas. T h i s w i l l a f f e c t the gas d y n a m i c a l l y , which has not been t r e a t e d q u a n t i t a t i v e l y i n t h e i r models. These models can be improved by t r e a t i n g more of the p h y s i c a l c o n d i t i o n s i n d e t a i l and r e d u c i n g the number of assumptions needed. The thermal s t a t e of the gas can be determined once the h e a t i n g mechanism and the amount of ambient gas are s p e c i f i e d , s i n c e the c o o l i n g f u n c t i o n of nebular gas i s w e l l known. The d i f f e r e n t f o r c e s t h a t photons and e n e r g e t i c 6 p a r t i c l e s e x e r t on the gas can be determined. A time-dependent hydrodynamic c a l c u l a t i o n i s the obvious e x t e n s i o n of the steady s t a t e or d i s c r e t e models mentioned above. T h i s t h e s i s c o n s i s t s of a f i r s t attempt at t h i s problem, namely, s p h e r i c a l l y symmetric numerical hydrodynamic models of the response of a " t y p i c a l " a c t i v e nucleus to a s m a l l c e n t r a l source. The models presented here are too simple to p r e d i c t a l l o b s e r v a t i o n a l d e t a i l s . They i g n o r e important components of the n u c l e a r dynamics, such as r o t a t i o n and magnetic f i e l d s , and they give only a c u r s o r y e x p l a n a t i o n of a l l the s p e c t r a l components. However, they provide a c o n s i s t e n t p i c t u r e of the e n e r g e t i c s and e v o l u t i o n which agrees with other e s t i m a t e s of mass sources and n u c l e a r e n e r g i e s . The l a y o u t of the r e s t of the t h e s i s i s as f o l l o w s . In Chapter I I , the t y p i c a l a c t i v e nucleus i s d e s c r i b e d i n terms of o b s e r v a t i o n a l evidence so f a r a v a i l a b l e . (A l i s t o f symbols used i s g i v e n f o l l o w i n g the b i b l i o g r a p h y . ) C h a r a c t e r i s t i c p h y s i c a l parameters are summarized f o r the Standard A c t i v e Nucleus . Then a d i s c r e t e - c l o u d c a l c u l a t i o n i l l u s t r a t e s the d i f f e r i n g e f f e c t s of cosmic rays and photons i n a c c e l e r a t i n g t h e gas. Chapter I I I d e s c r i b e s the d e t a i l e d model t o be t r e a t e d n u m e r i c a l l y , i n terms of the geometry, the i n i t i a l s t a t e , and the m i c r o s c o p i c p h y s i c s i n v o l v e d . (Appendix I e v a l u a t e s the s p a t i a l .attenuation of the cosmic ray beam.) Chapter IV d e s c r i b e s the f i n i t e d i f f e r e n c e E u l e r i a n scheme that was w r i t t e n to s o l v e the time dependent hydrodynamic e q u a t i o n s of the gas. (Appendix I I d i s c u s s e s the accuracy of the c a l c u l a t i o n s , and Appendix I I I g i v e s a FORTRAN l i s t i n g of 7 the code.) S e v e r a l n u m e r i c a l models were produced, c o v e r i n g a range of the p h y s i c a l parameters, and they are d e s c r i b e d i n Chapter V. F i n a l l y , Chapter VI i n c l u d e s the probable f u t u r e e v o l u t i o n of the numerical models, and the i m p l i c a t i o n s of these models f o r o b s e r v a t i o n s and f o r the o v e r a l l p i c t u r e of a c t i v e n u c l e i . 8 CHAPTER I I SIMPLE MODELS OF THE OPTICALLY EMITTING REGION In t r y i n g to b u i l d models of a c t i v e n u c l e i , I w i l l c o n s i d e r mainly the o p t i c a l spectrum, s i n c e the most d e t a i l e d o b s e r v a t i o n s are of t h i s p o r t i o n of the spectrum. T h i s i n c l u d e s the e m i s s i o n and a b s o r p t i o n l i n e s , as w e l l as the nonthermal continuum. Much work and s p e c u l a t i o n has been devoted to the nature of the " o p t i c a l l y e m i t t i n g r e g i o n " (OES) r e c e n t l y ; Burbidge (1970) and Rees and Sargent (1972) g i v e good reviews. O b s e r v a t i o n s Many p h y s i c a l parameters of the OEE can be esitraated from o b s e r v a t i o n s . These i n c l u d e the nature and s p a t i a l d i s t r i b u t i o n of the gas, the nature of the energy source or i t s c o u p l i n g to the gas, and the n u c l e a r g r a v i t a t i o n a l f i e l d . These are d i s c u s s e d i n o r d e r . 1. Geometry Of The E m i t t i n g Gas The s p e c t r a of a c t i v e n u c l e i show f o r b i d d e n l i n e s from i o n i z a t i o n s t a t e s of widely d i f f e r i n g e n e r g i e s , from [ O i l ] to [ F e X I V ] , and a recombination spectrum of hydrogen and helium (Burbidge, 1970). T h i s probably i n d i c a t e s a s p a t i a l s t r a t i f i c a t i o n of i o n i z a t i o n . T h i s s t r a t i f i c a t i o n c o u l d come from a gas around a s m a l l c e n t r a l source of i o n i z i n g r a d i a t i o n w i t h i n a gas c l o u d . Models i n which the source i s assumed to be a c o n t i n u a t i o n of the o p t i c a l continuum to higher e n e r g i e s are g i v e n by Davidson (1972), MacAlpine (1972), and S h i e l d s and Oke 9 (1975). Osterbrock and Parker (1965) p o i n t e d out the absence i n NGC 1068 of the 01II f l u o r e s c e n t l i n e s XN3312 and 3444. These l i n e s are e x c i t e d by H e l l Ly4 i n p l a n e t a r y nebulae; t h e i r absence (assuming He to be present) means the H e l l Lyd i s dest r o y e d somehow. Osterbrock and Parker suggest e n e r g e t i c protons generated by c l o u d c o l l i s i o n s at many p o i n t s of the OER as the i o n i z a t i o n mechanism. T h i s would allow the n e u t r a l H and He to c o e x i s t s p a t i a l l y with the OIII. S o u f f r i n (1969) has c r i t i c i z e d the suprathermal proton i o n i z a t i o n mechanism on the grounds t h a t an i o n i z a t i o n balance c o u l d not occur i n the temperature and d e n s i t y range i m p l i e d by the f o r b i d d e n l i n e r a t i o s . The models d i s c u s s e d below i n v o l v e an inhomogeneous, h i g h e r d e n s i t y gas, and t h i s problem does not appear. A s m a l l c e n t r a l energy source has been p o s t u l a t e d because of the v a r i a b i l i t y - on time s c a l e s of a few days or a month, up to a year - observed i n the o p t i c a l continuum (Pacholczyk and Heymann, 1968 ) as w e l l as i n the hydrogen Balmer l i n e s ( E i l e k et a l . , 1973, ) and hydrogen and helium a b s o r p t i o n l i n e s (Anderson and K r a f t , 1971 ) . The i d e a of a " f i l l i n g f a c t o r " was f i r s t i n t r o d u c e d by Oke and Sargent (1968). The f i l l i n g f a c t o r i s d e f i n e d as the f r a c t i o n of the e m i t t i n g volume occupied by dense c l o u d s . T h i s f r a c t i o n i s l e s s than u n i t y , s i n c e the volume i n f e r r e d from the Balmer l i n e e m i s s i v i t y i s l e s s than the volume c o r r e s p o n d i n g t o the a p p a r e n t l y r e s o l v e d extent of the [ O I I I ] l i n e s . They estimated the f i l l i n g f a c t o r as 1/40; other e s t i m a t e s have been 10 as low as 10~ 3 (Bergeron and S a l p e t e r , 1973). D i r e c t evidence of the d u m p i n e s s of the gas i n g a l a c t i c n u c l e i comes from o b s e r v a t i o n s o f s t r u c t u r e i n the e m i s s i o n l i n e p r o f i l e s . (See, f o r i n s t a n c e , Walker, 1968, fiilek e t a l . , 1973, and Glaspey e t a l . , 1975, f o r NGC 1068; O l r i c h , 1973, and Walker, 1968, f o r NGC 4151; Anderson, 1971, f o r NGC 5548; Burbidge and Burbidge, 1965, f o r NGC 1275; Anderson, 1973, f o r NGC 7469.) These c l o u d s have very high r e l a t i v e v e l o c i t i e s , on the o r d e r of s e v e r a l hundred km s _ 1 . B l u e s h i f t e d hydrogen and helium a b s o r p t i o n l i n e s with up to t h r e e components have been seen i n NGC 4151 (Anderson, 1974) with a v e l o c i t y spread of 900 km s _ 1 . The very broad wings of the permitted l i n e s i n most S e y f e r t g a l a x i e s i n d i c a t e v e l o c i t i e s of a few thousand km s _ 1 , i f they are due to mass motions. Some r e p r e s e n t a t i v e v e l o c i t i e s of n u c l e a r c l o u d s w i t h i n a high r e s o l u t i o n l i n e p r o f i l e are l i s t e d i n Table I. For comparison, the escape v e l o c i t y a t 1 pc from a c e n t r a l mass of 10 8 MQ i s 950 km s - 1 , and the sound speed i s 10 km s-» at 10* K. The l i s t of S e y f e r t o b j e c t s given by Vorontsov-Vel*yaminov and I v a n i s e v i c (1974) i n d i c a t e s Balmer l i n e wing v e l o c i t i e s ( f u l l width a t zero i n t e n s i t y ) of up to 10,000 km s - 1 . I t i s worth n o t i n g that the s i x t y g a l a x i e s l i s t e d t h e r e with measured Balmer l i n e widths do not show a double peaked d i s t r i b u t i o n a t 5000 and 11,000 km s~ 1 , but r a t h e r a broad d i s t r i b u t i o n of v e l o c i t i e s from zero to 10,000 km s - 1 , with a peak at 4000 -5000 km s - 1 . (Comparison of t h e i r measured widths with other p u b l i s h e d values f o r w e l l known o b j e c t s i n d i c a t e s t h a t the Vorontsov-Vel•yaminov and I v a n s e v i c widths are lower by about Table I Maximum V e l o c i t y S e p a r a t i o n Between Components Of L i n e P r o f i l e s 11 | galaxy NGC 1068 | v e l o c i t y •+-1300 km s - i NGC 1275 | 400 | Hc< ,NII I 3000 1 | d NGC 4151 | 800 | OIII 1 b,c NGC 5548 j 1500 1 Ho( 1 e NGC 7469 l 200 1 Hc( I f I 4329A J 2400 1 H ^  I h 3C 390.3 | 4100 | Hd #Hp,HV 1 3 a) Walker 1968 b) E i l e k et a l . , 1974 c) U l r i c h 1973 d) Burbidge and Burbidge 1965 e) Anderson 1971 f) Anderson 1973 g) Glaspey 1974 h) Disney 1973 j) Burbidge and Burbidge 1971 l i n e s NeIII,OIII a",b source ten per c e n t . ) . Based on e a r l i e r , l e s s complete l i s t s of S e y f e r t l i n e widths, MacAlpine (1974) suggested t h a t such a double peaked d i s t r i b u t i o n e x i s t e d . I f t h i s were so, i t would tend to support tne Ptak and Stoner (1973) p a r t i c l e streaming model (see Chapter V I ) . The b l u e s h i f t e d a b s o r p t i o n l i n e s i n guasars show v e l o c i t i e s both w i t h i n the a b s o r p t i o n systems and r e l a t i v e to the emission system of up to an order of magnitude higher than the S e y f e r t v e l o c i t i e s . The quasar PKS 0237-23, f o r i n s t a n c e , shows a spread of Az=0.8 i n a b s o r p t i o n r e d s h i f t s , which corresponds to 12 &v=0.3c ( B a h c a l l e t a l . , 1968). Another o b j e c t , PHL 938, \ (Burbidge e t a l . , 1968) shows a spread, Az=1.3, which corresponds to Av=0.5c. Schmidt (1970) g i v e s a f u l l review. / / 2. Heating Mechanisms / A c e n t r a l l y l o c a t e d i o n i z a t i o n source w i l l a c c e l e r a t e the gas or c l o u d s outwards. T h i s has been pointed out i n c o n n e c t i o n with S e y f e r t g a l a x i e s and quasars by Weymann (1970), Mushotzky, Solomon and S t r i t t m a t t e r (1972), and T a r t e r and McKee (1973), among o t h e r s . More d e t a i l e d models have been c o n s i d e r e d by Matthews (1974). T h i s c a l c u l a t i o n w i l l be d i s c u s s e d i n d e t a i l below, and compared to the case of cosmic ray a c c e l e r a t i o n . I f the c e n t r a l i o n i z i n g source i s an e x t e n s i o n of the o p t i c a l s y n c h r o t r o n continuum to X-ray e n e r g i e s (as has o f t e n been assumed), the i o n i z i n g l u m i n o s i t y can be found by d i r e c t e x t r a p o l a t i o n of the power law. The o p t i c a l continuum l u m i n o s i t y i s t y p i c a l l y 10* 3 ergs s _ 1 (see, f o r i n s t a n c e , Anderson, 1970). In quasars i t i s higher, about 10* 6 e r g s s~l (Wampler and Oke, 1967; Wampler, 1968). X-rays have been observed i n the d i r e c t i o n of two S e y f e r t s , NGC 1275 (Davidsen §t a l . , 1975) and NGC 4151 ( K e l l o g , 1973; Margon e t a l . , 1975) with l u m i n o s i t i e s of 3 x 10** and 10* 2 ergs s _ 1 r e s p e c t i v e l y . For NGC -4151 a turnover a t 3 keV i s suggested. The e m i s s i o n i n the d i r e c t i o n of NGC 1275 i n c l u d e s the e n t i r e Perseus c l u s t e r . One QSO, 3C 273, shows an X-ray l u m i n o s i t y of 4 x 10* 6 ergs s~1 (Bowyer e t a l . , 1970). These o b s e r v a t i o n s may i n d i c a t e a separate X-ray source that i s not an e x t e n s i o n of the o p t i c a l 13 s o u r c e . In t h e o r e t i c a l c a l c u l a t i o n s , Matthews (1974) has taken 10 4* ergs s _ l as the i o n i z i n g photon l u m i n o s i t y f o r S e y f e r t n u c l e i and 1 0 4 6 ergs s~l f o r quasars. I f the i o n i z a t i o n i s due to e n e r g e t i c p a r t i c l e s , a measure of the p a r t i c l e f l u x based on observable q u a n t i t i e s i s h i g h l y model dependent. The d e t a i l e d d i s c u s s i o n of t h i s i s l e n g t h y , and i s postponed to the f o l l o w i n g s e c t i o n . The p a r t i c l e energy c o n t a i n e d i n the " t y p i c a l " a c t i v e o b j e c t i s e s t i m a t e d to range from 1 0 s 3 t o 1 0 5 7 e r g s , with unavoidable u n c e r t a i n t i e s of a couple o r d e r s of magnitude, f o r models of S e y f e r t n u c l e i and quasars. An a l t e r n a t i v e dynamical model i s proposed by Wolfe (1974). In h i s model a s u p e r s o n i c g a l a c t i c wind, a k i n to the s o l a r wind s o l u t i o n s by Parker (1965), accounts f o r the observed v e l o c i t i e s . The e n e r g y / i o n i z a t i o n sources are not e x p l i c i t l y c o n s i d e r e d but the c o n s t a n t temperature of 10 6 K which he assumes throughout the wind probably r e g u i r e s an extended, n o n - c e n t r a l heat source. P a r k e r ' s wind model does not r e q u i r e an i s o t h e r m a l gas, but does r e q u i r e t h a t T (r) decrease l e s s r a p i d l y than r - 1 . As shown below, the gas must be very hot, a t l e a s t 10 6 K, to reach the observed v e l o c i t i e s with the s u p e r s o n i c wind model. T h i s model t r e a t s a c o n t i n u o u s gas flow. Wolfe suggests, however, t h a t thermal i n s t a b i l i t y may l e a d to c o o l e r , dense clou d s c o - e x i s t i n g with the wind. Extended, n o n - l o c a l heating f o r the OEH would probably come from s t e l l a r c o l l i s i o n s and bow shocks, or c o l l i s i o n s of d i s c r e t e c l o u d s . Cloud c o l l i s i o n s have been mentioned i n c o n n e c t i o n with quasars by D a l t a b u i t and Cox (1972). For 14 S e y f e r t g a l a x i e s , t h i s mechanism has been suggested by Osterbrock and Parker (1965) and by Oke and Sargent (1968). However, the source of the c l o u d k i n e t i c energy which i s transformed i n t o gas h e a t i n g remains u n s p e c i f i e d . Star c o l l i s i o n s i n a dense s t e l l a r nucleus have been c o n s i d e r e d by S p i t z e r and Saslaw (1966), S p i t z e r and Stone (1967), and more r e c e n t l y by Shara and Shaviv (1974). In order f o r the f u l l energy output of S e y f e r t g a l a x i e s to be thus e x p l a i n e d , (Shara and Shaviv d i s c u s s energy r e l e a s e through c o l l i s i o n - i n d u c e d novae i n white dwarfs) an u n u s u a l l y dense s t e l l a r system i s needed - over 1 0 1 0 MQ w i t h i n 1 pc of the c e n t e r . A d e t a i l e d model must, of c o u r s e , a l s o c o n s i d e r the e f f i c i e n c y of t r a n s f o r m i n g the k i n e t i c energy to thermal energy of the gas, which w i l l depend on the o p a c i t y of the r a d i a t i o n produced i n the shocks. Another steady wind model which does not r e q u i r e an extended hot gas has been proposed by I p a v i c h (1975). In t h i s model, the i n t e r a c t i o n of e n e r g e t i c p a r t i c l e s with a random magnetic f i e l d of about 10-* gauss w i l l a c c e l e r a t e a c o l d gas up t o 1500 km s ~ l . 3. C e n t r a l Mass And Source L i f e t i m e The c e n t r a l mass i n the S e y f e r t nucleus i s hard to o b t a i n o b s e r v a t i o n a l l y . Oort (1971) has used microwave o b s e r v a t i o n s to measure the s t a r d e n s i t y i n the very i n n e r r e g i o n s of our galaxy and M31, f i n d i n g a d i s t r i b u t i o n f o r the i n n e r few pc, M (r) = 4 x 10 6r Mg,, where r i s i n pc, with a c e n t r a l mass of 15 3 x 10^ M g i n s i d e 0-1 pc. For N G C 1068, Walker (1968) found M = 3 x 10« M & w i t h i n 2000 pc of the c e n t e r ; f o r N G C 3227 Rubin and Ford (1968) found 3 to 4 x 10* H c i n s i d e 620 pc; f o r N G C 7469 Anderson (1973) found about 10 9 ft& i n s i d e 400 pc. Sanders (1970), c o n s i d e r i n g c l o u d motions, estimated the c e n t r a l mass of N G C 4151 as between 1-1 and 3.5 x 10 9 M 0 i n s i d e 4.5 pc. Improved S t r a t o s c o p e I I photographs of N G C 4151 ( S c h w a r z s c h i l d , 1973) show a nu c l e a r diameter of under 7 pc, c o n t a i n i n g an estimated c e n t r a l mass of 4 x 10 9 M ^ . Anderson (1973), based on the smoothness of the i n n e r p a r t of the r o t a t i o n curve of N G C 7469, concludes there i s , however, no c e n t r a l p o i n t mass g r e a t e r than 108 i n that galaxy. The higher of the nu c l e a r v e l o c i t i e s i n d i c a t e d by the l i n e widths and l i n e p r o f i l e s t r u c t u r e s are g r e a t e r than the escape v e l o c i t y f o r the r e g i o n ; f o r a c e n t r a l mass of 10 8 H Q, the escape v e l o c i t y i s 950/>Jr^ km s ~ 1 i f r ^ c i s the d i s t a n c e i n pc. T h e r e f o r e , the phenomena we are seeing may be q u i t e s h o r t l i v e d . A c l o u d t r a v e l l i n g a t 1000 km s - 1 w i l l c r o s s 10 pc i n 10* years. The sound t r a v e l time f o r the same re g i o n a t 10* K, i s 5 x 10 s years. The mass outflow i n v o l v e d i n Wolfe's (1974) steady s t a t e model i s 20 H Q per year; t h i s i m p l i e s a l i f e t i m e l e s s than 10 7 years, i f a l l the n u c l e a r mass feeds the wind. T h i s f i g u r e i s j u s t s l i g h t l y higher than the mass l o s s i n d i c a t e d by Walker's (1968) measurements of N G C 1063. Since the S e y f e r t phenomenon i s s t a t i s t i c a l l y estimated to l a s t 10 8 years ( W o l t j e r , 1959), some means of renewal of these processes i s necessary. E s t i m a t e s of the mass l o s s r a t e by normal s t a r s (see Sanders and Prendergast, 1974), perhaps augmented by a high c o l l i s i o n r a t e 16 such as S c h w a r z s c h i l d (1973) p r e d i c t s i n the nucleus of NGC 4151, seem t o account f o r the above f i g u r e s . ( T h i s i s d i s c u s s e d i n Chapter VI.) On the oth e r hand, Sanders and Prendergast (1974) have shown that a s i n g l e e x p l o s i v e event ( r e q u i r i n g 3 x 10 5« ergs and 10 s Me e j e c t e d , i n a numerical model of our galaxy) i n a r o t a t i n g g a l a c t i c d i s k w i l l produce an o s c i l l a t i n g r i n g . T h i s seems t o l a s t through s e v e r a l o s c i l l a t i o n s , much l o n g e r than i t s expected " c r o s s i n g time". Such a mechanism, based on the i n t e r a c t i o n of the n u c l e a r gas with the i n n e r g a l a c t i c plane, may reduce the mass'loss r a t e and the replenishment necessary. The " t y p i c a l " model d e s c r i b e d here i s summarized i n Table I I , f o r l a t e r r e f e r e n c e . I f the parameters are observed to be s i g n i f i c a n t l y d i f f e r e n t f o r S e y f e r t n u c l e i than f o r quasars, the quasar value i s given i n the second l i n e . E s t i m a t e s Of C e n t r a l Source E n e r g e t i c s I f the nonthermal l u m i n o s i t y i n some wavelength band of the a c t i v e nucleus i s e l e c t r o n s y n c h r o t r o n r a d i a t i o n , and i f the othe r source parameters are s p e c i f i e d , then t h a t l u m i n o s i t y depends uniquely on the t o t a l p a r t i c l e spectrum. In p r a c t i c e , these other parameters -- p a r t i c l e energy, magnetic f i e l d , p r o t o n / e l e c t r o n energy r a t i o , source geometry --'cannot be uniquely determined o b s e r v a t i o n a l l y . S i n c e many a c t i v e o b j e c t s emit i n the r a d i o and o p t i c a l r e g i o n s , and a few i n the IB and X-ray as w e l l , e f f o r t s have been made to combine some of these components i n t o a c a n o n i c a l model. As an example, Fi g u r e 0 shows the conti n u o u s s p e c t r a of 17 F i g u r e 0. Continuum o b s e r v a t i o n s of the two well-observed S e y f e r t n u c l e i are shown from the r a d i o to the X-ray. NGC 1068 i s from S h i e l d s and Oke, 1975, and from Jones and S t e i n , 1975. NGC 4151 i s from B a i t y e t a l . , 1975. 18 F i q u r e 0. S e y f e r t Nuclear S p e c t r a 1? T a b l e I I Parameters Of A Standard A c t i v e Nucleus r T / / - ••• i | Extent ( v a r i a b i l i t y ) | 0.1 pc | --(VLS I , r a d i o source) | 0.1 pc / I — ( o p t i c a l l y resolved) | 1-10 pc | C e n t r a l Mass | ( S e y f e r t s only) I 108 M o | E m i t t i n g Mass | (Balmer l i n e s ) I 102 M o - 103 M @ | Lx ( h ^ ~ 1-10 keV) | 1 0 * Z - 1 0 * * | 10* 6 ergs ergs s- 1 s - i | J L(opt + UV) | 1 0 * 3 - 1 0 * * | 10* 6 ergs ergs s _ 1 s - i | 1 L l R - 1 0 1 3 U z> | 10*5-10*6 | 10* 7 ergs ergs s - i s - i | | 1038 - 1 0*1 | 10* 5 ergs ergs s - i s - i I | 1 0 s 3 ergs \ 1054-1057 ergs L , . .. _. 1 1 NGC 1068 and NGC 4151. Demoulin and Burbidge (1968) c o n s i d e r the o p t i c a l continuum, a s c r i b e i t t o e l e c t r o n s y n c h r o t r o n , and assume e q u i p a r t i t i o n of e l e c t r o n and magnetic f i e l d e n e r g i e s . Bergeron and S a l p e t e r (1973) assume the IR f l u x (peaking about 1 0 1 2 Hz) to be s y n c h r o t r o n , and the X-ray peak to be due to i n v e r s e Compton s c a t t e r i n g of the IR photons. Jones, O ' D e l l and S t e i n (1974a,b) and Burbidge, Jones and O' D e l l (1974), c o l l e c t i v e l y known as JCES, assume the r a d i o e mission t o be s y n c h r o t r o n , and the o p t i c a l emission to r e s u l t from i n v e r s e Compton s c a t t e r i n g of the r a d i o a m i s s i o n . 20 One important r e s t r i c t i o n i n i d e n t i f y i n g the s y n c h r o t r o n - s o u r c e p a r t i c l e s with the i o n i z i n g mechanism f o r the o p t i c a l gas i s the s m a l l s c a l e of the gas, i m p l i e d by the r a p i d v a r i a b i l i t y - The s i z e s of the r a d i o sources of the JOBS models, as measured by VLBI, are c o n s i s t e n t with the t h e o r e t i c a l e s t i m a t e s and with t h e • v a r i a b i l i t y . The IE s y n c h r o t r o n source i n a c t i v e n u c l e i i s c a l c u l a t e d at a s i z e of under 1 pc, but o b s e r v a t i o n s are i n c o n f l i c t as to i t s s i z e - Reports of v a r i a b i l i t y from 5 t o 25 microns (Kleinmann and Low, 1970, Rieke and Low, 1972) are d i s p u t e d (Morrison and Simon, 1973; S t e i n , G i l l e t t and M e r r i l , 1974). Neugebauer, Garmine, Reike and Low (1971) r e p o r t r e s o l v i n g the n u c l e i of NGC 1068 at 1.6 and 2.2 microns. They suggest a p o i n t source dominating at l o n g e r wavelengths plus an extended source as l a r g e as 100 pc a t 1-2 microns. An a l t e r n a t e e x p l a n a t i o n of the IR emission i s thermal emission due t o dust g r a i n s . Bergeron and S a l p e t e r show t h a t a source at l e a s t 100 pc a c r o s s i s needed t o reach the high f l u x e s seen i n some o b j e c t s , up to 3 x 10* 6 ergs s~l at 10* 3 Hz i n NGC 1068 and NGC 4151 a c c o r d i n g to Low (1971). Recent s p e c t r a of NGC 1068 by Jameson, Longraire, McLinn and woolf (1974) resemble the d u s t l i k e s p e c t r a of some pl a n e t a r y nebulae. The measurement of reddening i n s e v e r a l S e y f e r t n u c l e i by Wampler (1968) a l s o i n d i c a t e s the presence of dust. Jones and S t e i n (1975) review a l l of the IR o b s e r v a t i o n s of NGC 1068 and conclude that thermal dust models best s a t i s f y the data. The necessary parameters f o r IR s y n c h r o t r o n models are d e r i v e d by Bergeron and S a l p e t e r . The s e l f a b s o r p t i o n turnover r e q u i r e s a magnetic f i e l d of 10 to 100 gauss.- T h i s i s much 21 g r e a t e r than the e q u i p a r t i t i o n f i e l d , which i s t y p i c a l l y 100 to 1000 m i l l i g a u s s , i m p l y i n g the magnetic f i e l d dominates the gas dynamics. A l s o , the p a r t i c l e energy i s r e l a t i v e l y low, 10* 8 ergs or so, and the i n d i v i d u a l p a r t i c l e l i f e t i m e only 10 3 seconds, so constant replenishment i s needed. F i n a l l y , Knacke and Capps (1974) d e r i v e a magnetic f i e l d of 100 t n i l l i g a u s s from Faraday r o t a t i o n measures. T h i s agrees with the dust model. On the other hand, a s c r i b i n g the r a d i o source to e l e c t r o n s y n c h r o t r o n emission, JOBS produce models with magnetic f i e l d s a f a c t o r of ten below the e q u i p a r t i t i o n v a l u e . T h i s i m p l i e s a p a r t i c l e dominated dynamics. T h e i r t o t a l e l e c t r o n e n e r g i e s range from 1 0 5 3 to 1 0 5 7 ergs i n t h e i r models of s p e c i f i c S e y f e r t n u c l e i and quasars. The r a t i o of p r o t o n / e l e c t r o n e n e r g i e s depends on the unknown a c c e l e r a t i o n mechanism, and i s taken by Demoulin and Burbidge to be i n the range 1 to 100. For comparison, e q u i p a r t i t i o n e s t i m a t e s using n u c l e a r s i z e s compatible with the observed s h o r t term v a r i a b i l i t y p r e d i c t 1 0 5 2 to 1 0 5 b ergs i n t o t a l p a r t i c l e energy i n a c t i v e n u c l e i . JOBS emphasize the u n c e r t a i n t i e s i n these energy e s t i m a t e s . The low energy c u t o f f of the p a r t i c l e spectrum as well as the p r o t o n / e l e c t r o n energy r a t i o must be assumed, and p h y s i c a l l y r easonable v a r i a t i o n s i n these parameters can a f f e c t the t o t a l energy by orders of magnitude. Estimates of the,magnetic f i e l d from d i f f e r e n t observed parameters are i n c o n s i s t e n t w i t h i n a s i n g l e o b j e c t , which r e f l e c t o b s e r v a t i o n a l e r r o r s or complexity of the source. The geometry of the source, or a n i s o t r o p y of the p a r t i c l e streaming, a l s o s t r o n g l y a f f e c t s the d e r i v e d parameters. 22 In view of the u n c e r t a i n t i e s even w i t h i n the r a d i o - s y n c h r o t r o n model — which seems more t e n a b l e than the IR s y n c h r o t r o n model — no d e f i n i t e value f o r the cosmic ray i o n z i n g f l u x , or the t o t a l cosmic ray energy, can be s p e c i f i e d . For purposes of the c a l c u l a t i o n s i n t h i s t h e s i s , a " t y p i c a l " r a t h e r than a s p e c i f i c model of S e y f e r t n u c l e i or quasars i s d e s i r e d , and any cosmic ray f l u x c o n s i s t e n t with the e n e r g i e s gi v e n by JCBS or by e g u i p a r t i t i o n w i l l be c o n s i d e r e d a c c e p t a b l e . A p a r t i c l e dominated dynamics w i l l a l s o be assumed. A c c e l e r a t i o n Of Gas Clouds By A C e n t r a l Source A t t e n t i o n has been given i n the l i t e r a t u r e t o the dynamical e f f e c t s of the i o n i z a t i o n sources proposed f o r S e y f e r t n u c l e i and quasars. Emphasis has been on the v e l o c i t i e s a t t a i n a b l e by ra-diation pressure on d i s c r e t e c l o u d s , with i n t e n t to e x p l a i n the l a r g e r e l a t i v e v e l o c i t i e s seen i n these o b j e c t s . The observed v e l o c i t i e s range from 100 km s - 1 to 0.5c, as mentioned above. Weymann (1973) has reviewed the s u b j e c t . He p o i n t s out t h a t r e s o n a n c e - l i n e a c c e l e r a t i o n i s not l i k e l y to produce high enough v e l o c i t i e s when c o n s i d e r e d r i g o r o u s l y . A c o n t i n u o u s , steady supersonic-wind flow i s a l s o u n l i k e l y to reach the observed v e l o c i t i e s . I f the wind i s at the temperatures i n d i c a t e d by the f o r b i d d e n l i n e s , about 10* K, then the q u a n t i t y L^/GH must i n c r e a s e d r a s t i c a l l y past the s o n i c p o i n t , where L i s the c e n t r a l source l u m i n o s i t y , ^ i s the c r o s s s e c t i o n between the quanta from the c e n t r a l source and the ambient gas, and M i s the c e n t r a l mass. T h i s r a p i d change i n <i i s u n l i k e l y {^ r e f l e c t s the i o n i z a t i o n s t r u c t u r e and the o p t i c a l depth). 23 Larger v e l o c i t i e s can be produced i n wind s o l u t i o n s i f the gas i s very hot — perhaps 10 8 K -- but then t h i s high temperature must be maintained throughout the flow. An a l t e r n a t i v e s o l u t i o n i s non-steady gas flow, s t a r t i n g with a s t a t i c gas. A d i s c r e t e c l o u d model s i m p l i f i e s the gas e q u a t i o n s and a l l o w s one to c a l c u l a t e as with b i l l i a r d b a l l s . The important f a c t o r i s the o p t i c a l depth of the c l o u d s , which determines how e f f i c i e n t l y they use the i o n i z i n g f l u x . The o p t i c a l depth depends on the i o n i z a t i o n s t a t e . The X-ray and p a r t i c l e i o n i z a t i o n mechanisms behave very d i f f e r e n t l y i n t h i s r e s p e c t , because a f u l l y i o n i z e d gas i s t r a n s p a r e n t t o X-rays but not to charged p a r t i c l e s . Matthews (1974) has c o n s i d e r e d X-ray a c c e l e r a t i o n of c l o u d s . He found t e r m i n a l c l o u d v e l o c i t i e s t h a t are s m a l l e r than observed, u n l e s s the c l o u d s are very s m a l l (<10 - 7 and are pressure bound so t h a t they do not expand. However, the s t r o n g e r c o u p l i n g of cosmic rays to the i o n i z e d gas w i l l produce higher v e l o c i t i e s . In t h i s s e c t i o n I w i l l summarize the c a l c u l a t i o n s r e g a r d i n g X-rays i n my- n o t a t i o n , then g i v e a more d e t a i l e d c a l c u l a t i o n f o r cosmic r a y s . The models d i s c u s s e d below are f a i r l y simple, and are meant as a good estimate of the a c c e l e r a t i o n processes but not as a comprehensive p i c t u r e . 1. X-ray a c c e l e r a t i o n . The i o n i z a t i o n s t a t e of the cloud i s determined from the balance eguation f o r a hydrogen gas, which i g n o r e s the f i v e t o ten per cent e f f e c t of c o l l i s i o n a l i o n i z a t i o n : V L n (1-x) = c< n 2 x 2 (1) 4Trr 2hv 24 where x i s the f r a c t i o n a l i o n i z a t i o n , <r i s the i o n i z a t i o n c r o s s s e c t i o n ( A l l e n , 1963), i s the recombination c o e f f i c i e n t , about 3 x 1 0 - 1 3 cgs (see Chapter I V ) , r i s the d i s t a n c e from the source i n a s p h e r i c a l geometry, and n i s the t o t a l number d e n s i t y . L i s the l u m i n o s i t y i n ergs s _ 1 ; L / h v / i s the photon l u m i n o s i t y . T h i s q u a d r a t i c equation has the two l i m i t i n g s o l u t i o n s x 1 when L oi h^ (2) 4i r r 2 n ' x « 1 when L // <x hi*. 4<nr2n ^ S T Rephrasing t h i s , we f i n d that x ^  1 f o r _ L _ 1 ° * S = 1.7 x 1Q5 LMt,o.s pc (3) jMiM 4ttn J n°.5 where i s the i o n i z i n g l u m i n o s i t y i n u n i t s of 1 0 4 6 ergs s _ i and hVSlOOeV. Thus, f o r 1.^=0.01 to 1.0 and n = 10* to 10<*, the c l o u d i s f u l l y i o n i z e d at l e a s t i n the i n n e r tens of pa r s e c s . The o p t i c a l depth of a hydrogen c l o u d of r a d i u s a to photon i o n i z a t i o n i s T ~ 2an<r(1-x) (3a) When x 1 i t i s convenient to r e w r i t e t h i s , using (1) , as T ~ 8lr°< h v a n 2 r 2 / L The c l o u d w i l l be o p t i c a l l y t h i c k at 100 eV, then, when " Z a > L = 9 x I P 2 * L<th cm-* (4) 8Tr«.hvr2 r 2 i f r^,0 i s the d i s t a n c e from the source i n pc. A c l o u d of one 25 s o l a r mass 1 and d e n s i t y 10 6, has n 2 a = 1 0 2 9 ; i t w i l l be t h i c k , f o r i n s t a n c e , i f = 0.01 and r ^ = 1. In the o p t i c a l l y t h i n case, the equation of motion i s du = -GM -f ~a 2.LT (5) dt r 2 4Trr 2M cc M i s the c e n t r a l mass, c i s the speed of l i g h t , and Mc the mass of the c l o u d . M = (4Tr/3)mna3 i f m i s the mean atomic mass. For x 2 « 1 , t h i s can be w r i t t e n , using (1), as du = -GM + 3*hv n , (6) dt r 2 2cra The important f e a t u r e of t h i s equation i s t h a t the r a d i a t i v e a c c e l e r a t i o n i n the t h i n case i s p r o p o r t i o n a l t o the d e n s i t y , but independent of the l u m i n o s i t y and the d i s t a n c e from the s o u r c e . T h i s means a b s o l u t e upper l i m i t s f o r the a c c e l e r a t i o n can be found which are r e l a t i v e l y model-independent. I t a l s o means expansion of the cloud w i l l lower the r a d i a t i v e f o r c e . I f the c l o u d does not expand as i t moves outward -- say i t i s c o n t a i n e d by pressure of a hot i n t e r c l o u d gas — i t w i l l become o p t i c a l l y t h i c k when i t reaches the d i s t a n c e r p c ' = [ 9 x 1 0 2 9 L ^ b / n 2 a ] 0 - 5 , given by e q u a t i o n (4). C o n s i d e r i n g t h i s c a s e , and a l s o s e t t i n g M = 0, w i l l g i v e the maximum estimate of the cloud v e l o c i t y up to t h i s p o i n t . I n t e g r a t i n g (6) from r s i t o r g i v e s 1 The mass of c l o u d chosen seems a r b i t r a r y . The Balmer emission l i n e s i n NCG 1068 r e q u i r e 10 to 10 2 M G of e m i t t i n g gas ( E i l e k , et a l . , 1973), and the a b s o r p t i o n system i n NGC 4151 r e q u i r e s about 10 2 M© (Anderson and K r a f t , 1969). On t h e other hand, Matthews (1974) uses q u i t e s m a l l c l o u d s , from 1 0 ~ 2 1 to 1 0 - 7 H G. T h i s i m p l i e s a model with many, many dense c l o u d s and a very s m a l l f i l l i n g f a c t o r . u(r) = [ (3«Wcin ) n(r - r^.) jo.s (7) = 0.93 (r^n) o.s km s-* i f r » r s t . I f r f C i s 0.3 and n = 0(10*), u (r) i s 510 km s-». Past r, the c l o u d f o l l o w s the o p t i c a l l y t h i c k e q u a t i o n of motion, du = -GM + T r a 2 L (8) dt r 2 4"tfr 2M ec I n t e g r a t i n g t h i s from r outwards, s t i l l with the s i z e a he l d constant and M = 0, g i v e s f o r the t e r m i n a l v e l o c i t y uz = u (r) 2 4 a 2 L ± (9) X 2M cc r = u ( r j 2 + 0.78x10^0 L n o (km s " 1 ) 2 (na) rf(_ i f na = 1 0 2 3 (see above), and L 4 ( j •= 0.01, and ?p t = 0.3, then u^. = 720 km s _ 1 . The e x t e n t o f the c l o u d a f f e c t s the v e l o c i t y , s i n c e (na)-r goes as a 0 - 5 . Thus, a model with many s m a l l c o n t a i n e d clouds can match the observed v e l o c i t i e s . However, i t seems more l i k e l y t h a t the pressure necessary t o c o n t a i n the c l o u d w i l l not be maintained throughout the r e g i o n and t h a t some expansion of the c l o u d w i l l occur. A simple case of t h i s i s f r e e expansion at the sound v e l o c i t y , g iven by a (t) = a D + c s t . Since the sound speed, c s , i s g r e a t e r than the outward v e l o c i t y i f the c l o u d s t a r t s a t r e s t , the c l o u d w i l l expand f a s t e r than i t moves out, and i f i t s t a r t s t h i n i t w i l l remain t h i n , thus never f e e l i n g a s i g n i f i c a n t f o r c e . A c l o u d which i s i n i t i a l l y t h i c k w i l l expand to become t h i n . The a c c e l e r a t i o n i n t h i s case i s again given by (8); however, a i s time dependent. Converting the independent v a r i a b l e from t to a , and d e f i n i n g v = u/c^, which g i v e s v = dr/da, we get 27 df_r = -GM + L a 2 (10) d a 2 c f r 2 4M cccf r 2 Again we w i l l c o n s i d e r M = 0, to get an upper l i m i t on the v e l o c i t y . The equation as i t stands i s not e a s i l y s o l v a b l e except with s p e c i f i c i n i t i a l c o n d i t i o n s . However, w r i t i n g dv/da = d 2 r / d a 2 and i g n o r i n g the v a r i a t i o n of r on the r i g h t hand s i d e , an upper l i m i t can be found. T h i s i s , u ^ L ( a 3 - a 3) (11) x 12M cc sc r 2 where a i s the r a d i u s where the c l o u d becomes o p t i c a l l y t h i n . T h i s r a d i u s can be found by i n v e r t i n g equation (4), g i v i n g u t-6 4.6 x 10-9 L c u * (na 3) o.z r-o.8 jc m s ~ * (12) i f c s = 10 km s _ 1 (a c o o l gas) and a 0<<a. I f n a 3 i s as l a r g e as 10*o (10 3 MG) , with =0.01 and r = 1, u t i s no l a r g e r than 650 km s - 1 . Past t h i s p o i n t , the c l o u d a c c e l e r a t e s as a t h i n c l o u d , s u b j e c t to d e c r e a s i n g d e n s i t y and t h e r e f o r e decreased r a d i a t i o n p r e s s u r e . 2. Cosmic Ray A c c e l e r a t i o n . In the case of X-ray a c c e l e r a t i o n , we have seen t h a t c o n d i t i o n s extreme i n terms of our model are j u s t a b l e to e x p l a i n the magnitude of the v e l o c i t i e s seen i n S e y f e r t n u c l e i . (The h i g h e r quasar c l o u d v e l o c i t i e s are even l e s s l i k e l y to a r i s e i n t h i s way.) The case of cosmic ray pressure seems more promising f o r two reasons: the o p t i c a l depth of a c l o u d depends only on the geometry, not on the i o n i z a t i o n ; and the f o r c e i s independent of the d e n s i t y i n a l l c a s e s . The " o p t i c a l depth" f o r a charged p a r t i c l e can be d e f i n e d i n terms of an e n e r g y - l o s s d i s t a n c e . T h i s i s because the 28 p a r t i c l e l o s e s only a s m a l l p a r t of i t s energy i n each hydrogen i o n i z a t i o n , and so p e n e t r a t e s much f u r t h e r than the mean f r e e path, (no-) - 1. The e q u a t i o n f o r the r a t e of energy l o s s with time i s given i n Appendix I, equation (1.1); i t can be expressed as a s p a t i a l l o s s , dE = -1. 5x10~32n (x + 0. 3) (13) dz E where x i s the f r a c t i o n a l i o n i z a t i o n and z i s the d i s t a n c e i n t o the c l o u d . T h i s i s equation (1.3), and emphasizes the r e l a t i v e independence of the i o n i z a t i o n s t a t e . T h i s can be i n t e g r a t e d from 0 to the c l o u d s i z e , a, to give E ( a ) 2 = Ef - 3.0x10-32na (x + 0. 3) (13a) i n a homogeneous c l o u d . A cloud of u n i t o p t i c a l depth can be d e f i n e d as having a column d e n s i t y such t h a t E(a) = 0, namely, na = 3.3x103i E z (x + 0.3)- 1 « 1020 c m - z (14) if? E e , the s t a r t i n g energy as seen by the c l o u d , i s 1 MeV. A c l o u d of s o l a r mass and d e n s i t y 10 6 c m - 3 w i l l have na = 1023 cm - 2; c l o u d s s t a r t i n g i n our model w i l l very l i k e l y be t h i c k . The " o p t i c a l " depth i s r e l a t i v e l y independent of i o n i z a t i o n , as r e f l e c t e d i n the (x+0.3) term. An o p t i c a l l y t h i c k c l o u d not s u b j e c t to expansion w i l l behave s i m i l a r l y under p a r t i c l e a c c e l e r a t i o n as under photon a c c e l e r a t i o n . However, the c l o u d can be t h i c k to cosmic rays i f i t i s i o n i z e d . T h i s i s i n c o n t r a s t to the photon case, when a c l o u d i s o p t i c a l l y t h i c k only i f i t i s n e u t r a l . The o p t i c a l depth to photons, then, depends on the d i s t a n c e to the source, s i n c e c l o u d s are i o n i z e d near the source, and n e u t r a l f a r from the source, as i n equation (3). A l s o , the momentum gai n f o r p a r t i c l e s i s more e f f i c i e n t , because L/c must be r e p l a c e d by 29 I - / V p i with v^> the p a r t i c l e v e l o c i t y . The " i n v e r s e - s q u a r e " approach i s the same, using equation (8) with a = c o n s t a n t and no g r a v i t y . I n t e g r a t i o n out to some r » rit g i v e s u ( r » r s t ) 2 = 31 _J_ (15) IrnrVpiii (na) rsi ~ 12 x 1Q3Q (km s ~ * ) 2 (na) r S f (pc) i f E c = 1 MeV and v ?= 10* km s ~ l ; and using (na) = 1 0 2 3 , L^j, = 0.01, g i v e s u = 1 100/r s r (pc) km s _ 1 . T h i s v e l o c i t y i s a n o u - n e g l i g i b l e f r a c t i o n of the proton v e l o c i t y ; o b v i o u s l y the e f f e c t of the c l o u d v e l o c i t y must be taken i n t o account. We see immediately however, the d i f f e r e n c e between the cosmic ray and X-ray c a s e s . The a n a l y s i s below of the expanding c l o u d s , which i s probably the more l i k e l y model, does i n c l u d e the r e l a t i v e v e l o c i t i e s . For b e t t e r numerical e s t i m a t e s , the equ a t i o n of motion was n u m e r i c a l l y i n t e g r a t e d i n the t h i c k c l o u d and t h i n c l o u d l i m i t s . Equation (13) can be used to d e r i v e a s o l u t i o n f o r the change i n p a r t i c l e v e l o c i t y upon passing through the c l o u d , -£v = v ( a ) - v 0 . The r e l a t i v e v e l o c i t y upon e n t e r i n g the c l o u d i s v o = v^ -u i f u i s the clo u d v e l o c i t y r e l a t i v e to the source, and v(a) i s the p a r t i c l e v e l o c i t y a f t e r l e a v i n g the c l o u d . The t h i n c l o u d case w i l l be d e f i n e d when Av<<v 0; from equation (13a), v 3 A v « 1. 3x10i 6 n a (16) The momentum t r a n s f e r r e d to the c l o u d per p a r t i c l e i s then mfAv. As the r e l a t i v e v e l o c i t y decreases, the momentum gain per p a r t i c l e i n c r e a s e s as ( v ^ - u ) - 3 , which r e f l e c t s the i n c r e a s e d h e a t i n g e f f i c i e n c y of s o f t e r cosmic r a y s . When the t h i c k case i s reached, given by equation (14), the momentum gain per 30 p a r t i c l e i s m t ( v r - u ) . The number f l u x of cosmic r a y s i s L / ( 4 t r E c r 2 ) . The equation of motion f o r a t h i n c l o u d i s du = L a 2 mpA v _ GM dt 4 E D r 2 4trm na 3/3 r 2 which becomes du = 3.0x10 2L^ 1 2.0x10-* lM (17) dt E 2 . s (1-u) 3 E 2 r z i f E 0 i s i n MeV, M i n s o l a r masses, r i n pc, u i n u n i t s of the ( r e s t frame) cosmic ray v e l o c i t y , and t i n u n i t s of ( p c / v p ) . T h i s was s o l v e d n u m e r i c a l l y f o r a range of the parameters LHifi , E 0 , H and r s t , the s t a r t i n g d i s t a n c e of the c l o u d from the s o u r c e . The r e s u l t s are summarized i n Table I I I . Since the g r a v i t a t i o n a l term i s very s m a l l f o r most expected n u c l e a r models, i t was omitted i n the c a l c u l a t i o n s . The upper l i m i t on the c e n t r a l mass c o n s i s t e n t with t h i s (the mass f o r which the g r a v i t a t i o n a l f o r c e i s one per cent of the cosmic ray f o r c e ) i s given i n the t a b l e . I t turns out that the s o l u t i o n s w i l l e i t h e r approach steady asymptotic v e l o c i t i e s , i f r 2 i n c r e a s e s f a s t e r than ( 1 - u ) 3 decreases or tend to blow up, i f ( 1 - u ) 3 dominates. The l a t t e r case i s i n v a l i d , of course, s i n c e the t h i n c l o u d assumption no longer h o l d s . The i n t e g r a t i o n was stopped i f the c l o u d reached a v e l o c i t y of 0.9Vj,. The v e l o c i t i e s i n Table I I I i n d i c a t e t h a t even f o r t h i n c l o u d s , v e l o c i t i e s of s e v e r a l thousand km s - 1 are o b t a i n e d . The eguation f o r the t h i c k case assumes the t o t a l momentum of the proton beam h i t t i n g the c l o u d i s t r a n s f e r r e d to the c l o u d . The equation f o r the c l o u d v e l o c i t y i s du = L a 2 m? (v P -u) GH dt 4 E e r 2 ilTmna 3/3 r 2 Table I I I Cosmic Ray A c c e l e r a t i o n Of F r e e l y Expanding Clouds - T 1 a - Thin c l o u d s — no g r a v i t y | L(x10**) v/v ? E0(MeV) v (km s - i ) \ ergs s ~ i | 10-3 1x108 0.078' 1 1100 | TO" 2 1 I x I C 0. 36 1 5500 I 1 0 - 1 10 1x10io 0. 25 1 3500 | 10-2 1 4x106 0.008 10 360 I 1 0 - 1 1 4x10? 0.04 10 1800 1 1 10 4x 10» 0.01 10 4500 I 1 1 4x108 0. 16 10 7200 J b. Thick c l o u d s - - no g r a v i t y | L(x10**) R*,r(Pc) v/v p E e (MeV) v(k m s~ i) | ergs s _ i | 10-2 <, 8x 106 0.073 T 1050 I 10-i 1 8x107 0. 170 1 2400 I 1 1 8x 108 0. 35 1 4900 I 10 1 8x109 0. 65 1 9100 I 10-i 1 2x107 0.04 10 1800 | 1 1 2x108 0. 20 10 9000 | 10 1 2x109 0.49 10 22000 I c. Thick c l o u d s -•- with g r a v i t y | 'L (x 10**) R 5 t(pc)' v/v f S 0 (MeV) v(km s - i ) I ergs s - 1 i 10 ! 1 0.65 1 9100 I 10 1 10 0.64 1 9000 I 100 1 100 0.83 1 12000 | 103 1 10 3 0. 80 1 11000 | 10 1 10 0. 47 10 21000 | 100 1 10 0. 82 10 37000 1 _ j C r i t i c a l Masses: T h i n : M r t^<1.3x10HL 4 4/E2.5 (1-u) 3 K q T h i c k : M < 1.3x10 i 3 L H H (1-u) / (na) E-o . s M G 32 which becomes, n u m e r i c a l l y , du = 2.8x10* L4i» (1-u) 2. Ox 1 0 - 1 1 H (18) dt E « - 5 (na) r 2 ~ r 2 with the same u n i t s as i n equation (17). The r e s u l t s of numerical i n t e g r a t i o n of t h i s equation are a l s o l i s t e d i n Table I I I . Large c l o u d s of d e n s i t y 10 7 cm - 3 and one s o l a r mass were assumed. These s o l u t i o n s a l l d i s p l a y a c h a r a c t e r i s t i c approach to a steady, asymptotic v e l o c i t y , which can be as high as tens of thousands of km s-». In a couple of cases the c l o u d became t h i n b e f o r e i t reached t h i s v e l o c i t y . Again, the g r a v i t a t i o n a l term was n e g l e c t e d , and the l a r g e s t allowed c e n t r a l mass i s i n d i c a t e d . Since these masses are s m a l l e r than those allowed f o r t h i n c l o u d s (compare equations 17 and 18) f u r t h e r i n t e g r a t i o n s were done i n c l u d i n g , a range of c e n t r a l masses. The nature of the s o l u t i o n s was not much a f f e c t e d by t h i s , nor, as Table I I I shows, were the r e s u l t a n t v e l o c i t i e s changed d r a s t i c a l l y . In the t h i c k c a l c u l a t i o n s , where the column d e n s i t y i s important, the c l o u d was allowed to expand at a sound speed of 10 km s _ 1 . T h i s w i l l decrease the a c c e l e r a t i o n somewhat compared to pressure-bound c l o u d s but the c a l c u l a t i o n s v e r i f y t h a t t h i s i s a s m a l l e f f e c t . Summarizing these r e s u l t s , we have found that c l o u d v e l o c i t i e s r e s u l t i n g from cosmic ray pressure w i l l g e n e r a l l y be higher than the photon-produced v e l o c i t i e s f o r the same model parameters. The a b s o l u t e upper l i m i t on the c l o u d v e l o c i t y , of c o u r s e , w i l l be the v e l o c i t y of the cosmic rays themselves, which i s 4 . 5 x 10* km s _ l f o r 10 MeV protons. 33 CHAPTER I I I \ CONTINUOUS FLOW MODELS The Model / / Based on the c o n s i d e r a t i o n s above, i n t h i s / t h e s i s I w i l l / e x p l o r e the dynamical e f f e c t s of a p o i n t source of r a d i a t i o n which i s turned on i n s i d e a s p h e r i c a l l y symmetric yas c l o u d . In order to s i m p l i f y the c a l c u l a t i o n s , and to understand the e f f e c t of the c e n t r a l source alone, I s h a l l i g n o r e f o u r important phenomena. The f i r s t i s extended h e a t i n g , as might come from s u p e r s o n i c s t e l l a r motions through the gas, or other sources of t u r b u l e n c e . The second i s r o t a t i o n ; there' i s both d i r e c t (Anderson, 1974, d e s c r i b e s the p o s s i b l e r o t a t i o n of the nucleus of NGC 4 151) and i n d i r e c t (Glaspey et a l - , 1975, f i n d a s u g g e s t i v e l y symmetrical emission l i n e p r o f i l e i n NGC 1068) evidence f o r r o t a t i o n of the n u c l e a r gas. T h i r d l y , I s h a l l i gnore r a d i a t i v e t r a n s f e r , and assume the gas i s t h i n to a l l secondary r a d i a t i o n . F i n a l l y , no magnetic f i e l d s w i l l be c o n s i d e r e d . As pointed out by I p a v i c h (1975), a cosmic ray-magnetic f i e l d i n t e r a c t i o n can be another source o f outward pr e s s u r e , thereby i n c r e a s i n g the dynamic e f f e c t s on the gas. However, l a c k i n g good i n f o r m a t i o n on the s i z e of the f i e l d , and any i n f o r m a t i o n on the o r d e r i n g , I assume B < B , which i s c o n s i s t e n t with the d i s c u s s i o n of Chapter I I . I t should be noted t h a t dust i s 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 s t r o n g reddening seen by Wampler (1968) i n S e y f e r t s i n d i c a t e s the presence of dust i n or around the nucleus — i n d e e d t h i s may be the IR source — but Mathis (1970), by 34 c o n s i d e r i n g the r a d i a t i v e t r a n s f e r , concludes the g r a i n s must be o u t s i d e the OER. The r e s o l u t i o n of the IR source a t 100 pc (Neugebauer, Garmire, Rieke and Low, 1971) supports t h i s c o n c l u s i o n . The model to be c o n s i d e r e d i n v o l v e s l e s s gas than would produce the emission l i n e s at " r e a s o n a b l e " temperatures and d e n s i t i e s . (The Balmer l i n e o b s e r v a t i o n s of S i l e k et a l . , 1973 suggest a d e n s i t y of 3 x 10 7 cm - 3 i n a re g i o n 1 0 1 7 cm i n radius.) At t h i s d e n s i t y , the i o n i z i n g quanta w i l l p e n e t r a t e only a few per cent of t h i s d i s t a n c e , as shown below, so to be s p a r i n g of computer time, I r e p r e s e n t only the i n n e r part o f t h i s r e g i o n n u m e r i c a l l y . T h i s covers a l l the i n t e r e s t i n g dynamical e f f e c t s . T h i s gas i s giv e n an a r b i t r a r y i n i t i a l c o n f i g u r a t i o n . A h y d r o s t a t i c e q u i l i b r i u m c o n f i g u r a t i o n around the massive c e n t r a l o b j e c t would be c e n t r a l l y condensed on a much s m a l l e r s c a l e than the p e n e t r a t i o n d i s t a n c e f o r the i o n i z i n g quanta. T h i s would be awkward t o handle n u m e r i c a l l y . The g r a v i t a t i o n a l f r e e f a l l time f o r t h i s gas i s t y p i c a l l y 10 5 years, much l e s s than the l i f e t i m e of the S e y f e r t s t a t e , but l a r g e r than the escape time f o r a gas c l o u d . The probable repeat time of the phenomenon suggested i n Chapter VI i s a l s o s h o r t . A l s o , r o t a t i o n of the nucleus w i l l support an extended d e n s i t y s t r u c t u r e . T h e r e f o r e , an a r b i t r a r y i n i t i a l d e n s i t y c o n f i g u r a t i o n i s used, which i s much l e s s c e n t r a l l y condensed than i n the h y d r o s t a t i c e q u i l i b r i u m case. The gas w i l l be i n i t i a l l y i s o t h e r m a l , at a moderate temperature of 5000 t o 10,000 K. Schwarz, McCray and S t e i n (1973) p o i n t out that the c o o l i n g time f o r gas below t h i s 35 temperature i s an o r d e r of magnitude l o n g e r than t h a t f o r h o t t e r gas. I f the gas comes from the s t e l l a r r e g i o n s of the galaxy and has condensed i n the c e n t e r , we would expect i t s t i l l t o be f a i r l y warm and s t i l l c o o l i n g . (Matthews and Baker, 1971, c a l c u l a t e the temperature of the gas c o l l e c t i n g i n the c e n t e r s of e l l i p t i c a l s , due to thermal i n s t a b i l i t y i n the wind, t o be about 6000 K.) In any case, t h i s i s not a s e n s i t i v e parameter s i n c e once the c e n t r a l source i s turned on the gas q u i c k l y reaches a new temperature e q u i l i b r i u m determined by the balance of i o n i z a t i o n h e a t i n g and r a d i a t i v e l o s s e s . I s h a l l assume a massive c e n t r a l energy source, with a few x 10 a s o l a r masses 1 c o n t a i n e d i n s i d e a n e g l i g i b l e r a d i u s . (The one X-ray model uses a lower c e n t r a l mass.) T h i s source w i l l suddenly t u r n on, producing quanta which heat the gas and p r o v i d e an outward p r e s s u r e . The outer boundary of the gas i s a zero d e n s i t y s u r f a c e . Since most e f f e c t s propagate outward, t h i s t r u n c a t i o n w i l l not a f f e c t the e a r l y e v o l u t i o n very much, although i n some cases the t r u n c a t e d outer boundary w i l l g i ve r i s e t o s p u r i o u s e f f e c t s . The i n n e r boundary i s a hard s u r f a c e of nonzero r a d i u s with zero gas v e l o c i t y . The e v o l u t i o n of t h i s gas i s f o l l o w e d u s i n g numerical hydrodynamic f i n i t e d i f f e r e n c e e q u a t i o n s u n t i l i t i s expected to become u n s t a b l e . Based on the c o n s i d e r a t i o n s i n Chapter I I , I i n v e s t i g a t e mainly the cosmic ray case but a l s o do one 1 T h i s c e n t r a l mass, p l u s a s t e l l a r . m a s s d i s t r i b u t i o n l i k e t h a t g i v e n by Oort (1971), i s lower than S c h w a r z s c h i l d • s (1973) estimate of the c e n t r a l mass of NGC 4151; however i t i s c o n s i s t e n t with the e s t i m a t e s of Anderson (1973) and of Wolfe (1974). 36 c a l c u l a t i o n with x-rays alone. The other important parameters, the cosmic ray l u m i n o s i t y and the r a t i o of cosmic ray d e n s i t y to gas d e n s i t y , w i l l be v a r i e d to i n v e s t i g a t e t h e i r e f f e c t s . What beh a v i o r would we expect f o r t h i s model, and how would i t be observed? The i n n e r l a y e r s of the gas c l o u d w i l l absorb the p a r t i c l e s (or photons) from the c e n t r a l source and w i l l be pushed outward, growing i n t o a dense s h e l l and l e a v i n g a r a t h e r empty c a v i t y i n s i d e . The l e s s dense outer r e g i o n s w i l l have a lower expansion v e l o c i t y , s i n c e the net f o r c e s a p p l i e d t h e r e w i l l be lower. I f the s h e l l d e n s i t y i s high enough t o p r e f e r e n t i a l l y guench the f o r b i d d e n l i n e s i n the s h e l l , an ob s e r v a b l e v e l o c i t y d i f f e r e n t i a l between the per m i t t e d and f o r b i d d e n l i n e s may r e s u l t . 1 Such a c o n f i g u r a t i o n has a l s o been suggested by MacAlpine (1974), i n c o n n e c t i o n with the p a r t i c l e - s t r e a m i n g model of Ptak and Stoner (1973 a,b). He has a l s o suggested t h a t the gas heated by the p a r t i c l e - d r i v e n shock w i l l c o o l by r a d i a t i o n , which t r a n s f e r s the shock k i n e t i c energy to the outer p a r t s of the gas and f u r t h e r i o n i z e s i t . A heavy l a y e r of gas supported a g a i n s t g r a v i t y by a l i g h t , l e s s dense gas i s s u s c e p t i b l e to the R a y l e i g h - T a y l o r i n s t a b i l i t y (see Chandrasekhar, 1961). As the s h e l l develops i n t h i s problem, i t resembles the c l a s s i c a l R a y l e i g h - T a y l o r u n s t a b l e case with two d i f f e r e n c e s . The gas i s not s t a t i c but i s moving outward, and t h i s motion may s t a b i l i z e some d i s t u r b a n c e s . A l s o , * T h i s v e l o c i t y d i f f e r e n c e i s i n the same sense as i n the o p t i c a l l y t h i c k f i l a m e n t model, as summarised by Rees and Sargent (1972), but i n the o p p o s i t e sense to the Av i n d i c a t e d by the o b s e r v a t i o n s of E i l e k et a l . , (1973). 37 t h e r e i s an outward f o r c e on some of the dense gas due to the cosmic r a y s , such t h a t the e f f e c t i v e g r a v i t y i s a c t u a l l y p o s i t i v e i n p a r t of the s h e l l . The f u l l p e r t u r b a t i o n e q u a t i o n s with nonzero v e l o c i t i e s are i n t r a c t a b l e , so an estimate of the s t a b i l i t y w i l l be made f o r each model. Should the s h e l l become unst a b l e at some d i s t a n c e away from the source, the e v o l u t i o n beyond t h i s p o i n t can be t r e a t e d with the d i s c r e t e c l o u d model, as d i s c u s s e d i n Chapter I I , and the r e s u l t s t h e r e should a p p l y . F i n a l l y , we note t h a t such a c o n f i g u r a t i o n would be very s h o r t - l i v e d . The shock w i l l empty the re g i o n i n something l i k e 10* y e a r s . The numerical c a l c u l a t i o n s w i l l not be extended to d i s c u s s i o n of the r e g e n e r a t i o n mechanism which would be needed to maintain the a c t i v e nuclear s t a t e f o r 10 8 y e a r s . Energy T r a n s f e r And P h y s i c a l C o n d i t i o n s 1. Heating Mechanisms The e n e r g e t i c p a r t i c l e s e m i t t e d by the c e n t r a l source t r a n s f e r energy and momentum to the surrounding gas by i o n i z a t i o n and by d i r e c t Coulomb i n t e r a c t i o n with the i o n s i n the gas. The t r a n s f e r mechanism f o r p a r t i c l e i o n i z a t i o n of hydrogen i n the i n t e r s t e l l a r gas i s d i s c u s s e d i n d e t a i l by Dalgarno and McCray (1972). The main point i s that only a s m a l l f r a c t i o n of the p a r t i c l e energy ends up as heat i n the gas a f t e r one i o n i z a t i o n . The suprathermal e l e c t r o n produced i n the f i r s t i o n i z a t i o n w i l l produce s e v e r a l secondary i o n i z a t i o n s and e x c i t a t i o n s , l o s i n g some energy each time, u n t i l i t s k i n e t i c 38 energy i s l e s s than 10.2 eV, whereupon the remaining energy i s c o l l i s i o n a l l y d i s p e r s e d as heat. (See a l s o Goldsmith, Habing and F i e l d , 1969, f o r a d i s c u s s i o n of t h i s process.) A t y p i c a l value quoted by Dalgarno and McCray f o r the energy thus t r a n s f e r r e d i s about 27 eV per primary i o n i z a t i o n i n an i o n i z e d (x = 0.1) gas. Most of the c a l c u l a t i o n s on t h i s h eating r a t e assume the gas i s o p t i c a l l y t h i n , and the recombination r a d i a t i o n i s l o s t . A l d r o v a n d i and Peguinot (1973) have c a l c u l a t e d t h e r m a l - e q u i l i b r i u m , cosmic ray heated models f o r the t h i n and very t h i c k c a s e s , and f i n d t h a t r e - a b s o r p t i o n of t h i s secondary r a d i a t i o n i s not a dominant e f f e c t . The heating and c o o l i n g r a t e s are changed by no more than a f a c t o r of two by r e - a b s o r p t i o n . I f the gas i s p a r t i a l l y i o n i z e d , as i s the case i n these g a l a c t i c n u c l e i , the dominant energy t r a n s f e r w i l l be through long-range Coulomb c o l l i s i o n s . Dalgarno and McCray g i v e a heating r a t e per hydrogen i o n i z a t i o n 1 by t h i s process of 10x times the i o n i z a t i o n h e a t i n g r a t e , where x i s the f r a c t i o n a l i o n i z a t i o n , n e / n H . Thus i f x i s g r e a t e r than 0.1, the Coulomb hea t i n g i s the more important p r o c e s s . The heating produced by i n c i d e n t photons, r a t h e r than charged p a r t i c l e s , resembles the cosmic ray i o n i z a t i o n h e a t i n g , i n t h a t the f i r s t i o n i z a t i o n produces a very hot e l e c t r o n , which produces s e v e r a l s e c o n d a r i e s . The d i f f e r e n c e i s that a l a r g e r p o r t i o n of the photon energy i s converted to heat. T y p i c a l l y 1 The h e a t i n g r a t e i s not o b v i o u s l y r e l a t e d t o the i o n i z a t i o n r a t e , i f the dominant process i s Coulomb i n t e r a c t i o n ; i n f a c t they t u r n out to be n e a r l y p r o p o r t i o n a l . See below. 39 about 70 per cent of the energy i s d e p o s i t e d as heat per primary hydrogen i o n i z a t i o n , f o r s o f t X-rays i n a gas of cosmic abundances. T h i s d i f f e r e n c e i n e f f i c i e n c y i s mainly due to the higher c r o s s s e c t i o n f o r helium i o n i z a t i o n by the^ photon (q-(He) 'S 20 T(H) f o r s o f t X-rays) so t h a t the helium atoms, even i f only ten per cent by number, c o n t r i b u t e 7 a l a r g e r amount to the energy g a i n ( d e f i n e d per hydrogen i o n i z a t i o n ) . The energy t r a n s f e r to the gas w i l l be reduced i f the gas i s very hot or i f i t i s moving away from the source a t n e a r l y the cosmic ray v e l o c i t y . T h i s r e d u c t i o n of the energy t r a n s f e r , d e s c r i b e d below, i s c o n t a i n e d i n the f a c t o r f ^ l , used i n Chapter V. The energy l o s s of an e n e r g e t i c p a r t i c l e i s independent of the temperature of the ambient gas as long as kTme/nip i s much below the p a r t i c l e energy. When these e n e r g i e s become comparable, one would expect the heating r a t e t o decrease. L a r k i n (1960) d e r i v e d a T ~ 1 - 5 dependence f o r the energy l o s s r a t e v i a Coulomb i n t e r a c t i o n s when the e l e c t r o n thermal v e l o c i t y i s much l a r g e r than the cosmic ray v e l o c i t y . In the c a l c u l a t i o n s , f a i r l y s o f t cosmic rays are assumed, and the t r a n s i t i o n temperature i s 10 7 K. As the gas v e l o c i t y , u, approaches the v e l o c i t y of the lower energy cosmic rays the i o n i z a t i o n r a t e w i l l decrease. The l o s s goes approximately as E(u) 0^, where E (u) i s the lowest energy of the power law spectrum to reach the gas, and o( i s the exponent of t h a t spectrum. T h i s comes from the dependence of the i o n i z a t i o n r a t e on the low energy c u t o f f (equation 19 below.) The energy l o s s per i o n i z a t i o n a l s o depends on the low 40 energy c u t o f f (Bergeron and C o l l i n - S o u f f r i n , 1973), i n t r o d u c i n g a second-order c o r r e c t i o n . These c a l c u l a t i o n s o n l y i n c l u d e the momentum and energy t r a n s f e r due to hydrogen i o n i z a t i o n and Coulomb processes, i n the cosmic ray case, and t o hydrogen and h e l i u m / i n the X-ray / case. Matthews (1974) has c a l c u l a t e d at l e n g t h the momentum t r a n s f e r r e d to a gas by a b s o r p t i o n of a power law photon spectrum, i n c l u d i n g both the c o n t i n u a and the s t r o n g e r l i n e s of hydrogen, carbon and oxygen, with cosmic abundances, and found t h a t the hydrogen continuum o p a c i t y i s the dominant f o r c e under most c o n d i t i o n s , with helium a l s o important. Thus, f o r the purposes of t h i s c a l c u l a t i o n , the average v a l u e s given at the s t a r t o f t h i s s e c t i o n are an a c c u r a t e enough estimate of the momentum t r a n s f e r and h e a t i n g r a t e . 2. I n c i d e n t Flux and U s e f u l R e l a t i o n s . The spectrum of i o n i z i n g p a r t i c l e s i s assumed to obey a power law, i n accord with most e n e r g e t i c a s t r o p h y s i c a l sources observed. The t o t a l energy f l u x i s , of cou r s e , the i n t e g r a t e d energy spectrum, and i s p h y s i c a l l y the most meaningful parameter to d e s c r i b e the c e n t r a l source. In order t o compare with other work, 1 however, I s h a l l c a l c u l a t e i n terms of the i o n i z a t i o n r a t e per hydrogen atom. T h i s i s needed f o r the i o n i z a t i o n balance e q u a t i o n , and can be r e l a t e d t o the t o t a l f l u x and to 1 In l o c a l s p i r a l - a r m work, the i o n i z a t i o n r a t e ~$ can be determined d i r e c t l y from o b s e r v a t i o n s where the cosmic ray f l u x cannot. T h i s makes i t a u s e f u l parameter t h e r e . Recent work by Meszaros (1974) i n d i c a t e s £ < 1 0 ~ 1 6 i n some nearby r e g i o n s . 41 the heating r a t e . I f <T(E) i s the c r o s s s e c t i o n f o r hydrogen i o n i z a t i o n , and j(E) i s the p a r t i c l e f l u x per cm 2 per second at an energy E, the i o n i z a t i o n r a t e i s g i v e n by / 1 ^ \ <T (E) j (E) dE. / (19) & ° The power law spectrum i s g i v e n by j (E) = j E - K. f o r E > E 0 , where E c i s the low energy c u t o f f . The exponent, <\ , i s about 2.7 f o r g a l a c t i c cosmic rays above 10 2 MeV ( L i n g e n f e l t e r , 1973). The r a p i d d e c l i n e of j(E) with i n c r e a s i n g energy means that the lower energy p a r t i c l e s w i l l be most important i n heating the gas. The c r o s s s e c t i o n as given by Bethe (1933) i s <J (E) = 9 x 1 0 - 2 *s (^ ) /E cm 2 (20) where s(fi) = 6.2 + l o g g 2 + 0.86P>2 ' * 1-p and Q = v/c, which i s v a l i d f o r s u b r e l a t i v i s t i c p a r t i c l e s . The energy l o s s e s f o r a charged p a r t i c l e p a s s i n g through an i o n i z e d gas can be represented-as b . ^ f E ) , the i o n i z a t i o n l o s s e s , and (E), the Coulomb l o s s e s . From Appendix I, equation (1.1), we see that b ^  (E) oc aur In ( E)/v (E) , and b ^ J E ) « n e l n (E) /v (E) . I t can be shown that the gas h e a t i n g r a t e i s p r o p o r t i o n a l t o the i o n i z a t i o n r a t e . T h i s i s d i s c u s s e d i n some d e t a i l by Goldsmith, Habing and F i e l d (1969), and I s h a l l summarize t h e i r argument. Because cosmic ray i o n i z a t i o n t r a n s f e r s only a f r a c t i o n of the energy l o s s to the gas, the i n t e g r a t e d heating r a t e can be w r i t t e n H = j ^ j (E) ( F b ^ ( E ) + b e U c (E))/v (E) dE (21) 42 where F < 1 r e p r e s e n t s the e f f i c i e n c y of the energy t r a n s f e r . The i n t e g r a n d here has the approximate f u n c t i o n a l form j ( E ) . E _ l ( i f F i s assumed to vary s l o w l y with E ) . From equation (20) we see t h a t t h i s i s a l s o the form of the most r a p i d l y v a r y i n g p a r t of the i n t e g r a n d i n e q u a t i o n (19). Thus Hex. J / even though the / energy l o s s e s are not d i s c r e t e , as with photons. The t o t a l energy f l u x over a sphere of r a d i u s r can a l s o be r e l a t e d to the i o n i z a t i o n r a t e , u sing t h e spectrum g i v e n above: °<3 F ^ = 4TrrfJ j (E) EdE. (21a) The i o n i z a t i o n c r o s s s e c t i o n i s r e p r e s e n t e d again as T (E) = T c E - i where 3.5x10-" f o r E near 1 MeV. The c o n v e r s i o n from £ to t o t a l cosmic ray l u m i n o s i t y i s a r b i t r a r y because the low end of the cosmic ray spectrum i s unknown. (The numerical c a l c u l a t i o n s i n t h i s t h e s i s , of course, depend only on - ) In energy e s t i m a t e s below I s h a l l assume ({-2.1 above E,=100 MeV, i n accord with measurements of the g a l a c t i c spectrum ( L i n g e n f e l t e r , 1973), and use the s o f t e r o(.= 1.0 from E Q to E H . E v a l u a t i n g both (19) and (21a), then, g i v e s the t o t a l f l u x , F ^ = 6x1052 X ^ ( E 0 / E , ) r 2 e r g s s ~ i (22) where rz i s now i n p c i For the X-ray c a s e , a power law l u m i n o s i t y spectrum w i l l be assumed, i n accordance with o b s e r v a t i o n s : L (v) = L 0 i / - C < - f o r V > Vc. T y p i c a l l y , °L •- 1 or 2 i n e n e r g e t i c s o u r c e s . F o l l o w i n g Dalgarno and McCray, h"0o i s taken as 100 eV i n these c a l c u l a t i o n s . The c r o s s s e c t i o n f o r photon i o n i z a t i o n of hydrogen i s g i v e n by A l l e n (1963) as Cj^  (-u) = (V ~3' The c r o s s s e c t i o n f o r s i n g l e i o n i z a t i o n of helium i s ^TWe_ (V) = 4 3 The h e a t i n g r a t e i n t h i s case w i l l again be expressed as p r o p o r t i o n a l t o the i o n i z a t i o n r a t e . For a h i g h l y i o n i z e d gas, the heat gain i s p r o p o r t i o n a l to the photon energy (the e f f i c i e n c y f a c t o r i s n e a r l y unity) and a s u i t a b l e average energy gain can be d e f i n e d . That i s , we can express H = j [L (v)/hv](<T H (v)+f fteV^Cv) ) E h (v) dD (23) i f f M e i s the r e l a t i v e helium abundance by number, as H = < E K > \ [L (v) /hv] r»(v) ^ = % <EW>-(See Dalgarno and McCray f o r some c a l c u l a t i o n s of < E K > , the average energy g a i n per i o n i z a t i o n . ) Because the p h o t o i o n i z a t i o n c r o s s s e c t i o n drops o f f so r a p i d l y with frequency, the lowest energy photons have a very s t r o n g e f f e c t on the h e a t i n g and i o n i z a t i o n of the gas. The low energy end of the spectrum chosen f o r the c a l c u l a t i o n i s a s t r o n g parameter a f f e c t i n g the e n e r g e t i c s . The i n t e g r a t e d X-ray l u m i n o s i t y can be r e l a t e d to the i o n i z a t i o n r a t e i n the same manner as done above. T h i s g i v e s F^ = 3x10"$' E* r2 5 y ergs s- 1 (24) where E 0 = hx>o i s the low energy s p e c t r a l c u t o f f i n MeV, and $ i s c l o s e to u n i t y . T h i s c o n v e r s i o n assumes the power law behavior c o n t i n u e s to the c u t o f f frequency, and t h a t the spectrum drops to zero below that frequency. As pointed out i n Chapter I I , X-ray o b s e r v a t i o n s of some S e y f e r t g a l a x i e s i n d i c a t e a l o c a l maximum i n the f l u x at s e v e r a l keV, meaning the s o f t X-ray spectrum i s not a simple e x t r a p o l a t i o n of the o p t i c a l power law continuum. These problems w i l l not a f f e c t the nu m e r i c a l c a l c u l a t i o n s d i r e c t l y , except through the o p t i c a l depths d i s c u s s e d below, and through the value of <EL,>, Only the 44 energy requirements are s e n s i t i v e t o them. S i m i l a r arguments of course hold f o r the cosmic ray spectrum assumed, but there i s as yet no d i r e c t o b s e r v a t i o n a l evidence of the p a r t i c l e spectrum w i t h i n the c e n t r a l parsec o r two. These approximate s p e c t r a are t h e r e f o r e s u f f i c i e n t f o r t h i s c a l c u l a t i o n . 3. O p t i c a l Depths. The r a t e of energy l o s s given above i n d i c a t e s t h a t a 1 MeV p a r t i c l e w i l l t r a v e l about 5 x 1 0 _ s pc i n an i o n i z e d hydrogen gas c l o u d of d e n s i t y 10 6 cm - 3. The mean f r e e path of a 200 eV s o f t X-ray photon i n the same gas, i n c l u d i n g hydrogen and helium i o n i z a t i o n o p a c i t y (Brown, 1971) i s 2 x 1 0 - 5 pc i f the gas i s n e u t r a l . These d i s t a n c e s are very s h o r t compared to the s c a l e s expected to be r e l e v a n t to S e y f e r t n u c l e i , 0.1 to 1.0 pc. The r e f o r e the i o n i z a t i p n r a t e w i l l drop o f f r a p i d l y as the d i s t a n c e from the i n n e r edge of the gas (where the i o n i z i n g f l u x i s assumed to o r i g i n a t e ) . The r a t e of t h i s d r o p - o f f w i l l a f f e c t the s i z e of the dense s h e l l which develops. The s p a t i a l dependence of ^ or ^ i n v o l v e s i n t e g r a l s over energy and d i s t a n c e t r a v e l l e d . The a t t e n u a t i o n w i l l be approximated i n the c a l c u l a t i o n as f o l l o w s : I ( R ( r ) = ^ ( r , ) ( r f / r 2 ) e x p ( - a Z ° . * 5 ) (25a) and fyr) = \(rs ) ( r f / r ? ) exp (-a^o-s) ( 25b) where r, i s the i n n e r edge of the gas, a i s a cons t a n t of order u n i t y , and Z and ZK are norm a l i z e d column d e n s i t i e s . These equations are d e r i v e d i n Appendix I . The a t t e n u a t i o n r a t i o , 45 ^ ( r) / ^ (T, ) i i s r e p r e s e n t e d by _ T L ( r ) r 2 / r 2 t o r both cosmic rays and X-rays i n the work f o l l o w i n g . We note t h a t drops o f f f a s t e r than TTC^, which r e f l e c t s the d i f f e r e n c e s i n the i o n i z a t i o n c r o s s s e c t i o n s . _/ / 4. C o o l i n g Mechanisms. ^ The gas l o s e s energy r a d i a t i v e l y . The c o o l i n g f u n c t i o n t a b u l a t e d i n the program has been taken from A l d r o v a n d i and Peguinot (1973). They c o n s i d e r e d l i n e s and c o n t i n u a from H, He, C, 0, Ne, Mg, S i , and S, with cosmic abundances, i n a gas with the i o n i z a t i o n s t r u c t u r e determined by an e x t e r n a l f l u x of 10 MeV e l e c t r o n s , and f o r a range of gas d e n s i t i e s . The c o o l i n g mechanisms c o n s i d e r e d were c o l l i s i o n a l e x c i t a t i o n by e l e c t r o n s and atomic hydrogen, c o l l i s i o n a l i o n i z a t i o n by e l e c t r o n s , recombination and bremsstrahlung. T h e i r o p t i c a l l y t h i c k c o o l i n g c urve, A (T), f o r n = 10* was put i n t o the c a l c u l a t i o n s i n t a b u l a r form, and i s shown i n F i g u r e 1 i n Chapter V. f\ (T) i s d e f i n e d such t h a t /\(T)ngn^ = energy l o s s i n ergs s _ 1 cm - 3. They t e s t e d the e f f e c t s of o p t i c a l l y t h i c k and t h i n nebulae (the t h i c k case i n v o l v e s r e - i n s e r t i o n of some of the c o o l i n g r a d i a t i o n i n t o the gas, n o n - l o c a l l y ) and found l i t t l e d i f f e r e n c e i n the c o o l i n g f u n c t i o n i n the two cases. Nor i s the gas d e n s i t y a s e n s i t i v e parameter, above 10* K, u n t i l n reaches 10 8 cm - 3, when the peak of A at T 10 s K i s suppressed. I t i s worth comparing t h e i r c a l c u l a t i o n s with those summarized i n Dalgarno and McCray (1972), which assumed onl y thermal i o n i z a t i o n . The two curves agree w e l l above T = 10* K, and the d i f f e r e n c e s may be due to d i f f e r e n c e s i n atomic 46 parameters or l i n e s i n c l u d e d ; below 10* K, the i o n i z a t i o n i s h i g h e r i n the p a r t i c l e - i o n i z e d case, and the c o o l i n g f u n c t i o n , which a r i s e s from e l e c t r o n impact e x c i t a t i o n of t r a c e elements, i s higher as w e l l . I t i s a l s o below 10* K t h a t t h i s f u n c t i o n i s s e n s i t i v e to the gas d e n s i t y (the h i g h e r d e n s i t y suppresses i o n i z a t i o n ) . The dominant c o o l i n g mechanisms above t h i s p o i n t are as f o l l o w s . The sharp r i s e at 10* K comes from hydrogen e x c i t a t i o n . Hydrogen c o o l i n g drops again when i t becomes i o n i z e d , a t a few times 10* K; other l i n e s , mainly from OIII to OVI and NelV to NeVIII, provide the s t r u c t u r e of the peak up to 10 6 K, a c c o r d i n g to Cox and Tucker (1969). Above t h i s p o i n t hydrogenic bremsstrahlung i n c r e a s e s and /\ (T) r i s e s again. Past T = 10 8 K, A(T) i s simply e x t r a p o l a t e d as T ° » 5 , the form f o r bremsstrahlung. 5. I o n i z a t i o n S t a t e ; Time S c a l e s . The hydrogen and helium i o n i z a t i o n s t a t e s are assumed to s a t i s f y an e q u i l i b r i u m e g u a t i o n , wherein the r a t e s of cosmic ray (or photon) i o n i z a t i o n plus e l e c t r o n c o l l i s i o n a l i o n i z a t i o n are balanced by recombination. T h i s means only a q u a d r a t i c equation {or a coupled s e r i e s of equations) must be s o l v e d a t each s t e p , r a t h e r than the f u l l r a t e equations. T h i s s i m p l i f i c a t i o n r e q u i r e s that the r a t e s f o r a l l t h r e e processes be r a p i d compared with the dynamical time s c a l e they are summarized i n T a b l e V I I . And the time s c a l e f o r the i n t e r n a l energy to change, t (en). The time step i n the program, as d i s c u s s e d below, i s u s u a l l y chosen to be 0.1e/|&e/dt| ~ 0.1t(en). T h i s i s t y p i c a l l y St = 1 0 - 5 ta, based on the dense gas where |de/^t| i s the l a r g e s t , which g i v e s t (en) = 10-*t ete 3 x 1 0 7 r c seconds as the t i m e s c a l e f o r changes i n the i n t e r n a l energy- (Areas o f t h i n n e r gas w i l l have a l a r g e r t ( e n ) . ) The i o n i z a t i o n time s c a l e can be estimated from the recombination r a t e o((T) = 3 x 1 0 _ 1 i T - o ^ n x 2 s~l. In an i o n i z e d r e g i o n t h i s g i v e s t (rec) = 3 x 1 0 i ° T O « s / n s . i f T = 10* and n = 10*/r o# t (rec) = 3 x 10*r o s j t ( i o n ) = t (rec) (1-x)/x. <5> 48 CHAPTER IV NUMERICAL METHODS The b a s i c e q u a t i o n s of f l u i d dynamics are d i f f e r e n t i a l forms of the c o n s e r v a t i o n laws f o r mass, momentum and energy. (cf. Thompson, 1972, f o r d i s c u s s i o n . ) Written i n c o n s e r v a t i v e form, f o r s p h e r i c a l l y symmetric geometry, the eguation of mass c o n s e r v a t i o n i s \f + 1 ! ( r 2 p u ) = 0. (27) Si * The momentum equation i s K? u ) + l l ( r 2 P u 2 ) = _ A_p + ^G(r) (28) G (r) r e p r e s e n t s the e x t e r n a l body f o r c e s : r a d i a t i o n pressure and g r a v i t y i n t h i s case. For a cosmic ray source, G(r) i s the sum of the momentum t r a n s f e r from hydrogen i o n i z a t i o n ("ion"), and that from Coulomb i n t e r a c t i o n s ("e"), minus the g r a v i t a t i o n a l f i e l d : G(r) = ^ ( r ) < Air >^P(HI)/^m (29) + ^ ( r ) < A l r >c ^(HII)/^m - GM c/r 2 Here, p (HI) and P (HII) are the atomic and proton d e n s i t i e s ; £J i s the t o t a l d e n s i t y , and p, the i s o t r o p i c p r e s s u r e . < &fif> i s the average momentum t r a n s f e r per hydrogen i o n i z a t i o n due to each p r o c e s s : <6Y> = 0 ( A E / v p ) . Tc^( r) ^ s t n e i o n i z a t i o n r a t e per hydrogen atom, using the f u n c t i o n a l form d e r i v e d i n Appendix I. The energy equation i s 49 fc( o e+0.5 p u*) + 1 ) ( p r 2 u ( e + 0 . 5 u H p / ^ )) (30) St K K X * \ = p R ( p ,r) + ^ uG (r) where e i s the i n t e r n a l energy per gram, and R(^,r) i s the net / h e a t i n g r a t e per gram. R (p,r) i s expressed as the heating due to i o n i z a t i o n , minus the r a d i a t i v e l o s s e s : / *<?'R> = T^(r) ^ A E ^ f l H I ) / ^ + < A E > e P ( H I I ) / ^ )/n — p/\(T,^(HII)/^)(^>(HII)/^)/m 2 (31) The < A E> terms are the average energy l o s s per i o n i z a t i o n through i o n i z a t i o n and through Coulomb i n t e r a c t i o n s . T h i s form i s only s t r i c t l y v a l i d as long as the gas i s t h i n to the c o o l i n g r a d i a t i o n , because r e a b s o r p t i o n of the secondary r a d i a t i o n i s i g n o r e d . In both e q u a t i o n s , f l u i d v i s c o s i t y and heat c o n d u c t i o n have been omitted, s i n c e they w i l l be very s m a l l f a c t o r s i n a hot t h i n gas. The i o n i z a t i o n s t a t e i s determined from an e q u i l i b r i u m e q u a t i o n . I f x = ^ (HII) / ^  •= n e / n 4 and 1-x = n H 1 / n ^ , ^ ( r ) < o ( 1 - x ) + C ( T ) ^ 2 x ( 1 - x ) = ( 4 ( T ) / m ) ^ 2 X 2 (32) f o r the case of a pure hydrogen gas i o n i z e d by cosmic r a y s . The hydrogen recombination c o e f f i c i e n t , o( (T) = 3 x 10-n T - O . S C M 3 s-lf (32 a) i s g iven by Seaton (1960). The mean atomic mass i s m. The e l e c t r o n c o l l i s i o n a l i o n i z a t i o n r a t e , C (T), i s given by HacAlpine (1972) f o r a Maxwellian e l e c t r o n spectrum, as C (T) = 3 x 10-io T o . 5 exp(-y) (1-exp (-y) ) (32b) i f y = 13.6 eV/kT. 50 To complete the system, an equation of s t a t e i s needed. For the pure hydrogen case (which w i l l be used f o r cosmic rays) t h i s i s j u s t the i d e a l gas law, with allowance f o r i o n i z a t i o n , P = p kT ( I n ) /m . (3 3) The i n t e r n a l energy per gram i s d e f i n e d r e l a t i v e to atomic hydrogen at 0 K, as e = 1.5(kT/m) (1+x) + (I H/m)x (33a) I H i s the i o n i z a t i o n energy of hydrogen-, 13.6 eV. When c a l c u l a t i n g with X-rays as the energy source, the helium i o n i z a t i o n s t a t e i s important. The s t a t e equations are extended, then, t o i n c l u d e helium abundance and i o n i z a t i o n energy. Before a t t a c k i n g the e q u a t i o n s n u m e r i c a l l y , the dependent v a r i a b l e s and source parameters are put i n d i m e n s i o n l e s s form. T h i s i s h e l p f u l f o r two reasons. F i r s t , proper s c a l i n g of the flow v a r i a b l e s ( d e n s i t y , r a d i u s and so on) means we are hand l i n g numbers around u n i t y , which makes the c a l c u l a t i o n more t r a n s p a r e n t . Second, the p h y s i c a l parameters ( c e n t r a l mass, cosmic ray f l u x and so on) are expressed i n terms of t h e i r r a t i o s to the flow parameters, which are the important q u a n t i t i e s d etermining the gas flow. We accomplish t h i s by choosing s c a l e s (which are f o r the moment a r b i t r a r y ) f o r the r a d i u s , time, v e l o c i t y , d e n s i t y , pressure and i n t e r n a l energy, l a b e l l e d r 0 , t e , u 0 , 0& , p„ and e c r e s p e c t i v e l y . Because only mass, l e n g t h and time are d i m e n s i o n a l l y independent q u a n t i t i e s , these s i x s c a l e s are redundant. We w i l l express three i n terms of the other t h r e e : t 0 = r 0 / u 0 , p c = ^ u c 2 / ^ , e,, = u 2 . ( Y i s t n e 51 a d i a b a t i c exponent.) S c a l i n g the equations (27) through (32b) i n terms of these u n i t s g i v e s us the p h y s i c a l parameters i n terms of r a t i o s : 1 = "W1'^* r°/u* (ZETA) C,<r V r / r. , <* 2 >^ / u« 3 A = A(T,x) f i r c / u > 2 ( A * CFLAM) (34) A •= ( o( (T)/m ) j> &r 0/u 0 <6 = C(T) r c ^  / u D G M c / r 0 u c 2 (AMS) The names i n parentheses are the names used i n the F o r t r a n program, and w i l l be r e f e r r e d to i n Chapter V. Of course, i f X-rays are the energy source, the d e f i n i t i o n s of ^ and H w i l l be c o r r e c t e d with the a p p r o p r i a t e <Zfr> and <AE> . Since "standard abundances" are being assumed, the s e n s i t i v i t y of to heavy elements i s suppressed, and A(x,T) as we l l as c{ (T) and C (T) depend only on atomic numbers, not the g a l a c t i c model chosen. For a given value of u 0 , ^ and H are d i r e c t l y r e l a t e d . The f u n c t i o n a l form I c a j r / r ^ ) depends on the proton spectrum, as given i n Appendixl. In order to s p e c i f y a unique p h y s i c a l system, t h e r e f o r e , once the spectrum (the low energy c u t o f f E ) i s known, we must a l s o s p e c i f y the p h y s i c a l parameters and M c, and the flow parameters, r c , u D and ^ D . However, from (34) i t i s apparent that these parameters do not occur independently. In order to produce a unique numerical 52 model f o r a given p a r t i c l e spectrum, i t i s s u f f i c i e n t to s p e c i f y only f o u r q u a n t i t i e s . These fo u r I w i l l choose as u D , ( |>0 r 0 ) , ^ a n d ^ . Thus, nowhere i s the l e n g t h s c a l e d i r e c t l y s p e c i f i e d . I t i s an i m p l i e d s c a l i n g i n the c o n v e r s i o n of these program parameters to p h y s i c a l parameters. T h i s i s summarized i n Table IV, which l i s t s the parameters needed f o r a unigue numerical model and f o r a unique p h y s i c a l model. In d e s c r i b i n g the c a l c u l a t i o n s below, the p h y s i c a l model w i l l be d e s c r i b e d i n terms of the t o t a l cosmic ray f l u x , F (equation 22), and the low energy c u t o f f E o , r a t h e r than the l e s s common q u a n t i t y , the i o n i z a t i o n r a t e . Table IV Determining Parameters For A Computed Model Numerical X^{r ( ) /rc (or F a,/r o2; E c) M c/r 0 ( M o ) P h y s i c a l The system to be s o l v e d i s d e r i v e d from the equations above using the d e f i n i t i o n s (34), with the flow v a r i a b l e s expressed i n terms of the b a s i c s c a l e s , r e , u c , and 0o, and using the 53 momentum equation to express the energy e q u a t i o n i n terms of the \ i n t e r n a l energy alone. The a r t i f i c i a l momentum and mass d i f f u s i o n terms ( i n v o l v i n g numerical c o e f f i c i e n t s and r e s p e c t i v e l y ) which w i l l be u s e f u l f o r the s t a b i l i t y of the .' / numerical s o l u t i o n are added e x p l i c i t l y . The f i n a l e q u ations are d i s p l a y e d i n Table V. / These equations have been s o l v e d as a numerical i n i t i a l - v a l u e problem, using the us u a l f i n i t e d i f f e r e n c e i n t e g r a t i o n schemes. The o v e r a l l viewpoint of such s o l u t i o n s i s d e s c r i b e d well by Richtmeyer and Morton (1967) or by Roche (1972). The s p e c i f i c method of s o l u t i o n used was a m o d i f i e d form of the I m p l i c i t Continuous E u l e r i a n (hence the acronym ICS) method due to Harlow and Amsden -(1971 and r e f e r e n c e s t h e r e i n ) . T h i s method was chosen, i n s p i t e of i t s apparent clumsiness (or o v e r k i l l of t h i s problem), because i t s i m p l i c i t f o r m u l a t i o n i s more l i k e l y to be s t a b l e , and because i t has worked w e l l with t r a n s s o n i c flow (which i s u s u a l l y handled with separate a l g o r i t h m s which are matched a t the s o n i c point) and with two—dimensional flow. The S o l u t i o n Algorithm The gas i n one space dimension and one time dimension i s repr e s e n t e d by i t s parameter values at a s e t of d i s c r e t e g r i d p o i n t s . In computation the values of d e n s i t y , pressure and energy a re taken at the ce n t e r of the c e l l s , and the v e l o c i t y 54 Table V The System Of Equations Mass Co n s e r v a t i o n ( r *P u > = X 1 ( r 2 $ f ) / Momentum l(pu) + 1 ^(r 2c>u 2) = - l i p - J^S + | ^ ( r ) (1-x)^+ %,(r)(>x - W(f/r = Energy (35) (36) 3t ^ V )) (37) = tf, ( r ) p ( l - x ) + C ( r ) p x + u J»(P+q) - A ? 2 x Sow. \ & \ v J J . I o n i z a t i o n ^ (1-x)^+ £<fx(1-x) = A f x 2 (38) S t a t e (39) p = ^  XT (1+x)/m A r t i f i c i a l V i s c o s i t y q = Ap3_u (40) values between the c e l l s . * In l a b e l l i n g , s u p e r s c r i p t s r e f e r to the time and s u b s c r i p t s to the space i n d i c e s . The d i f f e r e n t i a l e quations are r e p l a c e d by a s e r i e s of a l g e b r a i c equations i n i Welch et a l . , (1966) s t a t e that t h i s p a r t i c u l a r l o c a t i o n of f l u i d parameters r e l a t i v e t o the mesh i s the most workable one, and c i t e the case of c o n s e r v a t i o n of momentum and energy. 55 , U # J J . V.-Y,, <• U.v| u . . F u-F l = PL' ei» XC> G r i d Layout Used In The C a l c u l a t i o n which the d e r i v a t i v e s are represented as c e n t e r e d d i f f e r e n c e s between the a p p r o p r i a t e g r i d p o i n t s . In the f o l l o w i n g d i s c u s s i o n , i t w i l l be assumed t h a t v a l u e s of a l l v a r i a b l e s at the n ^ time st e p — which may be the i n i t i a l time — are known; the values at t , the "advanced" v a l u e s , are to be c a l c u l a t e d . The f i r s t s t e p of the s o l u t i o n i s to" i s o l a t e the mass and momentum f i n i t e d i f f e r e n c e e q u a t i o n s , d e r i v i n g a second order i m p l i c i t eguation i n the advanced pr e s s u r e . The momentum equation i s w r i t t e n ( p u ) ^ i - ( f u ) ^ = <t> (p. - P U ) + (1-6) (pt-PU, ) + C where the "source", + 1 / i r ( u ^ t ( r f (, u." - r 2 u*. ) / r*k) - - q? M ) / ^ r , c o n t a i n s only known terms from the pr e v i o u s c y c l e . The c o n v e c t i v e terms i n R ^ a r e w r i t t e n here i n Z I P - d i f f e r e n c e d form; d o n o r — c e l l form i s a l s o s t a b l e , and may be used i n s t e a d i f i t s lower order, 0(&r), e r r o r i s d e s i r e d to s t a b i l i z e 56 f l u c t u a t i o n s . 1 (See l a t e r d i s c u s s i o n o f d i f f u s i o n - e r r o r \ i n s t a b i l i t y . ) The term 0 i s u s u a l l y taken as 0.5, which, g i v e s an exact t i m e — c e n t e r i n g of the s e n s i t i v e pressure g r a d i e n t term. The c o n t i n u i t y equation i s d i f f e r e n c e d i n the same f a s h i o n , using © f o r the r e l a t i v e t i m e — c e n t e r i n g . / / + T (( P?;, - PD r 2.,yr. 2 - ( - ^ ) r. 2.,/r 2) We have w r i t t e n p ; , not p1?1 , i n equation (4 1) ; t h i s i s to i n d i c a t e t h a t the advanced pressure i s manufactured from the advanced d e n s i t y i n the h y b r i d form p. = ?t + c.* (p»«-i - £.*)- (4 4) T h i s r e l i e s on the changes i n the equation of s t a t e being mainly dependent on the d e n s i t y changes; cf = (op/^ )?~ £ C p*/^* f o r the equation of s t a t e of a p e r f e c t gas. T h i s i s r e l a t e d to the square of the sound speed. T h i s c o n s t a n t ^ 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 . I t s value w i l l s t r o n g l y a f f e c t the convergence 1 That i s , f o r ZIP d i f f e r e n c i n g , u 2 —> u. u- ,, I >- '2. > * It. and f o r donor c e l l , u 2 _x X u 2„ : I i f {u^+u.,,, ) > °" i f (u,v/i+u:.vJ < 0. H i r t (1968), as well as Roche (1972), d i s c u s s e s i m p l i c a t i o n s of these forms i n d e t a i l . 57 of the s o l u t i o n . In p r a c t i c e , s h o r t t e s t runs v a r y i n g C can be made, and an optimum value s e t t l e d on; I use C = 1 . 5 . At f i r s t s i g h t , one might t h i n k t h i s unnecessary, s i n c e an e x p l i c i t scheme would not need t h i s approximation; however f o r r a p i d l y changing systems t h i s i m p l i c i t method should be/more a c c u r a t e . / / The i n t e r n a l energy changes, a p p a r e n t l y ignored' i n ( 4 4 ) , can be t r e a t e d e x p l i c i t l y l a t e r . Combining ( 4 1 ) , ( 4 2 ) , ( 4 3 ) and ( 4 4 ) , we d e r i v e an equation which can be s o l v e d immediately f o r a l l i n t e r i o r p ; : P, ( V c f + ©_<t>§r2 ( r . ^ + r . ^ j / r 2 ) = . ( 4 5 ) where the v a r i o u s source and c o n v e c t i v e terms are c o l l e c t e d as + © Sj 2 (C^/r? - H ^ / r f ) ( 4 6 ) + St 2 6d-6) ( ( P £ , -P* ) =i?.,/rf - ( P J - P - H ) r r f . , y r f ) Thus, ( 4 5 ) p r o v i d e s a system of eq u a t i o n s , i n number two l e s s than the number of s p a t i a l g r i d c e l l s , N. I f the boundary c o n d i t i o n s are such t h a t p, and p^ are known, o r can be expressed i n terms of i n n e r p« 's, the system can be s o l v e d d i r e c t l y ; otherwise a r e l a x a t i o n procedure i s necess a r y . Once 58 a l l p ; are known, the advanced d e n s i t y can be found d i r e c t l y from equation (44), A * I = ( P j - p f j / c - * (4 7) The nature of t h i s s o l u t i o n i s i n d i c a t e d from i n s p e c t i o n of (45) and (46) . The dominant term i n G|\ s i n c e "V i s s m a l l or zero i n / most c a s e s , i s the c o n v e c t i v e term, V«(^u), which i s f i r s t order i n ot. Then, e q u a t i o n (47) i n d i c a t e s that the d e n s i t y change i s n e a r l y independent of C, namely, ^ * ^ + G* » ^ - \7>(pu)* i± (47a) The next step i n the s o l u t i o n i s to s o l v e d i r e c t l y f o r the advanced v e l o c i t y . One c o u l d use equation (41) as w r i t t e n , s i n c e pT and are computed a l r e a d y ; o r , as was found necessary i n t e s t i n g r a r e f a c t i o n c ases, (4 1) can be r e w r i t t e n i n ncn-conser v a t i v e form 1 and then used t o f i n d (u^+l): u** l-u*, = £ (r. „ ) (1-x.\ ) + (r- „ ) x.A„ - u. . u. . -u> - (q. , -q . ) (48) - % - JL [^(Prn - P . - ) + n-<t?) ( P ^ - P ? ) 3 At t h i s p o i n t the i o n i z a t i o n e q u a t i o n , (38), i s s o l v e d as a simple q u a d r a t i c equation f o r x*^ , using o**1 and T* . F i n a l l y , 1 Of course the non — c o n s e r v a t i v e form does not e x p l i c i t l y conserve momentum or energy when the f i n i t e d i f f e r e n c e equations are summed over the mesh; the a d i a b a t i c t e s t s d i s c u s s e d l a t e r do i n d i c a t e that the energy i s nonetheless w e l l conserved. A l s o , Gary (1964) d i s c u s s e s t e s t s of the two FD forms, and f i n d s the non-conservative form g i v e s more a c c u r a t e r e s u l t s i n the case of a plane r a r e f a c t i o n wave. 59 the advanced energy can be s o l v e d f o r e x p l i c i t l y . Again, r a t h e r than use the c o n s e r v a t i v e form, (37), we s u b s t i t u t e d the form e.n+i _ e.*- = 2S.«V"» St/ (p.** 4 ^ ) / (49a) where / s.»*i = ^ > u / + i { e ^ _ e ^ ) / J r - 2u.«* MP-, +/q,*+ 1 )/tfr; + ( u . ^ - u . ^ ) (p. 4 q ^ ) / ^ r (49b) + <rT ) ( 1 " V + 1) ft*- 1 + ^ e <r? > X ; W 1 f r l - A (T ,(X;^ H x r ) / 2 ) <X.*-M ( ^ ) 2 The temperature, T.*11-1 , i s determined from the advanced energy and i o n i z a t i o n from equation (33a), and s t o r e d i n a se p a r a t e a r r a y . With t h i s s t e p , a l l dependent v a r i a b l e s have been advanced through one c y c l e . I t e r a t i v e Procedures The c o n t i n u i t y e q u a t i o n i s nowhere s p e c i f i c a l l y used i n the s o l u t i o n a l g o r i t h m , although i t i s of course i m p l i c i t i n equation (45). I t was decided, t h e r e f o r e , to check how w e l l equation (43) i s s a t i s f i e d at each s t e p of the program. A f t e r the advanced d e n s i t y and v e l o c i t y are c a l c u l a t e d , the e r r o r i n the c o n t i n u i t y e q uation (the d i f f e r e n c e between the r i g h t hand s i d e of (43) and the l e f t hand side) d e f i n e s a f u n c t i o n , g^, at each p o i n t . I f 2T 19:1/^; > 0.01 (found to be a workable t o l e r a n c e ) , new v a l u e s of p; are c a l c u l a t e d from g ; + % lH, / ^  ty= o (50) where the matrix c o e f f i c i e n t s are e v a l u a t e d from a n a l y t i c d i f f e r e n t i a t i o n of (43) with r e s p e c t to a l l p. • s. &p i s he l d to 60 zero a t the boundaries, and equation (50) can be s o l v e d f o r the i n t e r i o r d e n s i t y c o r r e c t i o n terms. These new r e s u l t s are used to r e c a l c u l a t e the advanced v e l o c i t y . I t t u r n s out t h a t t h i s procedure i s r a r e l y needed i n p r a c t i c e , as the e r r o r i n the c o n t i n u i t y e quation i s u s u a l l y q u i t e s m a l l . The i o n i z a t i o n c a l c u l a t i o n (equation 38) i s based on the p r e v i o u s - c y c l e temperatures. However i f the temperatures are f l u c t u a t i n g — and the e x p o n e n t i a l terms i n the c o l l i s i o n a l i o n i z a t i o n c o e f f i c i e n t are very s e n s i t i v e — the s o l u t i o n x.*'*'1 (T.rt) may be i n c o n s i s t e n t with T**1. I f the average d e v i a t i o n , | x^ 1 (T^'1) - x.Atl (I*) | , i s above one percent, xl1*1 i s r e c a l c u l a t e d as an average of the two v a l u e s , and the energy c a l c u l a t i o n i s redone. T h i s u s u a l l y converges i n one or two i t e r a t i o n s ; i t i s necessary once every f i f t y c y c l e s or so. While the value of ^>>+1 i s not very s e n s i t i v e to c. w f o r any one time step, the value of p"; i s s e n s i t i v e to i t ; from e q u a t i o n (47) , p- K p* + C;A ^ (pU).*£t T e s t s of the program i n d i c a t e d t hat a s l i g h t change i n c- would produce a l a r g e change i n ~p- , and a change i n the o p p o s i t e sense i n e.** 1-e ; v V, from the -Vp term i n (49) . T h i s l e a d s to a r a p i d non—convergence of the equation of s t a t e , s t r i k i n g l y apparent a f t e r one or two hundred c y c l e s . Test runs with a range of the c o e f f i c i e n t C i n d i c a t e d the range of C where the e r r o r s Ap; = Pr - kT.vl"1 (1 + x^vl) /m were minimized, namely f o r C £ ^ to 1.5Y- These t e s t s i n d i c a t e d a s t r o n g s e n s i t i v i t y to C. Rather than attempt t o f i n d the best 61 C by t r i a l and e r r o r f o r each epoch of each model, an i t e r a t i o n s t e p based on the s i z e of ^p was d e v i s e d . Performing t h i s i t e r a t i o n at each computational c y c l e which d i s p l a y s a ^ p l a r g e r than some t o l e r a n c e w i l l have the same e f f e c t as o p t i m i z i n g C. Bather than a d i r e c t s o l u t i o n f o r the optimum c/", an i t e r a t i v e s t e p based on the s i z e of Ap should e f f e c t t h i s o p t i m i z a t i o n . The technique chosen to improve the p; s o l u t i o n d e r i v e s a p( from the equation p'. - ^ i k T ^ ( 1 + x ^ l ) / i = 0 (51) and a new p^  which i s the average of these two v a l u e s , p"* and P;' , i s used to repeat the c a l c u l a t i o n s of equations (48) and (49), to d e r i v e new v e l o c i t i e s and i n t e r n a l e n e r g i e s . The d e n s i t i e s are l e f t unchanged. The c r i t e r i o n i n use i s t h a t <Ap/p>, the e r r o r averaged over the mesh, be l e s s than 0.01; t h i s proved s a t i s f a c t o r y i n the t e s t s . T h i s i t e r a t i o n i s needed once i n about twenty computing c y c l e s . T h i s i t e r a t i o n a l s o a l l o w s f o r the e f f e c t of the change i n i n t e r n a l energy on the pressure, as mentioned f o l l o w i n g e q u a t i o n (44). A s i m i l a r i t e r a t i o n scheme, a l s o based on the energy equation, i s mentioned by Colgate (1964). Marker P a r t i c l e s Throughout the c a l c u l a t i o n , a s e t of L a g r a n g i a n — l i k e massless "marker" p o i n t s are f o l l o w e d along t h e i r paths through the f l u i d . At the s t a r t of the program they are d i s t r i b u t e d evenly i n a l l c e l l s , g e n e r a l l y with two per c e l l at an even h a l f g r i d s p a c i n g . A f t e r each c y c l e , each marker i s moved by an 62 amount c o r r e s p o n d i n g to the p r e v i o u s v e l o c i t y f i e l d at i t s \ p o s i t i o n m u l t i p l i e d by the time s t e p . The markers do not a f f e c t the flow of the gas, but simply r e c o r d i t s c o n f i g u r a t i o n , p r o v i d i n g f o r an obvious v i s u a l r e c o r d of the s o l u t i o n . They / have been found very u s e f u l i n d e s c r i b i n g f r e e s u r f a c e boundaries — as d i s c u s s e d below — and i n g e n e r a l two-dimensional problems. Welch et a l . , (1966) g i v e a f u l l d e s c r i p t i o n of the method, dubbed by them the MAC method ("Marker And C e l l " ) . Boundary C o n d i t i o n s The pressure and d e n s i t y at the l a s t p o i n t of the mesh are set to zero. There may be an e x t r a - n u c l e a r gas d e n s i t y i n a re"al galaxy, but t h i s i s probably very s m a l l i n comparison to the core d e n s i t i e s p o s t u l a t e d , at l e a s t u n t i l the core has expanded to many times i t s o r i g i n a l s i z e . As pointed out i n Chapter I I I , the outer.boundary does not extend f a r enough out to i n c l u d e a l l the mass necessary f o r the OER. The r a d i u s of the i n n e r edge i s always g r e a t e r than zero, f o r both p h y s i c a l and computational reasons. ( P h y s i c a l because the energy source has a f i n i t e s i z e ; computational because of the divergence as r approaches zero.) The v e l o c i t y i s s e t to zero at the i n n e r surface.. T h i s i s a more debatable c o n d i t i o n . (Free flow away from the i n n e r s u r f a c e i s another p o s s i b i l i t y . ) The z e r o v e l o c i t y c o n d i t i o n a l l o w s c l o s e comparison with known a d i a b a t i c s o l u t i o n s , as shown i n Appendix I I . The pressure and d e n s i t y values a t the i n n e r s u r f a c e are e x t r a p o l a t e d from i n t e r i o r g r i d 63 p o i n t s . Since the v e l o c i t y i s d e f i n e d between the c e l l s of the mesh, e x t r a v a l u e s are needed o u t s i d e the f i r s t and l a s t p o i n t s , and are found by l i n e a r e x t r a p o l a t i o n . With the pressure at the i n n e r s u r f a c e d e f i n e d i n t h i s way, the system of e q u a t i o n s (45) f o r p. i s p r o p e r l y s p e c i f i e d , and can be s o l v e d d i r e c t l y . The q u e s t i o n of what boundary c o n d i t i o n to s e t f o r the energy i s not p h y s i c a l l y obvious. Reasonable r e s u l t s seem to be obtained by using a z e r o d e r i v a t i v e at both the i n n e r and outer edges (s i n c e equation 46 holds only f o r i n t e r i o r p o i n t s ) . Tests with the energy held constant at the outer boundary, co r r e s p o n d i n g to an ambient, c o o l e x t r a - n u c l e a r medium, seem t o i n t r o d u c e i n s t a b i l i t i e s . S i n c e x i s undefined when o = 0, at the outer edge x i s s e t egual to i t s value at the l a s t p o i n t c a l c u l a t e d . Otherwise i t i s d e f i n e d everywhere. I t i s necessary to keep t r a c k of the p o s i t i o n of the f r e e (p = 0) s u r f a c e as the gas expands or c o n t r a c t s . S i n c e the model i s s p h e r i c a l l y symmetric, we are spared the problems of the s u r f a c e c o n f i g u r a t i o n which make.most MAC c a l c u l a t i o n s awkward. (See, f o r i n s t a n c e , H i r t and Shannon, 1968, or Alsop and Goodman, 1972.) The p o s i t i o n of the outermost marker p a r t i c l e i s d e f i n e d as the s u r f a c e . In each c y c l e , as the markers are advanced i n p o s i t i o n , i t i s easy to check whether the outer boundary has passed the l a s t box of the mesh. I f i t has, another p o i n t i s added to the mesh, and new values of a l l the dependent parameters are c a l c u l a t e d at the outer two p o i n t s . The best method f o r e x t e n s i o n was determined by t r i a l and e r r o r . 64 s e a r c h i n g f o r t h a t one which would l e a s t d i s t u r b the nature of the flow. The s t e p s i n the e x t e n s i o n r o u t i n e a r e : 1. I n c r e a s e the space g r i d by one p o i n t , from i = N to i = N+1: ^ 2. Set the d e n s i t y ^ ( = 0; and interpolat/e ^ = 3/8 p^ ,, (the r a t i o 3/8 was found to i n t r o d u c e minimum d i s t u r b a n c e s ) . 3. Set u = u, ; q u a d r a t i c a l l y i n t e r p o l a t e using u ,, u,., and u. c. to f i n d u „. L i n e a r l y e x t r a p o l a t e , as u s u a l , to u ^ . 4. Set e w + ( = ; l i n e a r l y i n t e r p o l a t e to f i n d e y ( . 5. C a l c u l a t e x as u s u a l . 6. Solve f o r p w ~ ^kT,(1+x)/m. T h i s method f o r e x t e n s i o n c o u l d a l s o be formulated f o r the case of i n f a l l , where the outer g r i d p o i n t would be dropped from the c a l c u l a t i o n as the l a s t marker l e f t i t , and the outer few p o i n t s r e - a d j u s t e d . A r t i f i c i a l D i f f u s i o n ; S t a b i l i t y The a d d i t i o n of an " a r t i f i c i a l v i s c o s i t y " term (suggested o r i g i n a l l y by von Neumann and Richtmeyer, 1950) to the FD equations, i n a form p r o p o r t i o n a l t o the v e l o c i t y g r a d i e n t , i s standard- p r a c t i c e . The form used here i s given i n equation (40); the e x p l i c i t d e n s i t y dependence allows a l e s s s t r i c t s t a b i l i t y c r i t e r i o n i n the very r a r e f i e d outer r e g i o n s . T h i s v i s c o s i t y resembles an a r t i f i c i a l momentum d i f f u s i o n . In t h i s computation, as i n the ICE method, an a r t i f i c i a l mass d i f f u s i o n 65 term i s a l s o b u i l t i n t o the system (see e quation 35). Both y and )\ can be s p e c i f i e d to any value with dimensions of l e n y t h ; u s u a l l y i n w e l l behaved c o n d i t i o n s they are s e t to z e r o , to a v o i d p o s s i b l e s p u r i o u s e f f e c t s . / H i r t (1968) g i v e s an i n t e r e s t i n g d i s c u s s i o n 7 of the e f f e c t only q u a s i - l i n e a r , the usual Fourier-component a n a l y s e s are of no use. However, H i r t suggests expanding each term i n the FDE i n a T a y l o r s e r i e s about i t s c e n t r a l mesh p o i n t , and r e t a i n i n g the lowest order even and odd d e r i v a t i v e s i n time and space, i n the f i n a l e x p r e s s i o n ; t h i s w i l l i n d i c a t e the DE a c t u a l l y r e p resented by the FDE. I t may not be the same as the s t a r t i n g DE; i f not, e r r o r s may be i n t r o d u c e d i n the s o l u t i o n . If i t t u r n s out that the d i f f u s i v e terms, ^p/^r; 2 andV"u/^r 2, appear and have negative c o e f f i c i e n t s , i t i s easy to v e r i f y t h a t s p a t i a l l y o s c i l l a t i n g components of the s o l u t i o n w i l l grow e x p o n e n t i a l l y i n time; i f these c o e f f i c i e n t s are p o s i t i v e these components w i l l d i e out, l e a v i n g a s t a b l e s o l u t i o n . The a d d i t i o n of a p o s i t i v e T ^ / ^ r " or /X^d/^ can t h e r e f o r e s t a b i l i z e an o s c i l l a t i n g s o l u t i o n . Thus, equation (43) can be expressed. of these terms on a FD equation. S i n c e the f l u i d e q u a t i o n s are - ( 1 *u+ tl & r 2 - Q./2 ( o ^ Lu+uVf ) £ t + Using (35) and (36) terms, t h i s becomes V + j l j ( r 2 p u ) =-Q (52a) 66 with V = T (1-(D-1) u/r)&t - '(2Y/r)St (52b) - (u/r+bu/^r) Sr 2/2 + (§-1) (u 2+c 2) ht from which the c o n s t r a i n t on T can be d e r i v e d , using V> 0: T > ( i / r + W ^ r ) or 2/2-K 1-6/2) (u 2+c 2) St (53) ^ (H(1-0) (u/r) St) Operating i n the same way with equations (41) and (42) g i v e s the c o n d i t i o n on X , , x (2u/r+Wdr) &r 2/2+ (3u24- (1-Q/2) c 2 ) £t (54) A ^ ( 1 + ( (1/0)^ 2^/«^r 2+(1/2r) «^/^r) ^ r 2 ) The a r t i f i c i a l v i s c o s i t y has long been found u s e f u l i n c a l c u l a t i n g shock t r a n s i t i o n s , where |6u/3r| goes very l a r g e and the n e a r - d i s c o n t i n u i t i e s i n t r o d u c e unstable p e r t u r b a t i o n s . The mass d i f f u s i o n has been found to s t a b i l i z e cases with very steep d e n s i t y g r a d i e n t s , as w i l l be d i s c u s s e d below. Eguations (53) arid (54) are used to estimate the values of the c o e f f i c i e n t s at each time s t e p . T y p i c a l l y , 'V and X have a value of a few times the g r i d s i z e o"r, f o r steep g r a d i e n t s . A value of T or X which i s too l a r g e c o u l d a l s o i n t r o d u c e i n s t a b i l i t i e s . S chulz (1964) has pointed out that s c a l a r forms of these d i f f u s i o n c o e f f i c i e n t s can cause s p u r i o u s momentum or mass t r a n s f e r . T h i s has not been made evi d e n t i n t h e s e c a l c u l a t i o n s . Although no s p e c i f i c t e s t s f o r t h i s e f f e c t have been made, the success of the numerical t e s t s i n Appendix II suggests that t h i s e f f e c t i s not important here. Although the Courart r e s t r i c t i o n on the s i z e of the time s t e p , <5t(|uj+ c s ) < £r, does not o b v i o u s l y apply t o a time centered FD f o r m u l a t i o n , a s i m i l a r c o n d i t i o n has been chosen to pick the s i z e of each time s t e p , which w i l l t h e r e f o r e decrease 67 as the gas a c c e l e r a t e s , Harlow and Amsden (1971) suggest such a c h o i c e t o improve the n u m e r i c a l accuracy- A second St i s e v a l u a t e d from e/|de/dt|, which can be s m a l l i f the gas i s not at i t s e q u i l i b r i u m temperature; t h i s w i l l happen i f the i n i t i a l c o n d i t i o n s have not yet s e t t l e d themselves, or i f an o s c i l l a t i o n about the e q u i l i b r i u m temperature occurs i n r e g i o n s where d A / d T i s l a r g e . The s m a l l e r of these two St's ( m u l t i p l i e d by 0.1) i s used to c a l c u l a t e the next c y c l e . In p r o d u c t i o n runs, &t ~ 1 0 _ s t o 1 0 - 6 i s u s u a l l y found. Computing Cy c l e The procedure f o r a computational c y c l e , i n c l u d i n g p o s s i b l e i t e r a t i o n s , i s summarized i n T a b l e VI. A l i s t i n g of the program i s g iven i n Appendix I I I . Numerical T e s t s The system of e q u a t i o n s (35) to.(40) can be reduced to the i s e n t r o p i c case by s e t t i n g the heat sources and c o o l i n g terms to z e r o . Then the r e s u l t s can be compared with numerical s o l u t i o n s e x i s t i n g i n the l i t e r a t u r e , and with a few known a n a l y t i c s o l u t i o n s . The most s e n s i t i v e p a r t s of an E u l e r i a n scheme are the a d v e c t i o n and pressure terms; these t e s t s w i l l i n d i c a t e t h e i r accuracy without i n v o l v i n g the h e a t i n g - f u n c t i o n e f f e c t s . A l s o i n these t e s t s the c e n t r a l mass and the i o n i z a t i o n a r e i g n o r e d , and only the i s e n t r o p i c expansion of a monatomic gas 68 10. Table VI Procedure For One Computational C y c l e 1. Determine i t and the s t a b i l i t y c o e f f i c i e n t s T and , depending on p r e v i o u s s o l u t i o n . / 2. Solve (45) v i a a r e c u r s i o n r e l a t i o n f o r (p. }. 3. Find from (47). 4. Find [uf*1} from (48) , or from the c o n s e r v a t i v e form i f d e s i r e d . N 5. Solve f o r the i o n i z a t i o n , using a steady s t a t e s o l u t i o n with the advanced d e n s i t i e s . 6. Check the equ a t i o n of c o n t i n u i t y . C o r r e c t t h e C^1} and repeat step 4. i f i t i s not well s a t i s f i e d . 7. Solve f o r the advanced energy, using (49a) or the c o n s e r v a t i v e form. 8. Check the c o n s i s t e n c y of {X**1} and {T;n*1} ; repeat step 7. using an averaged [x**1} i f needed. 9. Check t o see of the equation of s t a t e i s s a t i s f i e d t o wit h i n a s p e c i f i e d e r r o r ; i f not r e d e f i n e {p."""1} using the s t a t e equation and (p ; } and r e p e a t step 3. and st e p s f o l l o w i n g . Advance a l l marker p o s i t i o n s , and check whether the o u t e r boundary has abandoned the . l a s t box of the g r i d ; i f so, extend a l l flow v a r i a b l e s as p r e s c r i b e d above. i under i t s own pressure i s c o n s i d e r e d . The d e t a i l s of these t e s t s ace presented i n Appendix I I ; q u i t e good agreement i s achi e v e d . There are no e x i s t i n g s o l u t i o n s t o which the c a l c u l a t i o n s with the h e a t i n g , c o o l i n g and g r a v i t y i n c l u d e d can be compared. The only checks t h a t can be done are s e l f c o n s i s t e n c y (when 69 h a l v i n q o r , f o r i n s t a n c e ) and e s t i m a t e s of mass, energy and \ momentum c o n s e r v a t i o n . These t e s t s of the models c a l c u l a t e d i n the f o l l o w i n g chapter are a l s o presented i n Appendix I I . 70 CHAPTER V \ NUMERICAL CALCULATIONS T h i s numerical program was used to generate s e v e r a l n u c l e a r models r e p r e s e n t i n g a range of v a l u e s of p h y s i c a l ' parameters. / / X-ray Model One model using X-ray a c c e l e r a t i o n was c a l c u l a t e d to v e r i f y the d i s c u s s i o n i n Chapter I I . The expected behavior of the model can be i n v e s t i g a t e d i n some d e t a i l beforehand. I f the gas i s maintained i n thermal e q u i l i b r i u m determined by the balance of X-ray heating and r a d i a t i v e c o o l i n g , the temperature w i l l depend o n l y on the atomic parameters. Equating heating and c o o l i n g , ^ ( r ) <&E> (1-x) = A ( T ) ^ x 2 / m ( 5 5 ) where the n o t a t i o n i s the same as i n Chapter IV. Then s u b s t i t u t i n g from equation (32) and i g n o r i n g the s m a l l c o l l i s i o n a l term, < A E > 4 0 T - ° - 5 p x 2 = / \ ( T ) o x 2 T h i s g i v e s the temperature i m p l i c i t l y , T«.5 = < A E > < . ( 5 6 ) The s o l u t i o n shown i n F i g u r e 1, where the l i n e f (T) = <AE> =C,T-°- 5 i n t e r s e c t s the /\ (T) c u r v e , i s 30,000 K. The f (T) l i n e comes very c l o s e to the l\ (T) curve a g a i n a t T ~ 1 0 6 K. The c o o l i n g curve i s probably s e n s i t i v e enough to the gas composition and atomic parameters t h a t another i n t e r s e c t i o n i s p o s s i b l e . T h i s means equation ( 5 6 ) may have more than one s o l u t i o n . In t h i s c a l c u l a t i o n , the lowest temperature w i l l 71 / i F i g u r e 1. The c o o l i n g r a d i a t i o n from a gas of cosmic abundances i s g i v e n by / \ ( T ) n e n 4 . The i n t e r s e c t i o n of the /\ (T) curve with the dashed l i n e determines the temperature of the i o n i z e d gas heated by X-rays, which i s approximately 30,000 K i n t h i s case. The shaded areas are the t h e r m a l l y u n s t a b l e temperature r e g i o n s i n our model (as d i s c u s s e d i n Chapter V). 72 F i g u r e R a d i a t i v e C o o l i n g Curve 73 s t i l l be a t t a i n e d ; were the gas somehow heated by some t r a n s i e n t " k i c k " , the higher temperature phase c o u l d a l s o occur i n a two phase s t a t e . Equation (56) shows t h a t the e q u i l i b r i u m temperature i s otherwise independent o f r a d i u s . The i n t e r n a l energy c o r r e s p o n d i n g to t h i s temperature i s much s m a l l e r than the g r a v i t a t i o n a l energy per gram i n these models. The dp/or term (equation 36) w i l l be unimportant i n the a c c e l e r a t i o n as long as the g r a v i t a t i o n a l and pressure s c a l e s are s i m i l a r , and we can d i s c u s s the s o l u t i o n of the equations i n terras of the X-ray a c c e l e r a t i o n and g r a v i t a t i o n a l f o r c e . We w i l l use the f a c t t h a t the X-ray f o r c e per gram i n the o p t i c a l l y t h i n case i s p r o p o r t i o n a l to x 2 ^ , and .independent of ^ ,_£l(r) as long as C (T) i s s m a l l . The net f o r c e , then, which can be c a l l e d G (r) , i s given by G(r) = 6£(T)<AE> x 2 p ( r ) - G_M (57) m2 c N r 2 X ~ \ (r ) r 2 i f r i s the inn e r r a d i u s of the gas, and £l(r) < 1 i s the a t t e n u a t i o n . For a monochromatic beam i n a gas of o p t i c a l depth n (1-x) dr, £L(r) = e-^; i n our case with a power law spectrum, Q_ (r) oc exp-[ a ^  n (1-x) dr ]°• 5 , as d i s c u s s e d i n Appendix I . The s i g n of G (r) , f o r a given M, \ and r , depends on ^ ( r ) r ^ (r) and on x. The i o n i z a t i o n x can be approximated from the i o n i z a t i o n equation with C(T) = 0 i n two l i m i t s . a. - " t h i n " — x % 1 i f T°<Tl(r)/nr 2 >> 4*(T) i n which case G(r) ^ ol(T) <A5> 0 - GM (58a) m 2c \ r 2 which i s independent of ^ 9 and £l(r). b. " t h i c k " -- x X, [ \l £L (r)/oc(T) n r 2 ] ° - 5 << 1 i f 74 ^ f U r ) / n r 2 « 4*(T), g i v i n g G (r) - t c <AB>S)-(r) - _GJ1 m c r 2 r 2 (58b) which i s independent of x and P , except as they are c o n t a i n e d i n The t r a n s i t i o n from t h i n to t h i c k r e g i o n s i s r e m i n i s c e n t of c l a s s i c a l Stromgren t h e o r y , which d e a l s with a monochromatic f l u x i n a homogeneous gas. The t r a n s i t i o n from _ft(r) % 1 to $\ (r) << 1 occurs over a very narrow r e g i o n i n the Stromgren model. I f ^ (r) f a l l s o f f more s t e e p l y than r - 2 , i t i s p o s s i b l e t h a t the o p t i c a l depth never reaches u n i t y and the gas s t a y s t h i n and i o n i z e d as r co . With a power law X-ray spectrum, one would expect the t r a n s i t i o n d i s t a n c e t o be smeared out. C a l c u l a t i o n s of x (r) i n the power-law case were c a r r i e d out i n the manner of the Stroragren-radius c a l c u l a t i o n s i l l u s t r a t e d by Sobolev (1967). The i o n i z a t i o n e quation i n c l u d i n g the e x p o n e n t i a l o p a c i t y term i s converted to an o r d i n a r y d i f f e r e n t i a l equation f o r x ( r ) . T h i s equation can then be n u m e r i c a l l y i n t e g r a t e d . A l t e r n a t i v e l y , s i n c e _ f l (r) depends only on the value of x i n t e r i o r to the p o i n t r , the i o n i z a t i o n equation can be c o n s i d e r e d as a q u a d r a t i c at each p o i n t , working out from the i n s i d e p o i n t . F i g u r e 2 shows such i n t e g r a t i o n s f o r both monochromatic and power law s p e c t r a , £or ^ ( r ) h e l d c o n s t a n t and f o r ^ (r) oc r - 2 . As expected, the t r a n s i t i o n r e g i o n from t h i n to t h i c k i s broader i n the case of the power law spectrum. The case of £Y (r) oc exp (-r° * 2 5) , which a p p l i e s to a power law cosmic ray spectrum, was a l s o e v a l u a t e d , s i n c e i t w i l l be important l a t e r . The t r a n s i t i o n i n t h i s case i s very g r a d u a l . 75 F i g u r e 2 shows the t r a n s i t i o n from the f u l l y i o n i z e d gas (x=1.0) to the n e u t r a l gas (x=0.0), f o r the cases of constant and of d e c r e a s i n g d e n s i t y . The extent of the t r a n s i t i o n r e g i o n r e f l e c t s the a t t e n u a t i o n of the i o n i z i n g quanta. A monochromatic photon beam i s s h a r p l y a t t e n u a t e d , a power law photon spectrum i s l e s s s h a r p l y a t t e n u a t e d , and the cosmic ray power law spectrum produces g r a d u a l l y d e c r e a s i n g i o n i z a t i o n . 76 F i g u r e 2 I o n i z a t i o n S t r u c t u r e 77 with a very broad p a r t i a l l y i o n i z e d r e g i o n , as seen i n F i g u r e 2. The flow v e l o c i t y does not a f f e c t the i o n i z a t i o n c a l c u l a t i o n s , except as i t determines ^ ( r ) -I t i s convenient to rephrase the l i m i t s g i v e n above i n terms of the parameters a c t u a l l y used i n the program. The c r i t e r i o n f o r an o p t i c a l l y t h i n -- f u l l y i o n i z e d — gas i s , ( ZETAX \ f U r K v , 1 23 / CFLAM ~\ (59) \ 10& / r 2 ^  y' TO. 5 ^ 1027 / where ^ and r are now c o n s i d e r e d to be s c a l e d v a r i a b l e s , as i n the numerical c a l c u l a t i o n s . In these terms, the c o n d i t i o n G (r) > 0, necessary f o r an outward f o r c e on the gas, i s as f o l l o w s . a. In the t h i n case, G(r) > 0 i f AMS y 4x103 [ CFLAM \ T~°. 5 = 20 / CFLAM \ (60a) r 2 ^ ^ V 1 0 2 7 / \^  1 0 2 7 J i n the l a s t e q u a l i t y , T = 30,000 K i s s u b s t i t u t e d . b. In the t h i c k case, G(r) > 0 i f AMS < 0.9x103 ( ZETAX\ ft (r) . (60b) V \Ob / For example, using CFLAM •= 1 0 2 7 (corresponding to a d e n s i t y s c a l e of 1 0 6 / r o cm"3) the " t h i n " c o n d i t i o n i s AMS < 2 0 r 2 P £ 1. The c o n d i t i o n to maintain the gas f u l l y i o n i z e d i s : 2 £ 10-3. / ZETAX\ £1 (r) > 1 . 7 x 1 0 - 2 p r -The model c a l c u l a t i o n s show t h a t Si(r) i s i n the range 10~ 3 t o 10~ 5 f o r the values of CFLAM used here. Outward a c c e l e r a t i o n , then, r e q u i r e s AMS < 20 f o r the model below. T h i s l i m i t i s independent of the X-ray l u m i n o s i t y , as was found i n the d i s c r e t e c l o u d c a l c u l a t i o n s of Chapter I I . I f ZETAX • (r) i s below t h i s l i m i t , the gas i s t h i c k , and outward a c c e l e r a t i o n 78 r e q u i r e s AMS < 3x10 3x10~ 3 - 3. i 1 | X-ray Model I I 1 | F = 8 x 1 0 * 2 r o erqs s _ 1 | | n* = 7 x l 0 5 / r c cm- 3 | | M = 2x10 6r p I J ZETAX = 2x106 | | AM S = 1 | | CFLAM = 1 0 2 7 | | 40 p o i n t g r i d | | u 0 = 100 km s - i | | E„ = 100 eV | One numerical c a l c u l a t i o n was done with X-rays as the i o n i z i n g mechanism. A r e l a t i v e helium abundance n (He) =0.1 n (H) was assumed. The low value f o r the c e n t r a l mass was chosen to g i v e an outward a c c e l e r a t i o n without v i o l a t i n g observed f l u x e s . The i n i t i a l d e n s i t y s t a t e of the gas was a r b i t r a r i l y chosen to be ^= 1. a t the i n n e r boundary (r = r ( ) , d e c r e a s i n g g e n t l y to ^- 0.4 at r = 1.2r t. T h i s was taken a r b i t r a r i l y from * the numerical s o l u t i o n to the i s o t h e r m a l h y d r o s t a t i c e q u i l i b r i u m equation i n c l u d i n g the g r a v i t y of the gas i t s e l f and of the c e n t r a l source, (kT/m)dpydr = -G (M + m (r) ) ^ / r 2 with dm (r) = 4?rpr 2 and with the s m a l l e r c e n t r a l mass, M/HQ= r ( (pc)T(K) f o r an i n n e r boundary c o n d i t i o n . T h i s i n i t i a l s t a t e was used f o r a l l the models c a l c u l a t e d below, except Model 1, Case B. The model was run f o r 10,000 computing c y c l e s , r e p r e s e n t i n g a time span of 0.04t o = 400 r d years. In a l l - models, the time 79 s c a l e depends on the d i s t a n c e s c a l e as t 0 (years) = 1 0 * r o ( p c ) . Over t h i s time, the gas maintained a temperature averaging 29,200 K and was e n t i r e l y i o n i z e d (x = 0.998 to 1.00 everywhere). The model as i t stands i s a d e n s i t y bounded HII r e g i o n . I f the gas i s c o n s i d e r e d to extend beyond the l a s t / / computational p o i n t , the i o n i z a t i o n edge occurs f u r t h e r o u t, i f i t o c c u r s at a l l . The t o t a l i o n i z a t i o n means t h a t X-ray beam i s only very s l i g h t l y a t t e n u a t e d . T h i s s l i g h t a t t e n u a t i o n produces a decrease i n the momentum t r a n s f e r r e d to the gas with i n c r e a s i n g r a d i u s . T h i s r e s u l t s i n a maximum v e l o c i t y r e g i o n which g r a d u a l l y b u i l d s i n t o a s h e l l . T h i s i s seen i n the l i n e diagrams of d e n s i t y and v e l o c i t y i n F i g u r e s 3 and 4. Here, as i n a l l diagrams i n t h i s c h a p t e r , the u n i t s of v e l o c i t y and d e n s i t y are u Q and n^, given i n the boxes f o r each model. Time ( i n c r e a s i n g downward) I s i n u n i t s t c = r 0 / u o = 10*r o years i f r Q i s i n pc, and r a d i a l d i s t a n c e i s i n terms of the a r b i t r a r y s c a l e r 0 . By the l a s t epoch c a l c u l a t e d , the a c c e l e r a t i o n i s steady at &u/A,t ^ 10 i n s c a l e d u n i t s ; by t = 0.04t o the gas has reached v w o^ = 0.3u o = 30 km s ~ l . P r o j e c t i o n of t h i s c o n s t a n t f o r c e i n d i c a t e s a v e l o c i t y of 1-4u 0 = 140 km s - 1 when the s h e l l h i t s the outer edge of the g r i d . T h i s i s i n good agreement with the est i m a t e s of Chapter IT; using ( r - r S r ) n = 0.1 x 7 x 10 s i n egu a t i o n (7) g i v e s u (r) = 250 km s _ 1 . This i s a low v e l o c i t y i n terms of those d e s i r e d i n S e y f e r t models; and c o n s i d e r i n g the low c e n t r a l mass i n t h i s c a l c u l a t i o n , i t appears t h a t continuous flow X-ray models may net produce the observed v e l o c i t i e s . 80 F i g u r e s 3 and 4. F i v e s e l e c t e d epochs of the X-ray c a l c u l a t i o n are shown, i n order of i n c r e a s i n g time. The gas d e n s i t y i s shown i n F i g u r e 3 and the v e l o c i t y i s shown i n F i g u r e 4. Here, as i n a l l of the diagrams, the v e l o c i t y u n i t s are 100 km s e c - 1 , and the r a d i a l d i s t a n c e i s g i v e n i n a r b i t r a r y u n i t s ; n u m e r i c a l l y , the i n n e r r a d i u s i s one h a l f of the s c a l i n g d i s t a n c e r D . The d e n s i t y s c a l e i s n D = 7 x l O 5 / ^ cm - 3, where r 0 i s i n pc. The time s c a l e i s t =" 10*1^ years here and i n a l l the diagrams. Each c i r c l e r e p r e s e n t s a computed p o i n t . The growth of the s h e l l i s apparent i n the high d e n s i t y r e g i o n , which a c c e l e r a t e s and moves outward as the c a l c u l a t i o n advances. The i n c r e a s e i n v e l o c i t y a t the outer edge i n F i g u r e 4 i s due to the expansion of the f r e e s u r f a c e i n t o the vacuum. 81 82 03 83 Cosmic Ray Models Apart from the photon-or-proton c h o i c e , the most important f a c t o r i n the development of the models i s the temperature of the gas- Si n c e the gas i s i n thermal e q u i l i b r i u m most of the time (apart from the most r a p i d dynamical changes), the temperature i s determined by the r a t i o of cosmic ray d e n s i t y t o gas d e n s i t y . T h i s occurs because the balance equation f o r d i r e c t h e a t i n g by cosmic rays i s T° H ( r ) <&E> xo - A(T) x> 2 x = 0 (61) \e. r 2 m m2 * ^ l ( r ) i s now the a t t e n u a t i o n f a c t o r f o r the cosmic r a y s ; 1°^ = T C R ( r i ) r f a g a i n - D i r e c t heating -- th a t due to Coulomb i n t e r a c t i o n s — dominates the energy t r a n s f e r f o r i o n i z a t i o n s above ten per cent. T h e r e f o r e the i o n i z a t i o n h e a t i n g i s ignored here. Thus, t h i s e q uation l e a d s to \° SUr) = <AE>e< A (T) = <L l\ (T) (61a) r 2 ^ m The temperature s o l u t i o n depends only on the r a t i o ^ J ^ f L C r ) / ^>r2, which i s p r o p o r t i o n a l to the r a t i o of cosmic ray d e n s i t y to gas d e n s i t y at a r a d i u s r . Si n c e the Coulomb h e a t i n q per hydrogen i o n i z a t i o n i s <AE>^ = 5 X 1 0 — 1 0 e r g at proton e n e r g i e s of 1 MeV, t h i s equation i n terms of program v a r i a b l e s i s /\(T) = 2. 6x10~ 2 1 f ZETA\ £). (r) / 1 0 2 7 ) ' (62) r 2 D ^ 10& j \^Oru\t\ j The nature of the c o o l i n g curve, /\ (T) , i s c r i t i c a l i n determ i n i n g the i m p l i c i t s o l u t i o n , T , of t h i s e q u a t i o n . As seen i n Fiqu r e 1, the c o o l i n q curve based on cosmic abundances has two l o c a l extrema. There i s a l o c a l maximum, A j ( T ) # at an about 5 x 105 K due to l i n e s from the heavy elements, a l o c a l minimum a t 6 x 10 6 K and above t h i s temperature the curve approaches the T ° - 5 b e h a v i o r due t o hydrogen bremsstrahlung. T h e r e f o r e , a c o o l gas f o r which i t < LQ.(r) / P r 2 < £/\, w i l l heat u n t i l a steady temperature T < 5 x 105 i s reached.* I f / ^ r 2 > &A., # the temperature w i l l c o n tinue to climb u n t i l the c o r r e s p o n d i n g e q u i l i b r i u m p o i n t above 6x10 6 K i s . reached. However, t h i s p o i n t w i l l not be as high as the temperature c o r r e s p o n d i n g to the p r o j e c t i o n of A, °n the hot end of the curve. L a r k i n (1960) shows t h a t the energy t r a n s f e r to the gas i s reduced i n e f f i c i e n c y i n p r o p o r t i o n to T - 1 • 5 when T > Eme/knij, (10 7 K i f E = 1 MeV). Thus the temperature a t t a i n e d w i l l be somewhat lower. The g r a v i t a t i o n a l energy of an atom i n these models i s u s e f u l l y expressed as a " v i r i a l temperature", g i v e n by k TVur = GMm/r. For M = 108 n& t T g L r = 5 x 1 0 7 / r ( p c ) K. When T > Tv'f , the gas pressure term i n the momentum equation w i l l be l a r g e r than the g r a v i t a t i o n a l term. Since the cosmic ray pressure term i s a l s o s m a l l i n comparison, the equation then reduces to du/dt =-Vp, which i s f a m i l i a r as a d e s c r i p t i o n of a g a l a c t i c wind or an a d i a b a t i c expansion. Thus, i n t h i s "hot mode", the gas should approach a s t a t e of smooth flow where the v e l o c i t y i n c r e a s e s m o n o t o n i c a l l y with r a d i u s , and the d e n s i t y decreases m o n o t o n i c a l l y . i T h i s r e s u l t i s s e n s i t i v e to the s t a r t i n g c o n d i t i o n s . Had a very hot i n i t i a l temperature been chosen the gas would reach a steady temperature on the h o t t e r part of the /\ (T) curve, T > 6 x 10* K, given the same value of T^fr ( r ) / ^ r 2 . 85 The other case, when T < T v' < , sees the dynamic terms (cosmic ray f o r c e and g r a v i t y ) dominating the f l o w . e q u a t i o n . The e v o l u t i o n i n t h i s " c o o l mode" i s s i m i l a r to the d i s c r e t e c l o u d models d i s c u s s e d i n Chapter I I . The gas d e n s i t y and flow v e l o c i t y w i l l have narrow s p a t i a l maxima because the r a p i d decrease of _£l(r) l o c a l i z e s the momentum t r a n s f e r . T h i s and the s p h e r i c a l symmetry c r e a t e a narrow, dense s h e l l i n the con t i n u o u s flow. The snowplowing of the gas o u t s i d e the s h e l l w i l l a f f e c t the beh a v i o r of the s h e l l . T h i s " c o o l mode" seems more l i k e l y t o r e l a t e to o b s e r v a t i o n s of S e y f e r t n u c l e i . The gas i s l o c a l i z e d and f a i r l y dense, t h e r e f o r e r e l a t i v e l y c o o l and capable of producing e m i s s i o n or a b s o r p t i o n l i n e s . However e i t h e r u;ode may be s u s c e p t i b l e to fragmentation and subsequent c o o l i n g . T h i s w i l l be i n v e s t i g a t e d i n Chapter VI. I have c a r r i e d two "hot mode" models f a r enough i n the numerical c a l c u l a t i o n s to i n d i c a t e t h e i r f u t u r e e v o l u t i o n . Then s e v e r a l " c o o l " models are d i s c u s s e d , a l l evolved u n t i l t h e i r c h a r a c t e r i s t i c s h e l l v e l o c i t i e s are d e f i n e d . They are summarized i n Tab l e V I I , which l i s t s them i n order of t h e i r computational parameters, and give s the v e l o c i t i e s , i n n e r c a v i t y and s h e l l temperatures of the steady flow s t a t e a t t a i n e d i n each case. Each model i s then d i s c u s s e d i n the t e x t . 86 Hot Mode Models In the hot case, where the e q u i l i b r i u m temperature i s above the v i r i a l temperature, both A(T) and £ (T) have a n a l y t i c e x p r e s s i o n s , a l l o w i n g an e x p l i c i t s o l u t i o n f o r T. S u b s t i t u t i n g y\ (T) = 2 x 10-23 T o „ s a n (3 £ ( T) = ( T / 1 0 7 ) - i . s (chapter II) i n t o e q uation (62) , we get . T£ = 120 ( Z E T A W I P 2 7 \ (63) V 1 0 6 i^CFLAM / assuming as t y p i c a l s c a l e d values o r 2 = 0-05; where i s the temperature i n u n i t s of 10 a K. Higher Flux Hot Model | Hot Model 1 | j -I | F^ = 4x10* 5r o erg s ~ l I l n 0 = 7x10 5/r c cm- 3 j | M = 2 x 1 0 8 r o | | ZETA = 2x10* | | AMS =100 J | CFLAM = 102? | | 40 p o i n t g r i d j | u 0 = 100 km s - 1 | | E 0 = 1 MeV | L I The e q u i l i b r i u m temperature c a l c u l a t e d f o r t h i s f i r s t model i s T <^> 14. As the gas near the cosmic ray source approaches t h i s temperature, i t s t r a n s p a r e n c y to the cosmic rays i n c r e a s e s . T h i s i s i n marked c o n t r a s t to the " c o o l mode" b e h a v i o r d i s c u s s e d 87 below, where a lower temperature produces a narrow s h e l l which absorbs the e n t i r e cosmic ray f l u x . The gas everywhere i n t h i s hot mode model heats r a p i d l y toward the high e q u i l i b r i u m temperature. The c a l c u l a t i o n was stopped b e f o r e t h i s e q u i l i b r i u m was a t t a i n e d ; as shown i n F i g u r e 5, the v e l o c i t y and d e n s i t y curves had approached a steady form by t = 0.0185t o = 185r c y e a r s , and the c a l c u l a t i o n s were stopped then. F i g u r e 5 shows the d e n s i t y g r a d i e n t of the gas d e c r e a s i n g as the outer edge of the gas expands. The v e l o c i t y approaches the l i n e a r behavior with r a d i u s c h a r a c t e r i s t i c of the i s e n t r o p i c expansions c o n s i d e r e d i n Appendix I I . T h i s would be expected, s i n c e the thermal energy per gram i s much l a r g e r than the cosmic ray and g r a v i t a t i o n a l f o r c e s . The s l o p e of the v e l o c i t y curve v (r) decreases with time once i t i s e s t a b l i s h e d as l i n e a r , while the i n d i v i d u a l marker p a r t i c l e s are a c c e l e r a t e d outward. The p o s i t i o n of the outer edge 1 i n c r e a s e s with time f a s t e r than l i n e a r l y , which i s c o n s i s t e n t with the a c c e l e r a t i o n of each marker. (The high expansion v e l o c i t y made i t necessary to rezone, by d o u b l i n g &r, every so o f t e n , to save computer time.) The outer edge had reached a v e l o c i t y of 100u o = 10,000 km s - 1 when the run was t e r m i n a t e d , although most of the gas i s 1 The l a s t but one marker is. taken as r e p r e s e n t a t i v e of the outer edge. The a r t i f i r a l d e n s i t y g r a d i e n t imposed by the c u t o f f to zero pressure a t a low number of g r i d p o i n t s imposes a r a p i d expansion on the outer l a y e r of gas. In these hot mode c a l c u l a t i o n s , then, the l a s t marker i s h a l f again as f a r out as the l a s t - b u t "ORG iucl r k e r ; t h i s produces an extended, extremely low d e n s i t y outer envelope, which i s not p h y s i c a l l y important. 88 d i s t r i b u t e d r a t h e r e v e n l y between t h i s v e l o c i t y and the zero v e l o c i t y of the i n n e r edge. The high v e l o c i t y gas w i l l probably not be observed, t h e r e f o r e . The outer edge v e l o c i t y i s near the upper l i m i t imposed by the f i n i t e v e l o c i t y of the s o f t cosmic r a y s . Lower Flux Hot Model i 1 | Hot Model 2 I ^ ^ | F^ = 4 x 10 4 3 r 0 ergs s ~ 1 I | n G = 7x10*/r o c m - 3 \ I M = 2x10«r o | | ZETA = 2x10* | *"' | AMS = 100 | I CFLAM = 1 0 2 6 j | 40 p o i n t g r i d I | u 0 = 100 km s - i | | E„ = 1 MeV j I . I A lower f l u x model was a l s o c a l c u l a t e d . The e q u i l i b r i u m temperature i n t h i s case i s about 4 x 10 8 K. The behavior cf the s o l u t i o n i s very s i m i l a r t o the p r e v i o u s model, so no separate diagrams need to be shown. The lower temperature had produced an outer edqe v e l o c i t y of 30u o = 3000 km s - 1 when the c a l c u l a t i o n was stopped at t = 0.022t o = 220r c years. Aqain, the d e n s i t y d i s t r i b u t i o n of the gas i s l e v e l l i n g out, so most of the gas i s at lower v e l o c i t i e s . 89 F i g u r e 5. The e v o l u t i o n of the h i g h e r f l u x hot model i s shown- V e l o c i t y i s i n u n i t s of 100 km s e c - 1 , d e n s i t y i n terms of n 0= 7 x 10 5/r o cm - 3- Time i s i n u n i t s of 1 0 4 r o years- The d e n s i t y s t a r t e d from a c e n t r a l l y condensed s t a t e and i s approaching a more u n i f o r m l y spread out c o n d i t i o n . The v e l o c i t y i n c r e a s e s approximately l i n e a r l y with r a d i u s , and the r a t e of i n c r e a s e of v e l o c i t y with r a d i u s decreases with time as the outer edge of the gas expands. 91 D i s c u s s i o n Each of these models should continue t o ev o l v e i n the manner i n d i c a t e d i n F i g u r e 5, s i m i l a r to the standard i s e n t r o p i c expansion s o l u t i o n . T h i s e n t i r e r e g i o n o f gas i n c l u d e d i n the c a l c u l a t i o n s i s f u l l y i o n i z e d . I f the n u c l e a r gas i s assumed to extend f a r past the l a s t computational poin t (see Chapter III) there may be a t r a n s i t i o n t o a n e u t r a l r e g i o n a t some l a r g e r r a d i u s , or the i o n i z e d r e g i o n may be d e n s i t y - l i m i t e d , depending on the d e n s i t y s t r u c t u r e of the gas. I f the r e g i o n i s i o n i z a t i o n - l i m i t e d , then these "hot mode" models are extreme cases of the s h e l l models. The most i n t e r e s t i n g e f f e c t s w i l l occur at the i o n i z a t i o n edge. These e f f e c t s are i l l u s t r a t e d i n the Cool Models given l a t e r . I f the gas i s f u l l y i o n i z e d throughout, the i s e n t r o p i c expansion model d e s c r i b e s the e v o l u t i o n f u l l y . 92 Cool Mode Models The seven c o o l mode models d i s c u s s e d here r e p r e s e n t a range of f o u r orders of magnitude i n cosmic ray f l u x , from 4 x 1 0 * 3 r o to 4x10* 7r0 ergs s - 1 . T h i s f l u x i s converted to the t o t a l energy c o n t e n t r e q u i r e d by e s t i m a t i n g the escape l i f e t i m e . The lowest f l u x models, then, are c o n s i s t e n t with the lower e n e r g i e s i n Table I I , which apply t o S e y f e r t n u c l e i . The h i g h e s t f l u x models are c o n s i s t e n t with the higher energy content i n Table I I , which may apply to quasars. | Cool Model 1 1 \ — ^ | F c c = 4x10* 3r c. ergs s _ I I \ n0~ - 7x10*/r o c m - 3 I | M = 2x10 ar c | | ZETA = 2x10* | | AMS = 100. | | CFLAM = 1 0 2 6 | J 40 and 60 p o i n t g r i d s { | u c = 100 km s - i I | E~ = 1 MeV | i : i Model 1 was one of the f i r s t models run. I t r e p r e s e n t s the lowest v a l u e s f o r the cosmic ray f l u x and f o r the gas d e n s i t y t h a t are being c o n s i d e r e d f o r the nu c l e a r models. Because the i n t e g r a t i o n time step i s l a r g e r , the c a l c u l a t i o n s are more c o n s e r v a t i v e of computer time, and t h e r e f o r e these models were run a t l e n g t h to i n v e s t i g a t e the nature of the gas flow and s h e l l f o r m a t i o n . 93 Table VII Cool Mode Models Low F l u x l i c a v i t x l TJshel.ll, y j s h e l l ) 1. ZETA = 2x10* Case A 10 7 few x 10* 700 NP=40 and NP=60 ZETA = 2x10* Case B 10? few x 10* 550 ZETA =0.0 e v o l u t i o n ZETA = 2x10* h a l f g r i d t e s t I n t e r m e d i a t e Flux 2. ZETA = 2x105 3. ZETA = 2x106 3x107 108 5x10* 3-6x10* 1000 2000 High F l u x 4. ZETA = 2x10? 5. ZETA = 2x10? A MS = 1000 6. ' ZETA = 2x107 AMS = 1000 0.1 MeV 7. ZETA = 2x107 AMS = 1000 10 MeV 108 10« 108 108 5x10* 5x10* 5x10*-105 5x10*-105 3500 + 3400 + 2000 + 8000 + — a l l models assume E. 1 MeV and AMS = 100 u n l e s s s t a t e d , — p l u s s i g n by v ( s h e l l ) i n d i c a t e s top v e l o c i t y not y e t reached, probably another 20 per cent or so g a i n , based on lower f l u x c a l c u l a t i o n s . -- v e l o c i t i e s i n km s - 1 ; temperatures i n K. 94 The i n i t i a l s t a t e of the gas, as i n a l l the models, i s a g e n t l y d e c r e a s i n g d e n s i t y f u n c t i o n as g i v e n above,.with c o n s t a n t temperature (9000 K) and zero v e l o c i t y . Almost a l l of the cosmic ray f l u x i s absorbed over the f i r s t few p o i n t s . T h i s i n n e r gas i s heated and a c c e l e r a t e d outward. The o u t e r gas i s s l i g h t l y heated by the high energy end of the spectrum but not a c c e l e r a t e d enough to stop the i n f a . l l . (The i n i t i a l c o n d i t i o n does not s a t i s f y h y d r o s t a t i c e q u i l i b r i u m . ) T h i s behavior i s i l l u s t r a t e d i n F i g u r e s 6 and 7. The gas being a c c e l e r a t e d tends to p i l e up on the gas j u s t ahead of i t , forming a s h e l l . A low d e n s i t y r e g i o n i s c l e a r e d i n s i d e the s h e l l . The low d e n s i t y of t h i s i n n e r c a v i t y r e s u l t s i n the gas there being heated to s e v e r a l times 10 7 K. The denser s h e l l heats to a few times 10* K and maintains t h i s temperature. The outer r e g i o n s reach 10 4 K. The temperature s t r u c t u r e of the gas i s p l o t t e d i n F i g u r e 8a f o r s e v e r a l epochs. The temperature s t r u c t u r e determines the o p a c i t y of the gas, which i s p l o t t e d i n F i g u r e 8b f o r s e v e r a l epochs of t h i s model. The q u a n t i t y p l o t t e d i s f SI , where Si i s the a t t e n u a t i o n due to the gas i n t e r i o r to the p o i n t i n q u e s t i o n , and f < 1 i s a f a c t o r t o allow f o r r e d u c t i o n i n the c r o s s s e c t i o n s a t high thermal or flow v e l o c i t i e s . The a n a l y t i c form of f i s g i v e n i n Chapter I I I ("Heating Mechanisms"). The high temperature of the i n n e r c a v i t y i n t h i s model reduces the energy a b s o r p t i o n . The c o o l e r s h e l l then absorbs almost a l l of the remaining p a r t i c l e energy, l e a v i n g only a f r a c t i o n a l amount to a f f e c t the gas ahead of the s h e l l . The s h e l l , once d e f i n i t e l y formed, reaches a c o n s t a n t 95 v e l o c i t y , about 7 u 6 , and maintains t h i s v e l o c i t y with a s l i g h t d e c e l e r a t i o n . As d i s c u s s e d below, t h i s i s due to snowplowing of the outer gas, p r o v i d i n g an inward f o r c e which reduces the net cosmic ray p r e s s u r e . T h i s i s apparent i n F i g u r e 11, which p l o t s the p o i n t of maximum d e n s i t y as a f u n c t i o n of time, and a l s o i n F i g u r e s 9 and 10, contour p l o t s d i s p l a y i n g the same data as i n F i g u r e s 6 and 7. The " i s l a n d " e f f e c t i n F i g u r e 10 i s due t o the d i s c r e t e r e p r e s e n t a t i o n , and r e f l e c t s the movement of the s h e l l between g r i d p o i n t s . The p o i n t of maximum d e n s i t y , 2 ( t ) , moves outward at a c o n s t a n t r a t e , which i s c o n s i s t e n t with the p o i n t of maximum v e l o c i t y c a l c u l a t e d . The same d i s c r e t i z a t i o n e f f e c t i s a l s o seen i n the p l o t s of thermal and (thermal+kinetic) e n e r g i e s with time, i n F i g u r e 11. The o s c i l l a t i o n s i n energy r e f l e c t the o s c i l l a t i o n s i n d e n s i t y (due to the thermal balance) as the s h e l l moves between g r i d p o i n t s . (This i s best i l l u s t r a t e d i n the a l t e r n a t e f l a t t e n i n g and " s p i k i n g " of the s h e l l i n F i g u r e s 6 and 12.) The n e a r l y e q u a l average s l o p e s of these two l i n e s , once the s h e l l i s e s t a b l i s h e d , r e f l e c t the constant v i I t u . The e v o l u t i o n of t h i s model i s soon contaminated by s p u r i o u s e f f e c t s due to the a r b i t r a r y i n i t i a l c o n d i t i o n s . These e f f e c t s are probably not important f o r models of g a l a c t i c n u c l e i , because the i n i t i a l c o n d i t i o n s were chosen p a r t l y f o r ease of computation. The e f f e c t s d i s c u s s e d below were i n v e s t i g a t e d at l e n g t h to gain an understanding of the behavior and accuracy of the program. The outer boundary c o n d i t i o n i s zero d e n s i t y . T r u n c a t i n g the i n i t i a l d e n s i t y curve to zero at the l a s t g r i d p o i n t r e s u l t s 96 i n a sharp g r a d i e n t at the edge ( f o r i n s t a n c e , the f i r s t curve i n F i g u r e 6 ) . T h i s s t e e p g r a d i e n t w i l l expand i n t o the vacuum, de v e l o p i n g a smoother, low d e n s i t y outer halo- T h i s t h i n gas w i l l reach h i g h e r temperatures, as d i s c u s s e d above. The r e g i o n / / j u s t i n t e r i o r to i t i s s t i l l c o l l a p s i n g inward, /as i s a l l the gas between there and the s h e l l , but the high e r temperature means the outward momentum t r a n s f e r from the remnants of the cosmic ray f l u x i s l e s s . Thus, t h i s gas develops a f a s t e r i n f a l l speed, and a secondary " s h e l l " b u i l d s up on the outer edge and moves inward. T h i s becomes apparent a t t ~ 0.Q08t o i n the contour diagrams. Even b e f o r e t h i s s h e l l meets the outgoing one, s m a l l a c o u s t i c waves moving inward appear behind the s h e l l . They appear as p o s i t i v a - s l o p e p a t t e r n s of the contour p l o t s , moving at the sound speed i n the hot gas, and can be seen i n the l i n e p l o t f o r v e l o c i t y , F i g u r e 7. Lasker (1966) f i n d s s i m i l a r waves generated i n h i s hydrodynamic models of young HII r e g i o n shock f r o n t s . The waves i n our case are probably due to the s l i g h t d i s t u r b a n c e of the s u p e r s o n i c s h e l l by the i n f a l l i n g matter (although numerical t r u n c a t i o n may be a t f a u l t , of c o u r s e ) . The ne g a t i v e momentum t r a n s f e r from t h i s i n f a l l probably a l s o accounts f o r the s m a l l d e c e l e r a t i o n apparent i n the R W O l ¥ (t) p l o t . When the i n f a l l i n g s h e l l f i n a l l y h i t s the outgoing s h e l l , the outward v e l o c i t y drops s h a r p l y . The t r a c k i n the d e n s i t y contour diagram i s n e a r l y v e r t i c a l at t h i s p o i n t . Much s t r o n g e r a c o u s t i c waves are generated now. (This a l s o appears at t = 0.014-0.016 t e i n the l i n e p l o t s . ) Once the s h e l l has absorbed t h i s momentum, under the c o n t i n u i n g cosmic ray f o r c e i t 97 again a c c e l e r a t e s , but by t ~ 0.016t o i t has-reached the "edge" of the gas, and the a d i a b a t i c - l i k e expansion of the hot t h i n edge degrades the s h e l l . The t r a n s i t i o n t o the hot mode i s apparent i n the l a s t c y c l e on the l i n e p l o t s . The time s c a l e f o r t h i s passage i s 0.02t o= 200r o years f o r the g r i d as c a l c u l a t e d . I f the e n t i r e e m i t t i n g mass were i n c l u d e d , t h i s t i r a e s c a l e would be a f a c t o r of ten higher. An i d e n t i c a l model with the g r i d extended to 60 p o i n t s and the same 6r was a l s o run. The d e c e l e r a t i o n and subseguent de g r a d a t i o n of the s h e l l are edge e f f e c t s . F i g u r e s 12, 13, 14 and 15 g i v e l i n e and contour p l o t s f o r the d e n s i t y and v e l o c i t y e v o l u t i o n of t h i s model. The s h e l l maintains the c o n s t a n t v e l o c i t y without the s t r o n g d e c e l e r a t i o n ( c f . F i g u r e 14). The o u t e r r e g i o n s remain c o o l , at 10 4 K, and the s h e l l i s w e l l d e f i n e d . The a c o u s t i c waves are s t i l l apparent, s i n c e the outer r e g i o n s are s t i l l c o l l a p s i n g , but they maintain a s m a l l amplitude r a t h e r than growing. The growth i n the wave amplitude i n the NP •= 40 case i s due to the outer s h e l l which w i l l take l o n g e r to reach the main s h e l l i n t h i s example. I t seemed p o s s i b l e t h a t the f i n a l v e l o c i t y of the s h e l l c o u l d depend on the i n i t i a l steepness of the d e n s i t y f u n c t i o n . T h i s was t e s t e d by running the same model as above on a s o l u t i o n with a d e n s i t y g r a d i e n t three times s t e e p e r than-the one d e s c r i b e d above (and a d e n s i t y s c a l e twice as l a r g e , to roughly compensate and hold the outer r e g i o n at a reasonable " c o o l mode" temperature). T h i s i s r e f e r r e d to as CASE B; the p r e v i o u s one as CASE A (both 40 and 60 p o i n t g r i d s ) . The r e s u l t s were almost unchanged; t h e r e f o r e the c a l c u l a t i o n s can be c o n s i d e r e d 93 i n s e n s i t i v e t o l e s s - t h a n - d r a s t i c changes i n O ( r , t = 0 ) . The major change was a r e d u c t i o n i n the t e r m i n a l v e l o c i t y , to 5.5 u c . T h i s model was only c a r r i e d out u n t i l the c o n s t a n t v e l o c i t y was e s t a b l i s h e d ; i t was not f o l l o w e d u n t i l the s h e l l reached the outer edge. An i n t e r e s t i n g t e s t i s t o f o l l o w the e v o l u t i o n of the s h e l l i f i t i s switched i n t o " a d i a b a t i c mode" by t u r n i n g o f f the cosmic ray f l u x and the r a d i a t i v e c o o l i n g . The NP = 40 model at t = 0.006 t 0 was l e t evolve f o r awhile i n t h i s mode, with r e s u l t s shown as l i n e diagrams i n F i g u r e s 16 and 17. The v e l o c i t y of the s h e l l a t t h i s p o i n t i s l e s s than the escape v e l o c i t y , v C S c = 1 4 u 0 / r 0 - 5 = 20 u e , so the s h e l l d e c e l e r a t e s , and by t = 0.017 t c the gas everywhere has negative v e l o c i t y . F i n a l l y , as a s t a b i l i t y t e s t , the f i r s t model was run with the g r i d s i z e halved, and a l l other parameters i d e n t i c a l . T h i s i s d i s c u s s e d i n d e t a i l i n Appendix I I . The r e s u l t s v e r i f y that the c a l c u l a t i o n s are d u p l i c a b l e , thus t h a t the program i s n u m e r i c a l l y s t a b l e . The main d i f f e r e n c e i n the model with s m a l l e r 6r i s a sharper s h e l l , moving with a somewhat lower v e l o c i t y . Both of these are known e f f e c t s of i n c l u d i n g an a r t i f i c i a l v i s c o s i t y i n the c a l c u l a t i o n s . 99 F i g u r e s 6 and 7. S e l e c t e d epochs i n the low f l u x c a l c u l a t i o n (Model 1, Case A) using a 40 p o i n t g r i d are shown. F i g u r e 6 shows the d e n s i t y and F i g u r e 7, the v e l o c i t y . The u n i t s of d e n s i t y are n 0= 7x10*/r o cm - 3. Other u n i t s are as i n F i g u r e 3. The d e n s i t y p l o t s show the development of the s h e l l as i t moves outward and ,gains mass. The low d e n s i t y r e g i o n i n s i d e the s h e l l i s at a high temperature. The a r b i t r a r y outer edge c u t o f f a f f e c t s the subseguent development. The c o l l a p s e of the outer l a y e r s of the gas i s seen i n the outer maximum i n the d e n s i t y p l o t , which moves inward and merges with the s h e l l . The i n f a l l i n g gas sends sound waves back i n t o the i n n e r r e g i o n as i t h i t s the s h e l l , as seen at l a t e r epochs. In the f i n a l epoch, at t=197r c years, the s h e l l has reached the outer edge and has been degraded by the r a p i d expansion t h e r e . The v e l o c i t y p l o t a l s o i l l u s t r a t e s these e f f e c t s . The h i g h s h e l l v e l o c i t y , the negative v e l o c i t y of the outer p a r t s and the high expansion v e l o c i t y of the outermost edge can be seen. 100 1.043 1.0B5 1.12B R A D I U S F i g u r e 6 - S h e l l Model 1 (40 Point) - Density 1.17] 1.213 1.256 101 F i q u r e 7 . S h e l l Model 1 (40 Point) - V e l o c i t y 102 \ / / F i g u r e 8a. The temperature of the gas at s e l e c t e d epochs of the Model 1, Case A c a l c u l a t i o n . We see t h a t at e a r l y times, the gas i s a t T ~ 10 s K i n the s h e l l and by the l a t e r times has c o o l e d to 3 x 10* K as the s h e l l d e n s i t y i n c r e a s e d . The i n n e r c a v i t y i s at approximately 10 8 K and i s h e a t i n g s l i g h t l y . The hot outer edge i s due to the expansion i n t o the vacuum. 103 / -2 0.30 x 10 -2 a. t b. t c. t d. t 1.39 x 10 0.80 x 10 / -2 1.13 x 10 -3 -1 1.043 1.0 T T 1.085 1.128 1.171 R A D I U S 1.213 F i g u r e 8a- S h e l l Model 1 (40 Point) - Temperature 1.256 104 \ F i g u r e 8b. The energy t r a n s f e r r e d to the gas from the cosmic ray beam i s r e p r e s e n t e d by fD., where O i s the a t t e n u a t i o n s u f f e r e d by the beam and f < 1 i s the e f f i c i e n c y o f energy t r a n s f e r . T h i s e f f i c i e n c y i s l e s s than u n i t y i n a hot gas. The peak, of f Q corresponds to the high v e l o c i t y p o i n t i n the s h e l l . The diagram shows f o u r epochs of Model 1. 105 F i g u r e 8b- S h e l l Hodel 1 {40 Point) - Cosmic Ray A b s o r p t i o n 106 F i g u r e s 9 and 10- The same c a l c u l a t i o n as i n F i g u r e s 6 and 7 i s shown i n contour p l o t s . The o r i e n t a t i o n i s the same as i n the l i n e p l o t s , with time i n c r e a s i n g downward and r a d i a l d i s t a n c e on the h o r i z o n t a l a x i s . Numbers on the contour l i n e s r e p r e s e n t the d e n s i t y (Figure 9) and v e l o c i t y (Figure 10) v a l u e s . The three d i g i t numbers i n both p l o t s are i n u n i t s of 0.001 times the d e n s i t y ( v e l o c i t y ) s c a l e , while the s i n g l e and double d i g i t numbers are i n d e n s i t y ( v e l o c i t y ) s c a l e u n i t s . The growth of the s h e l l , the c o l l a p s e of the ou t e r gas, the sound wave g e n e r a t i o n and the deg r a d a t i o n of the s h a l l when i t h i t s the outer edge are apparent i n Fig u r e 9. The v e l o c i t y contour p l o t . F i g u r e 10, r e f l e c t s the co m p l i c a t e d v e l o c i t y s t r u c t u r e accompanying the sound waves (the l i n e a r p a t t e r n s going down and to the l e f t from t=0.013t o) and the expansion of the s h e l l i n t o the outer vacuum (the high v e l o c i t y c o ntours i n the lower r i g h t ) . The " i s l a n d " e f f e c t i n F i g u r e 10 r e f l e c t s the d i s c r e t e nature of the c a l c u l a t i o n s . I t i s not a numerical i n s t a b i l i t y . 107 RRDIRL DISTRNCE ID F i q u r e 9 - S h e l l Model 1 (40 Point) - D e n s i t y Contour 108 f i q u r e 10 . S h e l l Model 1 (40 Point) - V e l o c i t y Contour 109 F i g u r e 11. The p o s i t i o n of the s h e l l i s i n d i c a t e d by the p o s i t i o n ( i n g r i d points) of the point of maximum d e n s i t y , R^ . T h i s data i s from Model 1, Case A with a 60 p o i n t g r i d . The constant s h e l l v e l o c i t y i s r e f l e c t e d i n the steady i n c r e a s e of R ^ v;ith time. The thermal energy and (thermal+kinetic) energy a l s o show a steady i n c r e a s e with time, once the s h e l l i s w e l l e s t a b l i s h e d at t v 6 x 1 0 _ 3 t . The o s c i l l a t i o n i s not an i n s t a b i l i t y , but r a t h e r i t i s due to the d i s c r e t e nature of the c a l c u l a t i o n . 110 F i q u r e 11 . S h e l l Model 1 - R h _ (t) And E (t) 111 F i g u r e s 12 and 13. Model 1, Case A with a 60 p o i n t g r i d i s shown. (The u n i t s are the same as i n F i g u r e s 6 and 7.) The growth of the s h e l l and i t s approximately constant v e l o c i t y are apparent i n the d e n s i t y p l o t , F i g u r e 12. The outer r e g i o n s are c o l l a p s i n g and the ou t e r d e n s i t y maximum i s seen t o move inward- The v e l o c i t y p l o t , F i g u r e 13, agai n shows the g e n e r a t i o n and inward p r o p a g a t i o n of sound waves behind the s h e l l due to the c o l l a p s i n g outer gas. The extreme outer edge i s again expanding i n t o the vacuum at high v e l o c i t y . 112 o CO. t = 0.22 x 10 >-C 0 ° . - 7 CD UJ Q o ID ' a CM " n n n n ~ • t = 0.71 x 10 t = 0.93 x 10 " " " « n n P • R -2 t = 1.16 x 10 " " n t = 1.39 x 10 J i n t , ^ n n " ~ " " " " " " n n " " ' > f ' n " f ' " n " n " T T 1.0 1.063 1.127 1.19 R A D I U S 1.254 1.317 1.381 F i g u r e 12 . S h e l l Model 1 (60 Point) - Density 113 114 \ F i g u r e s 14 and 15- The same c a l c u l a t i o n as i n F i g u r e s 12 and 13 i s shown i n contour p l o t s . The growth and steady motion of the s h e l l i s apparent i n the dense r e g i o n of F i g u r e 14. The beginnings of the c o l l a p s e of the o u t e r p a r t s i s a l s o seen. F i g u r e 15 shows the high v e l o c i t y of the d e n s i t y peak, and a l s o shows the inward propagation of low amplitude sound waves. The expansion of the outer edge i n t o the vacuum i s seen i n the high v e l o c i t y contours i n the lower r i g h t c o r n e r . 115 116 F i g u r e 15 - S h e l l Model 1 (60 Point) - V e l o c i t y Contour 117 F i g u r e s 16 and 17 show the c o n t i n u a t i o n of an e a r l y epoch of the low f l u x , 40 g r i d p o i n t model with the cosmic rays and r a d i a t i v e c o o l i n g turned o f f . The s h e l l e v o l v e s under the i n f l u e n c e of g r a v i t y and i t s own momentum. F i g u r e 16 shows the c a l c u l a t e d d e n s i t y a t s e l e c t e d epochs. The s h e l l comes to a stop and degrades as i t s t a r t s to f a l l back i n toward the c e n t e r . F i g u r e 17 shows the c a l c u l a t e d v e l o c i t y , which decreases from the i n i t i a l peak and goes negative everywhere but at the extreme outer edge. 118 R A D I U S F i q u r e 16 - S h e l l Model 1 ( a d i a b a t i c ) - Density 119 1.043 1.085 1.171 F i g u r e 17 -1.12B R A D I U S S h e l l Model 1 ( a d i a b a t i c ) - V e l o c i t y 1.213 1.256 120 I n t e r m e d i a t e Flux Models The models of the previous s e c t i o n were used as a s t a r t i n g p o i n t f o r a s e r i e s of models, based on i n c r e a s i n g the cosmic ray i o n i z a t i o n r a t e by an order of magnitude at each s t e p 1 , while keeping the c e n t r a l mass and the cosmic ray low energy s p e c t r a l c u t o f f c o n s t a n t , and choosing the d e n s i t y s c a l e t o maintain c o o l outer temperatures. The q u a l i t a t i v e e v o l u t i o n of these models matched t h a t d e s c r i b e d i n the previous s e c t i o n , with the main d i f f e r e n c e s being i n the s h e l l v e l o c i t i e s . Cool Model 2 I : ..j F c t L = Ux10**r o ergs s ~ 1 I n 0 = 7x10 s/r„ cm - 3 I M = 2x108r o I ZETA = 2x105 | AMS = 100 | CFLAM = l O 2 7 | 40 p o i n t g r i d | u 0 = 100 km s - 1 | E = 1 MeV I In order to save computer time, these c a l c u l a t i o n s were 1 At t h i s p o i n t , i t might seem more n a t u r a l to s c a l e the models by net cosmic ray energy f l u x i n s t e a d ; however the m i c r o s c o p i c a l l y s i y n i f i g a n t parameter i s the i o n i z a t i o n r a t e of a s i n g l e atom. I t i s more d i r e c t l y r e l a t e d to the energy and momentum t r a n s f e r per atom than i s the f l u x ; i t i s used because i t seems important to be able to e x t r a p o l a t e o u t s i d e the parameter range covered i n these models. 121 only c a r r i e d out u n t i l the steady s h e l l v e l o c i t y was e s t a b l i s h e d . T h i s i s the most important c h a r a c t e r i s t i c number i n the c a l c u l a t i o n s . The temperatures of the s h e l l and of the i n n e r , hot c a v i t y are a l s o important; the s h e l l temperature w i l l c o o l s l i g h t l y as ^ i n c r e a s e s and X i . ( r ) / r 2 decreases, and can be estimated from the temperatures of e a r l y e v o l u t i o n s t a g e s . The | Cool Model 3 n c = M = 4 x 1 0 4 5 r & ergs s ~ 1 7x10 6/r f cm ~ 3 2x108r c ZETA = 2x10* AMS = 100 CFLAM = 1 0 2 8 40 p o i n t g r i d s D = 100 km s~l 1 MeV i t e r m i n a l v e l o c i t y i n these models i s determined by the p o i n t a t which the envelope of the v (t) curve becomes f l a t . T h i s i s i l l u s t r a t e d i n F i g u r e 18, which summarizes the vvwo.y: curves f o r the f i v e models c o n s i d e r e d so f a r . The upper envelope of these curves i s the p h y s i c a l l y important number. The o s c i l l a t i o n s r e p r e s e n t the f i n i t e g r i d s i z e , as do the " i s l a n d s " i n F i g u r e s 10 and 15. They occur because the peak of the s h e l l i n v e l o c i t y as well as i n d e n s i t y i s narrower than the g r i d s p a c i n g . As the s h e l l passes between two g r i d p o i n t s the h i g h e s t v e l o c i t y p o i n t i s lower than the t r u e maximum. The 122 p e r i o d of the o s c i l l a t i o n s i n F i g u r e 18 i s j u s t equal t o ^ t h e time f o r the s h e l l t o t r a v e l one g r i d space. The i n i t i a l high v e l o c i t y i n the Model 1, Case A curve i s probably a t r a n s i e n t r e f l e c t i n g the e s t a b l i s h m e n t of the s h e l l ; i t does not appear i n / the other data. The approach of the envelope of these curves to / zero s l o p e i s the c r i t e r i o n f o r s t o p p i n g the c a l c u l a t i o n s . S i n c e the models were sometimes stopped on the s h o r t s i d e of the zero s l o p e p o i n t , the s h e l l v e l o c i t i e s may be underestimated by ten or twenty per cent. Model 2 with F 0, = 4x10**r o ergs s _ 1 was f o l l o w e d out to t = 0. 0024t c = 24r e y e a r s , and by then had e s t a b l i s h e d a s h e l l v e l o c i t y of 10u o = 1000 km s _ 1 . The c r o s s i n g time f o r the g r i d chosen, based on t h i s v e l o c i t y . I s 0.012t o = 120 r 0 years. Again, i t must be noted that the g r i d used does not c o n t a i n enough mass to account f o r the observed o p t i c a l e m i s s i o n . The i n n e r hot gas had reached T = 3 x 10 7 K by t h i s p o i n t , and i s s t i l l h e a t i n g , not having reached the e q u i l i b r i u m temperature yet. The s h e l l has T = 5 x 10* K, and the outer gas i s a t 3-4 x 10* K. Model 3 with FC|^ = 4 x 10* 5r o ergs s - 1 was f o l l o w e d to t = 0. 000851^,= 8.5r 0 years o n l y . A s h e l l v e l o c i t y of 200u o= 2000 km s ~ 1 was i n d i c a t e d a l r e a d y , as seen i n F i g u r e 18. The s h e l l temperature i s 3-6 x 10* K, the outer gas i s at 10* K and the inner c a v i t y has reached T = 10 8 K. The trend i n these models towards s h o r t e r time s c a l e s , which are of course c o n s i s t e n t with the higher v e l o c i t i e s , i s worth n o t i n g . The phenomenon d e s c r i b e d here w i l l be very s h o r t l i v e d — as noted elsewhere. 123 F i g u r e 18. The v e l o c i t y of the s h e l l i s shown as a f u n c t i o n of time. Model 1, Case A shows the high v e l o c i t i e s as the s h e l l e s t a b l i s h e s i t s e l f , then slow d e c e l e r a t i o n as the s h e l l accumulates the i n f a l l i n g o u t e r gas. Only s e l e c t e d times are shown; roughly 6000 computational c y c l e s are r e p r e s e n t e d . Model 1, Case B has a lower gas d e n s i t y ahead of the s h e l l and t h e r e f o r e the s h e l l i s not d e c e l e r a t e d as f a s t but maintains a steady v e l o c i t y . The o s c i l l a t i o n i s a d i s c r e t i z a t i o n e f f e c t as the true v e l o c i t y maximum passes between two computational p o i n t s . Models 2 and 3 are at higher cosmic ray f l u x e s . Model 2 has reached a steady v e l o c i t y at 10u o= 1000 km s e c - i a f t e r 4000 c y c l e s . Model 3 i s j u s t approaching a steady v e l o c i t y of 18-20u o at 10,000 c y c l e s . 124 F i q u r e 18 . S h e l l V e l o c i t y v(t) - Models 1A, IB, 2, 3 125 High Flux Models Four models were run with the same high i o n i z a t i o n r a t e , ZETA = 2 x 10 7. The e x t e n s i o n to t h i s ZETA of the s e r i e s which was s t a r t e d with the p r e v i o u s models produced an i n t e r e s t i n g l y high s h e l l v e l o c i t y , 3500 km s _ 1 , with a cosmic ray f l u x of 4 x 1 0 4 6 r o e r g s s - 1 and a d e n s i t y of 2 x 1 0 7 / r o cm - 3. These parameters are not unreasonable i n terms of S e y f e r t n u c l e a r models or quasar models. Ther e f o r e t h i s n umerical model was i n v e s t i g a t e d i n more d e t a i l , by e x p l o r i n g the e f f e c t s of the c e n t r a l mass and of the cosmic ray energy. The numerical models were run with a coarse g r i d of only twenty p o i n t s , and were run only u n t i l the v e l o c i t y c u rves began t o reach a steady v a l u e . Again, t h i s was done to save computer time. F i g u r e 19 shows these curves f o r a l l f o u r models. The g r i d was chosen so t h a t the twenty p o i n t s cover s e v e r a l times the a b s o r p t i o n l e n g t h f o r the proton energy i n q u e s t i o n , so the s h e l l c ould be f o l l o w e d t o the steady s t a t e . Model 4 r e p r e s e n t s the e x t e n s i o n of the p r e v i o u s s e r i e s . The development of the s h e l l i s s i m i l a r , but the higher i o n i z a t i o n r a t e produces an i n t e r e s t i n g e f f e c t , a double v e l o c i t y maximum at the b e g i n n i n g . T h i s i s seen i n F i g u r e s 20 and 21, l i n e p l o t s f o r the 1 MeV -model, ( A c t u a l l y the diagrams r e f e r to the AMS = 1000 model, Model 5, t o be d i s c u s s e d below; • but the -two models are n e a r l y i d e n t i c a l i n r e s u l t s . ) The f i r s t v e l o c i t y maximum occurs a t the p o i n t of s t r o n g cosmic ray energy a b s o r p t i o n . T h i s p o i n t scon reaches temperatures high enough to make the energy gain i n e f f i c i e n t , so the a b s o r p t i o n p o i n t r e s e t t l e s i t s e l f a few g r i d p o i n t s f u r t h e r out. T h i s generates 126 r Cool Models 4 and 5 4x10* 6 r c ergs s~ 1 n 0 = 2 x 1 0 7 / r o c m - i i M 2x108r t - 2x10«r 0 ZETA = 2x107 AMS = 100 (Model 4) AMS = 1000 (Model 5) CFLAM = 3x1028 20 p o i n t g r i d s Uc. = 100 km s - 1 E = 1 MeV L : J the second maximum. By t = 0.19 x 1 0 ~ 3 t t h i s double s t r u c t u r e i s apparent and at l a t e r times the i n n e r peak i s d i s a p p e a r i n g . I t can a l s o be seen i n the i n f l e c t i o n s of the contour l i n e s i n F i g u r e s 22 and 23. (Again, these are a c t u a l l y data from the Medel 5 run.) A s h e l l e v o l v e s from the outer peak, and has reached i t s steady growth by t = 0.5 x 1 0 ~ 3 t o = 5r 0 y e a r s . 1 The v e l o c i t y curve v V a >, (t) i n d i c a t e s that the s h e l l w i l l reach about 3500 km s - 1 ; a t t h i s r a t e i t w i l l escape the e n t i r e n u c l e a r gas i n 0.03t o = 300r o y e a r s . F i g u r e 23a shows the temperature of the gas at s e l e c t e d epochs of Model 5. T h i s behavior i s t y p i c a l of a l l of the high f l u x models. The outer r e g i o n s are at an e q u i l i b r i u m temperature of 10* K and the s h e l l i s at 5 x 10* K. The i n n e r 1 The contour p l o t s i n Figures 22 and 23 show only the beginning s t a g e s of t h i s model. Comparison with the Model 1 c a l c u l a t i o n s shows that the high d e n s i t y , steady v e l o c i t y s h e l l only j u s t appears i n the lower p a r t s of the contour p l o t s . 127 c a v i t y has not reached e q u i l i b r i u m ; i t i s at 1 to 3 x 10 9 K and s t i l l h eating s l o w l y . Repeating the c a l c u l a t i o n with AMS = 1000 i n Model 5 s e r v e s only t o reduce the s h e l l v e l o c i t y s l i g h t l y , as seen i n F i g u r e 19. T h i s model can be i n t e r p r e t e d i n terms of e n s e a l i n g d i s t a n c e which i s down by a f a c t o r of t e n . Tha't i s , i f the c e n t r a l mass i s c o n s i d e r e d w e l l known — and 2 x 10 8 M e seems a good value — then t h i s value i s maintained i f r c i s i n t e r p r e t e d as 0.1 pc, r a t h e r than 1 pc. T h i s s c a l e s the f l u x down and the gas d e n s i t y up, so a s i m i l a r s h e l l v e l o c i t y i s produced more cheaply i n cosmic ray energy. 1 1 Cool Model 6 J 1 = 4 x 1 0 4 S r „ ergs s - 1 n c = 6.x10Vr c cm~ 3 M = 2x10*^° ZETA = 2x107 • AMS = 1000 CFLAM = 8x1027 20 p o i n t g r i d 100 km s - i Eo = 0.1 MeV 1 . - 1 The low energy c u t o f f assumed f o r the cosmic ray spectrum i s the most e f f e c t i v e energy of the spectrum i n the heating and momentum t r a n s f e r to the gas. The heating per i o n i z a t i o n drops as E c r i s e s , but the column d e n s i t y of gas penetrated by the primary beam goes up. (The a b s o r p t i o n goes as 128 exp—[ E ~ 2 ^ n dr ] ° ° 2 5 , as shown i n Appendix I. The energy d e p o s i t e d per i o n i z a t i o n as a f u n c t i o n of E 0 i s c a l c u l a t e d by Bergeron and C o l l i n - S o u f f r i n (1973), and goes down by about a f a c t o r of ten as E 0 goes from 0.1 MeV t o 10 MeV.)/ Model 6 with the same i o n i z a t i o n r a t e and c e n t r a l mass and with E 0 =0.1 MeV, was e v a l u a t e d u n t i l the s h e / l e s t a b l i s h e d i t s e l f , at t = 0.93 x 1 0 ~ 3 t o = 9.3r Q y e a r s . T h i s model has a lower t o t a l i n p u t f l u x : because the i o n i z a t i o n c r o s s s e c t i o n goes i n v e r s e l y with the p a r t i c l e energy, the i n t e g r a t e d f l u x s c a l e s as E 2 times the i n t e g r a t e d i o n i z a t i o n r a t e . The temperatures of the flow here are about equal to those of the p r e v i o u s model, although the l o n g e r time e l a p s e d has allowed the i n n e r c a v i t y to reach 2 x 10 8 K. The e a r l y c y c l e s show i n d i c a t i o n s of the double v e l o c i t y peak which appeared i n the e a r l y s t a g e s of the 1 MeV model. By the end of the c a l c u l a t i o n s , the s h e l l has reached a v e l o c i t y of 20u o = 2000 km s _ 1 . F i g u r e s 24, 25, 26 and 27 g i v e the l i n e and contour p l o t s f o r t h i s model. Model 7, with a proton energy of 10 MeV and the same i o n i z a t i o n r a t e , has a c o r r e s p o n d i n g l y higher i n t e g r a t e d cosmic ray f l u x . T h i s model was run u n t i l the s h e l l v e l o c i t y had begun to e s t a b l i s h i t s e l f , as shown i n F i g u r e s 28, 29, 30 and 31. In t h i s model, a g a i n , o n l y the very e a r l y stages of the flow are i n d i c a t e d i n the c a l c u l a t i o n s . However the temperature e q u i l i b r i u m i s w e l l e s t a b l i s h e d by t h i s p o i n t i n the model — the s h e l l i s s t i l l at 5-10 x 10* K, the outer gas a t 10* K and the i n n e r c a v i t y has reached 10a K -- and the subsequent behavior can be assumed to resemble t h a t of the e a r l i e r models. 129 Cool Model 7 = 4 x 1 0 * 7 r o ergs s ~ l = 6 x 1 0 V r o cm" 3 M = 2 x 1 0 ^ ZETA = 2x107 AMS = 1000 CFLAM = 8x1028 20 p o i n t g r i d u t = 100 km s-» E, = 10 MeV The v e l o c i t y had reached 73u e = 7300 km s _ l by the end of the c a l c u l a t i o n s , and i f the behavior of v w a_ y (t) i s s i m i l a r t o the other models, the s h e l l w i l l h i t a steady v e l o c i t y of about 8000 km s _ 1 or l a r g e r . 130 F i g u r e 19. The maximum v e l o c i t y w i t h i n the s h e l l i s shown f o r the models with high cosmic ray f l u x . Models 4 and 5 are f o r the case of a 1 MeV low energy c u t o f f and c e n t r a l masses of 1 0 8 r o and 1 0 9 r o s o l a r masses, r e s p e c t i v e l y . The c e n t r a l mass has only a s m a l l e f f e c t , and the curves n e a r l y d u p l i c a t e each o t h e r . Both are approaching a steady v e l o c i t y of 3500 km s e c - 1 a f t e r 14,000 computational c y c l e s . Model 6 i s f o r the 0.1 MeV case. The s h e l l develops and reaches a v e l o c i t y of approximately 2000 km s e c - 1 a f t e r 8000 computational c y c l e s . Model 7 i s f o r the 10 MeV case. A f t e r 30,000 c y c l e s the s h e l l i s begi n n i n g t o approach a steady v e l o c i t y near 8000 km s e c - 1 . 131 40 30 20 Model 4 J L 40 i — 25 30 35 4 0 t / c i o - 4 g 5 0 55 30 Model 5 20 25 30 35 40 45 t / ( H f 4 t 6 ) 55 20, 15 Model 6 6.0 > 60 40 20 7.0 8'° t/Clo" 4 t j 132 F i g u r e s 20 and 21. S e l e c t e d epochs i n Model 5 (E c = 1 MeV) are shown. The c e n t r a l mass i s M = 1 0 9 r o M0, and the d e n s i t y s c a l e i s n p=2x10 7/r e cm - 3- Other u n i t s are as i n F i g u r e 3. The s h e l l develops somewhat away from the i n n e r edge while the i n n e r p o i n t s reach a high e g u i l i b r i u m temperature. T h i s i s seen i n the two maxima i n e a r l y epochs of both the d e n s i t y p l o t , F i g u r e 20, and the v e l o c i t y p l o t . F i g u r e 21. The s h e l l has reached a v e l o c i t y o f 3500 km s e c - 1 by the end of the c a l c u l a t i o n . The outer gas i s beginning t o c o l l a p s e , but at a much lower v e l o c i t y so l i t t l e e f f e c t of t h i s c o l l a p s e i s seen. 133 134 135 \ F i g u r e s 22 and 23. The same c a l c u l a t i o n as i n F i g u r e s 20 and 21 i s shown i n contour p l o t s . The d e n s i t y p l o t , F i g u r e 22, shows the growth of two maxima and decay of the i n n e r one i n the "s-shaped" contour curves i n the upper l e f t . The s t r o n g d e n s i t y peak a t r~1.04r t i s apparent i n the lower h a l f of the f i g u r e . The v e l o c i t y p l o t , F i g u r e 23, shows the same steady growth and motion of the s h e l l a f t e r t ~ 2 x 1 0 - * t o . F i g u r e 22 - S h e l l Model 5 - Density Contour F i q u r e 23 . S h e l l Model 5 - V e l o c i t y Contour 138 F i g u r e 23a. The temperature of the gas i s shown at s e l e c t e d epochs of Model 5. The i n n e r r e g i o n s are a t T ~ 10 8 K and are h e a t i n g s l o w l y . The s h e l l i s at approximately 5 x 10* K and the outer r e g i o n s are at an e q u i l i b r i u m temperature of 10* K. 139 o i n C M cn in CQ i n O Oo i n LO i n i n ' i n o 1.0 / -3 0;„06 x 10 J a. t b. t = 0.19 x 10 c. t = 0.31 x 10 d. t = 0.43 x 10 -3 -3 -3 1.022 1.044 1.066 R A D I U S 1.DB7 F i g u r e 23a. S h e l l Model 5 - Temperature 1.1D9 1.13] 140 F i g u r e s 24 and 25 show s e l e c t e d epochs i n Model 6 (Ec,=0.1 MeV). The c e n t r a l mass i s 1 0 9 r o M D and the d e n s i t y s c a l e i s n c = 6 x 1 0 3 / r o cm - 3; other u n i t s are as i n F i g u r e 3. The g r i d s i z e chosen i s s m a l l e r than i n pr e v i o u s models because of the s h o r t e r a b s o r p t i o n l e n g t h . A narrow s h e l l has developed by t=9.3x10 _*t o=9.3r 0 years and i s beginning to move out at a v e l o c i t y of 2000 km s e c - 1 . The outer r e g i o n s are beginning to c o l l a p s e . 141 i.O 1.014 1.02B 1.042 1.056 1.07 1.084 R A D I U S F i g u r e 24 - S h e l l Model 6 - D e n s i t y 142 143 \ i I F i g u r e s 26 and 27 show the same 0.1 MeV c a l c u l a t i o n as i n F i g u r e s 24 and 25, i n the form of contour p l o t s . The d e n s i t y contour, F i g u r e 26, shows the i n i t i a l d e n s i t y g a i n at r~1.0i4r l , and the l a t e r development of a s t r o n g e r s h e l l at r ~ 1 . 0 8 r ( . The v e l o c i t y p l o t , F i g u r e 27, a l s o shows the growth of the high v e l o c i t y s h e l l , as w e l l as the slow c o l l a p s e beginning o u t s i d e of the s h e l l . F i q u r e 26 . S h e l l Model 6 - Density Contour RRDIRL DISTRNCE o ID F i g u r e 27 . S h e l l Model 6 - V e l o c i t y Contour 146 F i g u r e s 28 and 29. S e l e c t e d epochs i n Model 7 (E o = 10 MeV) are shown. The c e n t r a l mass i s 109r..MG and the d e n s i t y s c a l e i s n o=6x10 7/r £ ) cm - 3; other u n i t s are as i n F i g u r e 3. The s h e l l has formed and gained a v e l o c i t y of 7500 km s e c - 1 very r a p i d l y , and i s j u s t b eginning to move outward. The longer a b s o r p t i o n l e n g t h of 10 MeV protons a c c e l e r a t e s the outer r e g i o n s s u f f i c i e n t l y to prevent the c o l l a p s e seen i n the other models. 147 ° 1.0 1.022 1.044 1.066 1.087 1.109 1.131 RADIUS F i q u r e 28 . S h e l l Model 7 - Density 148 149 F i g u r e s 30 and 31. The 10 MeV c a l c u l a t i o n of F i g u r e s 29 and 30 i s re p r e s e n t e d i n contour p l o t s . The growth of the s h e l l and ev a c u a t i o n of the r e g i o n behind i t i s apparent i n the d e n s i t y p l o t o f F i g u r e 30. The n e a r l y v e r t i c a l l i n e s i n the r i g h t part of the f i g u r e occur because the outer gas has har d l y begun to move. The v e l o c i t y p l o t shows the growth of the maximum v e l o c i t y p o i n t . The peak has j u s t begun to move outward by the end of the c a l c u l a t i o n . F i g u r e 30 - S h e l l Model 7 - Density Contour RADIAL DISTRNCE 1.0 1.063 1.125 l.iflfl 1. P - J 1 1 1 1 UJ o F i g u r e 31 . S h e l l Model 7 - V e l o c i t y Contour 152 D i s c u s s i o n Of Cool Mode Models Seven p h y s i c a l l y d i s t i n c t c o o l mode models have been run. The computational parameters of each model can be c o n v e r t e d to unique p h y s i c a l parameters by choosing a d e f i n i t e value f o r the remaining s c a l i n g parameter, r 0 . F o l l o w i n g Table IV, i n Chapter IV, the models are summarized i n Table VIII i n terms of p h y s i c a l parameters ( t o t a l cosmic ray f l u x using equation 22, gas d e n s i t y , c e n t r a l mass and p a r t i c l e e nergy). A l s o , the thermal bremsstrahlung from the hot gas, Lx, i s given as w e l l as the i n n e r c a v i t y temperature. At f i r s t s i g h t i t may seem odd t h a t the q u a n t i t i e s d e n s i t y , c e n t r a l mass and cosmic ray f l u x , depend on r c i n the forms given i n the t a b l e . (The q u a n t i t y r 0 appears because the DE's are s c a l e d i n terms of the dynamical time s c a l e , r 0 / u o . ) For i n s t a n c e , a given numerical model r e p r e s e n t s a s e t of p h y s i c a l models: one a t F ^ , M and n c , say; and others a t 10 p FC(^ , 10^ M and 10- prt o where p i s any number. T h i s i s because the p h y s i c a l l y s i g n i f i g a n t r a t i o s , which remain c o n s t a n t w i t h i n a s e t of models, are FCi^ /M and Fc^ / n - 1 = n c . As d i s c u s s e d i n Appendix I I , these models a l l conserve mass, momentum and energy. T h i s , p l u s the i n s e n s i t i v i t y to the g r i d s p a c i n g chosen f o r the c a l c u l a t i o n s , i s c o n v i n c i n g evidence t h a t the numerical models are p h y s i c a l l y a c c u r a t e . The dependence of the r e s u l t s on the i n i t i a l c o n d i t i o n s deserves comment, however. The s h e l l seems to reach a steady v e l o c i t y and then d e c e l e r a t e s l i g h t l y , due to the drag from the outer gas. The T a b l e VIII Cosmic Ray A c c e l e r a t i o n : S h e l l Models 153 Lx; Tx 4x10* 3r o 7x10 + / r ^ M = 2x108r c 1 MeV Fa> = 4x10 + + r t 7x105/r c M = 2x10«r e 1 MeV Fce = 4x10+5r o n 0 = 7x10&/r c M = 2x103r L 1 MeV Fc(> = 4x10**r 6 n o = 2x107/ r„ M = 2x10«r t >" 1 MeV Fc,= 4 x 10 * & r t n c = 2x10 7/r o M = 2x10 t»r o 1 MeV 5x 10+ ir„ 3x107 K 9x10 + 3 r c 10° K 8x10 + + r c 108 K 3x10 + + r c 10* K T ( s h e l l ) / V e l o c i t y / 3x103»r o 3-6x10+ K 500-700 10 7 K 5x10+ K 3-6x10+ K 5x10+ K 5x10* K 1000 2000 3500 + 3400 + M = F,„ = M 4x10+s r. 6x10&/r o 2x10»r t 0.1 MeV 4x10*7r c 6x107/r o 2x10 9r c 10 MeV 5x10*3r,, 10 a K 5 x 1 0 * 5 r c 108 K 5-10x10* K 5-11x10+ K 2000+ 8000 + — r„ = l e n g t h s c a l e i n pc — u n i t s : Ft^ (ergs s _ 1 ) n 0 (cm - 3) «(«©) v e l o c i t y (km s _ 1 ) Lx (ergs s _ 1 ) 154 drag f o r c e on the s h e l l goes as ^v 2$ D i s the d e n s i t y ahead of the s h e l l . T h i s i n c r e a s e s as the s h e l l speeds up, and may e v e n t u a l l y d e c e l e r a t e the s h e l l . The net f o r c e on the s h e l l can be approximated as G(r) = X t°<^>^-N/r 2 - GM/r 2 - 0 v 2 i f N i s the column d e n s i t y of the s h e l l , and the f r a c t i o n of the cosmic r a y s absorbed by the s h e l l . S e t t i n g t h i s e q u a l to zero f o r the constant d e n s i t y case (N = Pr/3) g i v e s the p r e d i c t e d maximum v e l o c i t y , v * a Y * 3 0 ° t n - r . V r 3 ) ( V « 2 ) ] ° ' s / r O c « s km s~ i i f r, i s the in n e r r a d i u s , T = 10«£ and M = 10~ 8M/M„. T h i s i s a lower l i m i t to v i U 0 L y. s i n c e the i n i t i a l c o n d i t i o n s used i n the program i n v o l v e ^(r) d e c r e a s i n g with r . For Model 1 (\' = 7 x 1 0 - a s _ 1 ) t h i s simple model p r e d i c t s v l V i C t y* 500 km s - 1 and f o r Model 7 (T = 7 x 1 0 - 5 ) , v „^8«00 km s ~ 1 , both of which agree d e c e n t l y with the numerical c a l c u l a t i o n s . The probable f u t u r e of these models i s as f o l l o w s . The s h e l l i s probably R a y l e i g h - T a y l o r u n s t a b l e , as i s shown i n Chapter VI, on a time s c a l e s i m i l a r t o the c r o s s i n g time. The i n s t a b i l i t y w i l l produce s m a l l , denser clou d s which w i l l r e t a i n the outward v e l o c i t y of the s h e l l at fr a g m e n t a t i o n . In most of the models, t h i s v e l o c i t y i s w e l l above the escape v e l o c i t y , so the c l o u d s w i l l not be contained i n the n u c l e a r r e g i o n . The dynamics at the point of fragmentation w i l l f o l l o w the b i l l i a r d - b a l l d i s c u s s i o n of Chapter I I . 155 CHAPTER VI DISCUSSION AND IMPLICATIONS OF THESE MODELS S t a b i l i t y Two common i n s t a b i l i t i e s c o u l d be r e l e v a n t to the models c a l c u l a t e d here, namely, i s o b a r i c thermal i n s t a b i l i t y , which occurs i f the net c o o l i n g f u n c t i o n v a r i e s i n v e r s e l y with temperature; and the R a y l e i g h - T a y l o r i n s t a b i l i t y , which o c c u r s at the i n t e r f a c e between a heavy f l u i d and a l i g h t e r one e x e r t i n g pressure on the he a v i e r one. The i s o b a r i c thermal i n s t a b i l i t y c r i t e r i o n i s given by F i e l d (1965) : { X \ - 1 I L * \ < 0 (64) where i s the net c o o l i n g f u n c t i o n per gram, i n our case equal to f\ (T) ^  - ^ ,<AE> SL (T) . (This assumes Coulomb l o s s e s to be dominant and t h a t x = 0 ( 1 ) ; Q(T) i s the a t t e n u a t i o n f a c t o r f o r cosmic r a y s , i n c l u d i n g the temperature dependence i n the case of kT^ 5 > Eme/m^. See Chapter V.) The c o o l p a r t s of our models, the s h e l l and outer r e g i o n s , are c l e a r l y s t a b l e , as they occupy the steep p o s i t i v e s lope of the /\ (T) c u r v e . The hot r e g i o n s , the i n n e r c a v i t y of the s h e l l models and the e n t i r e r e g i o n of the hot mode models, occupy a l e s s steep r e g i o n of the f\ (T) c u r v e , which would be unstable by t h i s c r i t e r i o n were i t not f o r the dec r e a s i n g e f f i c i e n c y of the s o f t cosmic ray term with i n c r e a s i n g temperature. T h i s s t a b i l i z i e s the gas, as f o l l o w s : 156 i f then ( i f / d T ) p = (n-m-1) Ho T ^ _ 1 i f the e q u i l i b r i u m s t a t e i s c h a r a c t e r i z e d by = 0- T h i s shows that the gas i s t h e r m a l l y s t a b l e i f n-m > 1, which i s s a t i s f i e d i n our case where n = 1/2 (due to the asymptotic form of the c o o l i n g curve a t T > 10 8 K) and m = -3/2 ( r e f l e c t i n g the energy t r a n s f e r at high temperatures, as given i n Chapter I I I ) . I f hard cosmic rays were i n v o l v e d , _Q (T) would be u n i t y below T ~ m ec 2/k ^ 1 0 1 0 K; then m = 0 and n-m < 1, making the 0 hot end of /\ (T) t h e r m a l l y u n s t a b l e . T h i s u n s t a b l e c o n d i t i o n could o b t a i n i n l a t e r phases of our models i f the cosmic ray source were to t u r n o f f . T h i s c r i t e r i o n (equation 64) shows t h a t c o o l e r r e g i o n s of the gas may be u n s t a b l e as w e l l . Reference t o F i g u r e 1 shows that the A(T) curve has p l a t e a u s with l o g a r i t h m i c s l o p e n < 1 as marked with shaded r e g i o n s at the bottom of the F i g u r e . The f a c t t h a t the curve i s smoothed, and i t s s e n s i t i v i t y to the i o n i z a t i o n mechanism and abundances make these temperature r e g i o n s only approximate. These r e g i o n s s a t i s f y ( i ^ / ^ T ) ^ < 0 because m = 0 below T = 10 7 K. We note, the s h e l l s (Table VIII) occupy mostly s t a b l e r e g i o n s of the c o o l i n g c u r v e . F i e l d g i v e s the growth r a t e f o r the i s o b a r i c i n s t a b i l i t y i n the l i m i t of s m a l l wavelength and low thermal c o n d u c t i v i t y , f o r a gas which has zero i n i t i a l v e l o c i t y . In our case, t h i s s o l u t i o n should be a p p l i c a b l e i f the growth time i s much s h o r t e r than any dynamic t i m e s c a l e . T h i s growth r a t e i s given by h i s 157 equation (31) as CO =- oJ^ilzH p ( i L f \ = oM(V-l) p A(T) = t - i (65) / xRT A" /-» ST \ mz + k where iO i s the growth r a t e i n s - 1 , c( = (1-n+ra) and R/u. i s the gas c o n s t a n t per gram. (B^r = k/m) T h i s a p p l i e s to the s m a l l e s t wavelengths p o s s i b l e , where the s m a l l e s t s i z e i s determined by the thermal c o n d u c t i v i t y which i s a s t a b i l i z i n g i n f l u e n c e . 1 N u m e r i c a l l y , t h i s i s t i V v ~ 2 x 1 0 ~ l s T n - 1 / \ ( T ) - 1 s e c , t a k i n g the value of A(?) from F i g u r e 1, Chapter V. E s t i m a t i n g the expansion time of the s h e l l moving at v e l o c i t y v as t ^ = r 0 / v , the c o l l a p s e time can be w r i t t e n as Wfr, 10-3TJ 1Q27 \ / 10-23\/ v \ (66) VCFIAM j [ ft (T )jVu„J ^ 500Tfc / 1 0 ~ 2 3 \ v ^ n 0 r 0 y A (T) J u 0 which can be much l e s s than u n i t y f o r the hot phases of our models. (For i n s t a n c e , f o r Model 3 i n t a b l e V I I I , t ^ / t ^ ^ 0.4) The R a y l e i g h - T a y l o r i n s t a b i l i t y i s one to which the d r i v e n flow i s s u s c e p t i b l e . T h i s o b t a i n s when a l i g h t f l u i d pushes on a heavy one, f o r i n s t a n c e i n the c l a s s i c textbook case (Chandrasekhar, 1961) of a s t a t i c s i t u a t i o n with a l i g h t f l u i d ' 1 In the macroscopic e q u a t i o n s , we have i g n o r e d the thermal c o n d u c t i v i t y , as heat t r a n s f e r by c o n d u c t i o n i s n e g l i g i b l e i n a t h i n plasma. However i f the s c a l e i s s m a l l enough, i t s e f f e c t s w i l l be f e l t . E v a l u a t i n g F i e l d ' s e x p r e s s i o n f o r the maximum un s t a b l e wavenumber, and using S p i t z e r ^ (1967) e x p r e s s i o n f o r thermal c o n d u c t i v i t y , g i v e s . A (rain) ^  1 0 - 3 T „-i f 10 - 2 7 y.500 T s / ( n o r 0 ) r„ x V CFLAM / where n Q i s i n cm - 3 and r c i n pc, f o r the s m a l l e s t u n s t a b l e s c a l e i n a hot gas. Since n 0 r o i s about 10 s to 1 0 8 cm _ 3pc i n our models,-\ . 32 5 x 10 - <*r to 5 x 1 0 - 6 r . 158 s u p p o r t i n g a heavy one a g a i n s t g r a v i t y ; o r , i n models of r a d i o g a l a x i e s (Blake, 1972) i n v o l v i n g a beam of e n e r g e t i c p a r t i c l e s c a r v i n g a hole out of the ambient i n t e r g a l a c t i c gas. The s i m p l e s t case i s two homogeneous s t a t i c , i n c o m p r e s s i b l e f l u i d s / s e p a r a t e d by an i n t e r f a c e , where ^>> P,. The growth r a t e of s i n u s o i d a l p e r t u r b a t i o n s of t h a t i n t e r f a c e i s given as a f u n c t i o n of the wavelength by 2 = i f t i . \ K £ (67) Here, G i s the f o r c e a p p l i e d by the l i g h t gas ( i n the s t a t i c example, g r a v i t y ) . Blake has i n v e s t i g a t e d the s t a t i c , c o m p r e s s i b l e case and found that c o m p r e s s i b i l i t y s e m i - s t a b i l i z e s the i n t e r f a c e , by r e d u c i n g the growth r a t e f o r s m a l l s c a l e d i s t u r b a n c e s . I f r 0 i s a t y p i c a l s c a l e , (68) 03 2 * llStZB-which i s an expansion of a complicated p o l y n o m i a l equation i n .*)2 , v a l i d when A / r o « 1 . Using the simple form f o r the growth r a t e , the c o l l a p s e time s c a l e (t = t o - 1 ) can be expressed i n terms of the expansion time s c a l e : hp = In terms of our models, G i s the net outward cosmic ray f o r c e ( i g n o r i n g g r a v i t y f o r s i m p l i c i t y ) ; thus, t_«,r = f 3x10-5 <v/ua)_2 \ >>s (6 9) . t-o [ < ZETA /106) r^J For s c a l e s )y ~ r , t h i s r a t i o w i l l be n e a r l y u n i t y i n the high f l u x models; s m a l l e r s c a l e s w i l l c o l l a p s e f a s t e r . The s m a l l e s t 159 s c a l e s w i l l be s e m i - s t a b i l i z e d by c o m p r e s s i b i l i t y , a c c o r d i n g to eguation (68). The very s m a l l e s t s c a l e s t h e o r e t i c a l l y w i l l be s t a b i l i z e d a l s o by v i s c o s i t y , as Chandrasekhar shows. T h i s i s a n e g l i g i b l y small e f f e c t i n our models, however, s i n c e the v i s c o s i t y of a plasma i s very low. / Summarizing, we expect the s h e l l i n the c o o l mode models, which i s being pushed by the cosmic ray gas and by the hot i n n e r c a v i t y , to fragment due t o the R a y l e i g h - T a y l o r i n s t a b i l i t y b e f o r e i t has escaped the n u c l e u s . Once t h i s f r a g m e n t a t i o n o c c u r s , the c o n t i n u o u s flow e g u a t i o n s of Chapter IV no l o n g e r apply; the c l o u d s now obey the " b i l l i a r d b a l l " e q u a t i o n s of Chapter I I . That i s , i f they are l a r g e enough to be o p t i c a l l y t h i c k (which cannot be a b s o l u t e l y determined without a f u l l e r R a y l e i g h - T a y l o r a n a l y s i s , but i s l i k e l y g iven the e s t i m a t e s of Chapter II) they w i l l obey equ a t i o n (10) as they are a c c e l e r a t e d outward. In a l l except the lowest f l u x models, the s h e l l has reached the escape speed b e f o r e i t fragments, so u n l e s s severe snowplowing d e c e l e r a t e s the c l o u d s they w i l l escape the n u c l e u s . Since the i n s t a b i l i t y i s not i s o b a r i c , the clouds w i l l not be p r e s s u r e c o n t a i n e d but w i l l expand at t h e i r sound speed. The d e n s i t y and r e s u l t a n t temperature of the c l o u d s cannot be estimated from t h i s l i n e a r a n a l y s i s . One n o n l i n e a r c a l c u l a t i o n of thermal c o l l a p s e f i n d s d e n s i t y i n c r e a s e s of 20 to 100 i n c l o u d s i n the i n t e r s t e l l a r medium (Schwarz, McCray and S t e i n , 1972); while t h i s has no necessary r e l e v a n c e to R a y l e i g h - T a y l o r c o l l a p s e , we can use p'/P~100 to estimate the c o n d i t i o n s i n the fragments of the s h e l l . From eguation (6 1), 160 / ^ ( T J c t ^ - 1 . F i g u r e 1 (Chapter V) then i n d i c a t e s that i f the s h e l l was i n e q u i l i b r i u m at 5 x 10+ Kf the new temperature w i l l be about 10* K. The sound speed at t h i s temperature i s only 10 km s - 1 . The expansion of the cloud s w i l l be much slower than t h e i r outward motion, so they w i l l d i f f u s e o n l y s l o w l y as they move outward. The r e l a t i v e l y slow expansion and r e s u l t a n t d e n s i t y decrease as the c l o u d moves away from the nucleus (at v e l o c i t i e s v >; v ^ ^ , qive n by equation 18) may account f o r the r e l a t i v e l y l a r g e extent (0(10-100) pc) of the nebular l i n e e m i t t i n g gas i n some S e y f e r t s . A t h i r d i n s t a b i l i t y to which the model i s s u s c e p t i b l e i s due to the motion of tne cosmic ray protons through the hot c a v i t y . T h i s streaming can a m p l i f y s m a l l f l u c u a t i o n s i n the gas and t r a n s f e r the beam energy t o the gas. T h i s two-stream i n s t a b i l i t y i s a converse of the Landau damping p r o c e s s . B e r n s t e i n and Trehan (1960) show that the most r a p i d growth r a t e i s on very s m a l l s c a l e s , producing t u r b u l e n t f l u c t u a t i o n s r a t h e r than l a r g e - s c a l e s t r u c t u r e . T h i s i n s t a b i l i t y w i l l t h e r e f o r e enhance the energy t r a n s f e r to the gas. I t i s d i s c u s s e d i n t h i s c a p a c i t y below. E n e r g e t i c s and T i m e s c a l e s . The models presented here have i n t e r e s t i n g i m p l i c a t i o n s with r e s p e c t to the o v e r a l l e n e r g e t i c p i c t u r e of an a c t i v e nucleus. As mentioned b e f o r e , the r a p i d mass l o s s r e s u l t i n g from the 161 high gas v e l o c i t i e s i n v o l v e s very s h o r t t i m e - s c a l e s . The ext e n t of the nucleus, under a parsec, and the s h e l l v e l o c i t i e s of s e v e r a l thousand km s - 1 , mean the n u c l e a r r e g i o n w i l l be emptied of gas i n l e s s than 10* ye a r s . (A c l o u d moving a t 3000 km s — 1 w i l l c r o s s 1 pc i n 300 y e a r s , f o r i n s t a n c e . ) S t a t i s t i c a l arguments i n d i c a t e , however, t h a t the S e y f e r t phenomenon l a s t s f o r about one per cent of the l i f e t i m e of a s p i r a l g alaxy. To account f o r t h i s , then, some r e c y c l i n g mechanism i s needed t o provide new mass near the n u c l e u s . The h i g h e r f l u x models here i n v o l v e a mass l o s s r a t e of about 1 M@ per year, which must be r e p l a c e d . I f t h i s mass comes c o n t i n u a l l y from the c e n t r a l energy source i t s e l f , i t could l a s t 10 s years (Chapter I I ) , which i s j u s t the estimated S e y f e r t l i f e t i m e ( H o l t j e r , 1959). However the energy f l u x from the source would s u r e l y decrease as the mass was d e p l e t e d , which might reduce the a c t i v e l i f e t i m e . A more s a t i s f y i n g theory i s that the mass c o l l e c t s i n the c e n t e r a f t e r being l o s t from the g a l a c t i c s t a r s . There i s evidence f o r a dense s t e l l a r system i n the c e n t e r of S e y f e r t n u c l e i , as d i s c u s s e d e a r l i e r . The d e n s i t i e s i n v o l v e d i f 10 8 MQ i n s t a r s of normal mass e x i s t w i t h i n a volume of the order of a p c 3 a t the c e n t e r w i l l r e s u l t i n very f r e q u e n t ' c o l l i s i o n s , as w e l l as a r a p i d g r a v i t a t i o n a l e v o l u t i o n of the c l u s t e r . Such s t e l l a r systems have been c o n s i d e r e d o f t e n i n the l i t e r a t u r e ( S p i t z e r and Saslaw, 1966, S p i t z e r and Stone, 1967, Sanders 1973, Shara and Shaviv, 1974, H i l l s , 1975). Est i m a t e s of the mass or energy r e l e a s e d i n the e v o l u t i o n of such a system depend c r i t i c a l l y on the unsolved problem of the dynamics of a two-star c o l l i s i o n . Recent e s t i m a t e s range from about 0.05 M G, 162 per year (Shara and Shaviv, from white dwarf-^M s t a r c o l l i s i o n s ) to 10 H& per year (Sanders, from c o l l i s i o n induced supernovae). A gas d e p o s i t i o n r a t e as low as 1 0 - 2 M@ per y e a r , i f c o n t i n u o u s , can s a t i s f y the requirements of the model i f the energy source i s on f o r one per cent of the time r a t h e r than c o n t i n u a l l y . In t h i s case the mass l o s s from s t a r s w i l l r e p l e n i s h the r e g i o n while the energy source i s q u i e t . Thus, the mass l o s s does not seem to be a problem. The o n - o f f c y c l e suggested f o r the c e n t r a l source does not i n v o l v e the same t i m e s c a l e s as the v a r i a t i o n s observed i n many of these o b j e c t s . These v a r i a t i o n s o f t e n i n v o l v e a ten to f i f t y per cent change above a steady base of r a d i o or o p t i c a l f l u x , over t i m e s c a l e s of a year or l e s s . (See Peterson and Dent, 1973 f o r 3C 273; or S e i e l s t a d , 1974 f o r 3C 120.) The c a l c u l a t i o n s here, on the other hand, assume the c e n t r a l source s t a y s on f o r the c l o u d c r o s s i n g time, with a p o s s i b l e " o f f " c y c l e of up to one hundred times the c r o s s i n g time. (If the energy source stayed on a f t e r the gas had been blown away, and i f i t were st r o n g enough to prevent accumulation of new gas to v i s i b l e d e n s i t i e s , the nucleus might appear as a continuum o b j e c t . ) The t o t a l n u c l e a r cosmic ray energy c o n t e n t , based on the p a r t i c l e s ( r e l a t i v i s t i c : t y p i c a l l y 100 MeV to 1 GeV) r e s p o n s i b l e f o r the r a d i o s y n c h r o t r o n source, i s probably 1 0 5 3 to 1 0 s 7 e r g s , as d i s c u s s e d e a r l i e r . I f t h i s i s a l s o the energy c o n t a i n e d i n the s o f t cosmic rays (1 MeV), the energy f l u x i n the models w i l l d e p l e t e the source i n about 300 y e a r s . Thus a p a r t i c l e a c c e l e r a t i o n mechanism i s needed. T h i s i s emphasized by the s y n c h r o t r o n l o s s e s of the p a r t i c l e s . Assuming a magnetic f i e l d 163 of 1 0 - 1 gauss, the 1 MeV cosmic rays have a s y n c h r o t r o n l i f e t i m e o f 1 0 1 1 sec (longer than t h e i r i o n i z a t i o n / C o u l o m b l i f e t i m e , however) and the hot (1 GeV) p a r t i c l e s l a s t only 10 9 sec. The con t i n u e d e x i s t e n c e of the r a d i o source i s an argument f o r i n - s o u r c e a c c e l e r a t i o n by some p r o c e s s . The net energy e x p e n d i t u r e of cosmic rays i n our models over 10 8 years i s 1 0 6 l r 0 to 1 0 6 5 r o e r g s . The mass l o s t from the nucleus i n 1 MeV cosmic r a y s , based on the f l u x e s i n Table V II, i s J = 0.6 ( ZETA / 1 0 6 ) r t E o s o l a r masses per year. T h i s ranges from 0.01r o to 100r 0 M 0 per year f o r the c a l c u l t e d models, which i s comparable t o the mass l o s s r a t e i n f e r r e d from the cl o u d motions. The s o f t e r cosmic rays (about h a l f the t o t a l mass f l u x ) t r a v e l with the s h e l l , where they have l o s t most of t h e i r energy, so the s h e l l mass w i l l be doubled i n a few dynamic t i m e s c a l e s . With t h i s high a cosmic ray mass d e n s i t y , the problem i s more p r o p e r l y t r e a t e d as a two f l u i d c a l c u l a t i o n . 1 O b s e r v a b i l i t y The important c h a r a c t e r i s t i c s of the o p t i c a l spectrum are the d i f f e r e n c e s between the per m i t t e d hydrogen l i n e s , which can have very broad wings, and the f o r b i d d e n l i n e s , which are sometimes of the same width as the Balmer l i n e s and sometimes nartower. S i n c e the f o r b i d d e n l i n e s are quenched a t d e n s i t i e s 1 I t should be noted that the charqe exchange model of Ptak and Stoner (1973) r e g u i r e s A0Z M 0 per year i n 200 keV cosmic rays to e x p l a i n the broad Ho<. wings i n three S e y f e r t g a l a x i e s (Katz, 1975), although the e f f e c t of t h i s p a r t i c l e f l u x on the gas i s not c o n s i d e r e d i n any d e t a i l . 164 above 10 6 c u r - 3 , r o u g h l y , the two emission systems may a r i s e i n p h y s i c a l l y s eparate p a r t s of the gas. In the c o o l mode models, t h i s s e p a r a t i o n c o u l d a r i s e i f the Balmer l i n e s come from the s h e l l or the c l o u d s i n t o which i t fragments, and the nebular l i n e s a r i s e from t h e / o u t e r l a y e r s of y gas. The outer gas i s maintained a t 10* K (see'' Chapter V) and a moderate s t a t e of i o n i z a t i o n by the high energy t a i l of the cosmic ray spectrum. MacAlpine (1974) has poi n t e d out t h a t secondary Lyman r a d i a t i o n from the s h e l l w i l l a l s o heat the outer gas, i n t h i s geometry. The Balmer l i n e s , i n t h i s c ase, w i l l have a width g i v e n by the s h e l l v e l o c i t y , t h at i s , up to s e v e r a l thousand km s - 1 . The nebular l i n e s w i l l be much narrower s i n c e the gas past the s h e l l or c l o u d s i s n e a r l y s t a t i o n a r y . Another source of low d e n s i t y gas (n**106) i s the fragments of the dense s h e l l . By the time they have reached l a r g e d i s t a n c e s from the n u c l e u s , they w i l l have expanded to below the quenching d e n s i t y . I f these are the main nebular l i n e source, the l i n e widths w i l l be comparable to the permitted l i n e widths, s i n c e i n t h i s case both l i n e systems come f roin . the same matter. T h i s model does not e x p l a i n the f a c t t h a t i n n u c l e i where the two l i n e systems are of s i m i l a r widths, those l i n e widths are f a i r l y low. ' The nebular l i n e r a t i o s i n S e y f e r t n u c l e i i n d i c a t e a space-averaged temperature of 10* K (Burbidge, 1970), and an average d e n s i t y of only 10* cm - 3. (This method of determining (n^, T ) as given by Seaton (1960), f o r i n s t a n c e , tends to weight low d e n s i t y and low temperature areas.) T h i s may be 165 lower than d e n s i t i e s t h a t s h e l l fragments would reach i n a reasonable time upon escaping the nucleus, which would f a v o r the f i r s t i n t e r p r e t a t i o n given above. The low d e n s i t y gas o u t s i d e of the s h e l l c o u l d s a t i s f y these o b s e r v a t i o n s . The mechanism proposed by Ptak and Stoner to produce the broad hydrogen l i n e wings i n v o l v e s charge t r a n s f e r r e a c t i o n s between the ambient gas and the streaming p r o t o n s . T h i s process produces broad asymmetric wings on the hydrogen l i n e s our to 10,000 km s - 1 . The s t r e n g t h of the wings depends on the t o t a l cosmic ray number f l u x which i s reduced by three o r d e r s of magnitude i n our models compared to those of Ptak and Stoner (Katz, 1975). The energy dependence of the charge t r a n s f e r c r o s s s e c t i o n (wittkower, Rydiny and G i l b o d y , 1966) a l s o a f f e c t s the r e s u l t a n t l i n e p r o f i l e s . The c r o s s s e c t i o n drops s h a r p l y above 200 keV. A l l but one of the models c o n s i d e r e d here have an i n i t i a l low energy c u t o f f above t h i s v a l u e . The dense s h e l l ( i g n o r i n g fragmentation) w i l l r a d i a t e i n the o p t i c a l r e g i o n . P r e d i c t i o n of the d e t a i l e d spectrum would r e q u i r e e x t e n s i v e i o n i z a t i o n c a l c u l a t i o n s . An e s t i m a t e , however, can be made from the t o t a l c o o l i n g curve of F i g u r e 1, s i n c e the main emission i s i n o p t i c a l l i n e s when 10* K< T < 10 s K. T h i s i s the r e g i o n of the s h e l l temperatures. The l u m i n o s i t y of the s h e l l i s ^ 4vr 2&r /\ (T) n en^, i f r i s the'width of the s h e l l . For Model 1 ( c f . Table VIII) t h i s g i v e s L 0 ^ r ^ 1x10* 2r c ergs s _ 1 . T h i s i s lower than the o p t i c a l l u m i n o s i t y of S e y f e r t n u c l e i , which i s expected s i n c e the s h e l l does not c o n t a i n the mass necessary to produce the o p t i c a l l u m i n o s i t y . Model 7 g i v e s 6 x 1 0 * 7 r a ergs s _ l . T h i s i s 166 I somewhat high e r than t y p i c a l quasar l u m i n o s i t y e s t i m a t e s . The hot phases of the gas w i l l not produce the us u a l o p t i c a l l i n e s , s i n c e a l l normally observed elements w i l l be t o t a l l y i o n i z e d a t 10 s K. T h i s means the "hot mode" models w i l l not have the us u a l l i n e spectrum, u n l e s s the i o n i z e d r e g i o n i s i o n i z a t i o n bounded, and the gas o u t s i d e of the i o n i z a t i o n f r o n t i s v i s i b l e . However the hot gas w i l l be a s t r o n g thermal X-ray s o u r c e . The net volume e m i s s i v i t y f o r f r e e - f r e e r a d i a t i o n ( S p i t z e r , 1967), j = 2x1 0 ~ 2 7 TO- 5 n 6 n p ergs cm" 3 s ~ 1 (70) combined with the c a v i t y volume, ( 4 ^ / 3 ) r 3 , g i v e s Lx = 9x10 + + ( CFLAM| 2 p 2 r 3 T ° . s r c ergs s ~ l (71) = 2.3x1029 ( n 0 r o ) 2 T ° - 5 p 2 r c 3 r o ergs s ~ i where p and r t are now s c a l e d i n terms of n, and r c , n c i s i n cm - 3, and r 0 i s i n pc. Table VIII uses the temperature a t t a i n e d when each model c a l c u l a t i o n was stopped t o compute Lx. I f the gas has reached the e q u i l i b r i u m temperature, t h i s becomes (using equation 63 ), Lx (eg) = 3x10*2 (72) = 1 x 10 3 2 (n r ) 7/'»Fi/+H-1/*r3/* ergs s ~ i i f i s the cosmic ray f l u x i n u n i t s of 10+* ergs s ~ 1 and f 2 r t 3 = 1 0 — 3 . I t may be, i n view of the slow h e a t i n g r a t e s obtained i n the c a l c u l a t i o n s . , t h a t the s h e l l w i l l escape the nucleus before t h i s e q u i l i b r i u m temperature i s a t t a i n e d . Some values of Lx f o r s p e c i f i c models are qiven i n Table IX, alonq with l i n e widths and observed f l u x e s of o b j e c t s where 167 an X-ray f l u x has been measured. The c h a r a c t e r i s t i c \ temperatures, Tx, are estimated from the best f i t of a thermal bremsstrahlung spectrum to the data. Table IX Comparison Of Models With O b s e r v a t i o n s | NGC 1275 y Lx=3x10** (a) Tx ---108 (f) hv ~300 0 (H + neb) NGC 4 151 | NGC 106 8 Lx=8x10*2 (e) 1X10*2 (fc) Tx-1-3x10 a (g) 6v-v800 (neb) -0(103) (H) Lx<3x10*2 ( C ) i\v~1 300 (neb) 3C 273 Lx=4x10** (fl) Tx - 3x107 -3x108 (g) Av~0 (103) (em) •oO(10*) (abs +-Model 3 Model 2 Model 2 Model 7 +- +-Lx=9x10* 3r c v=2000 T=1 0« .. Lx=5x10* i r c v=1000 1=3x10? Lx=5x10*»r„ v = 1000 T=3x107 Lx=5x10*5r c v=8000 1=10 8 (a) F r i t z et a l . , 1971 (b) K e l l o g g 1973 (c) K e l l o g g e t a l . , 1971 (d) Bowyer et a l . , 1970 (e) Bergeron and S a l p e t e r 1973 (f) Lea et a l . , 1973, Davidsen et a l . , 1975 (g) .Margon e t a l . , 1975 B a i t y et a l . , (1975) have observed the X-ray spectrum of NGC 4151 from 7 to 110 keV. They suggest a power law spectrum 168 but s t a t e t h a t a 10 9 K thermal spectrum i s a l s o p o s s i b l e . Jlargon et a l . , (1975) based on "y,2 f i t s of both power law and bremsstrahlung (around 10 s K) s p e c t r a t o t h e i r 1-5 keV d a t a , conclude n e i t h e r can be r u l e d out. The same authors conclude the X-ray spectrum of 3C 273 i s probably thermal. The X-ray emission i n the d i r e c t i o n of NGC 1275 (which Davidsen e t a l . , 1975, conclude i s thermal at 10 8 K) i n c l u d e s the Perseus c l u s t e r i n t r a c l u s t e r e m i s s i o n , and t h e r e f o r e i s a high upper l i m i t to the f l u x from NGC 1275 i t s e l f -The s p e c t r a of NGC 4 151, the Perseus c l u s t e r and 3C 273 seem to turn over at a few keV. Sargent (1973) suggests these t u r n o v e r s are due to s e l f a b s o r p t i o n . C a l c u l a t i o n s of the hydrogen and helium X-ray o p a c i t y (Brown, 1971) i n d i c a t e a column d e n s i t y of 1 0 2 6 cm - 2 i s necessary to produce u n i t o p t i c a l depth at 10 keV; the V - 3 dependence of the c r o s s s e c t i o n reduces t h i s to 1 0 2 3 c m - 2 r e q u i r e d at 1 keV. I f the n u c l e a r m a t e r i a l or s h e l l fraqments do not have too s m a l l a f i l l i n g f a c t o r , t h i s column d e n s i t y may be obtained i n our models. On the other hand, Davidsen et a l . , (1975) d e r i v e a column d e n s i t y of 1 0 2 1 cm- 2 f o r the Perseus c l u s t e r ; and Kargon et a l . , (1975) say t h a t a b s o r p t i o n i s not needed i n t h e i r s p e c t r a l f i t s . We see from t a b l e IX t h a t the c a l c u l a t e d models are not i n c o n s i s t e n t with the o b s e r v a t i o n s t h a t e x i s t so f a r . We emphasize that the model parameters were not chosen to match s p e c i f i c o b j e c t s , but simply by c o n s i d e r a t i o n of the Standard A c t i v e Nucleus of Chapter I I . In view of the t r i p l y i n f i n i t e set of important parameter c h o i c e s , t h i s minimal agreement can be c o n s i d e r e d very f o r t u n a t e . 169 L i n e p r o f i l e s i n NGC 4151 and NGC 1068, the two p r o t o t y p i c a l broad and narrow l i n e S e y f e r t g a l a x i e s , (and the presence of c o r o n a l l i n e s i n NGC 4151 but not i n other S e y f e r t s ) r e g u i r e an e x p l a n a t i o n o u t s i d e of t h i s simple model, such as a s t r o n g e r low energy proton f l u x i n NGC 4151- Nor i s the d i f f e r e n c e i n a b s o r p t i o n and emission l i n e systems, so c h a r a c t e r i s t i c of quasars, simply e x p l a i n e d - The simple p r e d i c t i o n of t h i s model that o b j e c t s with higher v e l o c i t y d i s p e r s i o n s w i l l a l s o have higher X-ray l u m i n o s i t y ( s i n c e both are c o r r e l a t e d with the i n p u t cosmic ray f l u x ) seems from the t a b l e to be upheld, however. The reasonable agreement of these models with o b s e r v a t i o n can only be taken as a c o n s i s t e n c y check, not a v e r i f i c a t i o n , because the models are f a r from a complete p i c t u r e of the p h y s i c s of the n u c l e a r r e g i o n . The l a c k of d i s c u s s i o n of the o t h e r s p e c t r a l bands, the o p t i c a l continuum, the other p o s s i b l e s o u r c e s of the X-ray f l u x , such as i n v e r s e Compton r a d i a t i o n , keeps the model from completeness, even w i t h i n the narrow bounds d e f i n e d at the s t a r t of the t h e s i s . 170 CHAPTER VII CONCLUDING 3EMA3KS The time dependent numerical hydrodynamic c a l c u l a t i o n s i n t h i s t h e s i s have confirmed the r e s u l t s of the s i m p l e r d i s c u s s i o n / based on cloud dynamics -- that a c e n t r a l source of s o f t cosmic / r a y s w i t h i n a c l o u d of dense gas w i l l a c c e l e r a t e that gas up to v e l o c i t i e s of s e v e r a l thousand km s _ 1 . A c e n t r a l X-ray source w i l l have a much weaker e f f e c t , due to the t r a n s p a r e n c y o f the i o n i z e d gas to photons. S e v e r a l n u c l e a r models were c a l c u l a t e d , u s i n g d i f f e r e n t p h y s i c a l parameters. A l l of the models where the e q u i l i b r i u m temperature of the gas i s i n the range 10+-10 5 K developed a dense s h e l l ( s u s c e p t i b l e to R a y l e i g h - T a y l o r fragmentation) which reached a high outward v e l o c i t y . The nature of the model e v o l u t i o n i s not c r i t i c a l l y dependent on the p h y s i c a l parameters, which are the proton energy, the t o t a l cosmic ray f l u x , the gas d e n s i t y and the c e n t r a l mass. A range of these parameters w i l l reproduce s i m i l a r s o l u t i o n s , with a s h e l l v e l o c i t y of 10 3 to 10 4 km s _ 1 (the d i s c r e t e c l o u d c a l c u l a t i o n s of Chapter I I i n d i c a t e even h i g h e r v e l o c i t i e s , near 10 s km s - 1 ) . The s h e l l maintains a temperature of 10+ t o 10 s K and l e a v e s a h o t t e r c a v i t y behind at about 10 8 K. The most important parameters i n terms of the numerical e v o l u t i o n are the r a t i o of cosmic ray d e n s i t y to gas d e n s i t y , which determines the temperature of the gas due to the (cosmic ray h e a t i n g - r a d i a t v e c o o l i n g ) energy b a l a n c e , and the t o t a l cosmic ray energy, which a f f e c t s the gas v e l o c i t y . T h i s type of model has i n t e r e s t i n g i m p l i c a t i o n s f o r 171 o b s e r v a t i o n s of a c t i v e n u c l e i . The hot, t h i n gas which r e s u l t s from the strong cosmic ray source nearby w i l l be a thermal X-ray source of observable s t r e n g t h . The X-ray power should c o r r e l a t e with the s h e l l v e l o c i t y , which may be observed as a c l o u d v e l o c i t y , t a k i n g i n t o account the f r a g m e n t a t i o n / o f the s h e l l - c a v i t y i n t e r f a c e under the Rayleigh-Taylo'r i n s t a b i l i t y . The d i s t i n c t i o n between the source r e g i o n s of the p e r m i t t e d and f o r b i d d e n o p t i c a l e mission l i n e s may be the d i s t i n c t i o n between the dense s h e l l , which i s above the quenching d e n s i t y , and the outer gas; or i t may depend on the expansion r a t e of c l o u d s as they l e a v e the n u c l e u s . The energy i n v o l v e d i n the cosmic rays i s c o n s i s t e n t with e s t i m a t e s of the t o t a l p a r t i c l e energy r e q u i r e d f o r the s y n c h r o t r o n emission seen, although p a r t i c l e s of d i f f e r e n t e n e r g i e s are i m p l i e d f o r the two p r o c e s s e s . The most i n t e r e s t i n g i m p l i c a t i o n i s t h a t such a model only J-l a s t s f o r a few thousand years, by which time the gas i s o u t s i d e of the n u c l e a r r e g i o n , whose s i z e i s known from v a r i a b i l i t y and d i r e c t r e s o l u t i o n . To maintain the S e y f e r t nucleus or quasar long enough to be o b s e r v a b l e , then, r e q u i r e s a r e p e t i t i v e mechanism; perhaps one where the energy source a l t e r n a t e l y t u r n s on and o f f , being o f f f o r as long or longer than i t was on- The q u i e s c e n t phase w i l l allow accumulation of new gas i n the c e n t r a l r e g i o n s from "s t a n d a r d " processes such as s t e l l a r c o l l i s i o n s , without r e q u i r i n g more e x o t i c mass sou r c e s . One important e x t e n s i o n of t h i s work i s to i n v e s t i g a t e the e f f e c t s of the magnetic f i e l d s which we know must be present i n these sources. C u r r e n t e s t i m a t e s are t h a t the f i e l d i s below the e g u i p a r t i t i o n v a l u e , and t h e r e f o r e that the gas dynamics 172 dominates the f l o w . Even i n t h i s regime, the proton i n t e r a c t i o n with the f i e l d v i a g e n e r a t i o n of magnetohydrodynamic waves (I p a v i c h , 1975) i n c r e a s e s the cosmic ray-gas c o u p l i n g (although the models c a l c u l a t e d by I p a v i c h i n v o l v e s i t u a t i o n s where the magnetic energy d e n s i t y g r e a t l y exceeds the cosmic ray energy d e n s i t y ) . T h i s h i n t s t h a t a lower i n p u t cosmic ray f l u x w i l l s t i l l produce gas v e l o c i t i e s as high as observed. T h i s has the v i r t u e of s o f t e n i n g the X-ray emission from the hot gas, which might b e t t e r agree with o b s e r v a t i o n s . R a d i a t i v e t r a n s f e r w i l l a l s o be important. Under 10 s K, much of the c o o l i n g r a d i a t i o n has a path l e n g t h of l e s s than a parsec i n dense gas. The r a d i a t i o n from the s h e l l w i l l c r e a t e an e x t r a outward pressure on the outer r e g i o n s where i t i s absorbed, and the o v e r a l l energy balance of the gas w i l l change. In summary, t h i s work prese n t s a f i r s t attempt at i n v e s t i g a t i n g the dynamical nature of a c t i v e g a l a c t i c n u c l e i . The e f f e c t i v e n e s s of cosmic ray a c c e l e r a t i o n of the gas through Coulomb i n t e r a c t i o n s has been demonstrated. The demands of s i m p l i c i t y and l a c k of d e t a i l e d s t r u c t u r a l knowledge of these o b j e c t s n e c e s s i t a t e d many s i n s of omission i n the c a l c u l a t e d models. 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No equation number i s given f o r standard p h y s i c a l c o n s t a n t s and f o r symbols used o n l y i n the body of the t e x t . a = s i z e of c l o u d (3a) a 6 = i n i t i a l c l o u d s i z e (11) a = s i z e at which i n i t i a l l y t h i c k c l o u d becomes t h i n (11) b (E) = cosmic ray energy l o s s r a t e (21) B = magnetic f i e l d c = speed of l i g h t c s = speed of sound (10) C (T) = e l e c t r o n c o l l i s i o n a l i o n i z a t i o n r a t e (32) e '- i n t e r n a l energy per gram (33a) ; normalized p a r t i c l e energy (in appendix I) E = energy of s i n g l e cosmic ray (13) E c = low energy s p e c t r a l c u t o f f (19); i n i t i a l p a r t i c l e energy (Chapter I I ; 13a) E, = energy a t which cosmic ray spectrum bends (22); i n i t i a l p a r t i c l e energy (Appendix I) E , <E^> = net heating per i o n i z a t i o n due to X-rays (23) <AE> = average energy t r a n s f e r per cosmic ray i o n i z a t i o n ( 3 1 ) E c^ = t o t a l energy i n cosmic rays f = e f f i c i e n c y of cosmic ray energy t r a n s f e r F = e f f i c i e n c y of h e a t i n g due to i o n i z a t i o n (21) FcR. 'Y = cosmic ray (X-ray) i n p u t energy f l u x , ergs s~* (22,24) G = constant of g r a v i t a t i o n 181 G (r) = net f o r c e on gas (57) h, -h = Planck's constant. H = h e a t i n g r a t e (21) 1^ = i o n i z a t i o n energy of hydrogen y j = p a r t i c l e number f l u x , number cm - 2 s - 1 (19) ( J = p a r t i c l e mass f l u x , M y r _ l (Chapter VI) • L = i o n i z i n g i n p u t l u m i n o s i t y , ergs s ~ 1 (Chapter I I ; 1) £ = net c o o l i n g r a t e per gram (64) Lx = X-ray l u m i n o s i t y from gas, ergs s - 1 (71) 1 T = emission from s h e l l , ergs s - 1 (Chapter VI) m = mean atomic mass (5) me = e l e c t r o n mass nip = proton mass M = c e n t r a l mass (5) M = c l o u d mass (Chapter I I ; 5); c e n t r a l mass (Chapter IV; 29) n,n^ = number d e n s i t y of the gas (passim) rig = e l e c t r o n number d e n s i t y (32) n^ = n e u t r a l hydrogen number d e n s i t y (32) N (E) = proton energy spectrum (1.8) N (r) , N (t) = column d e n s i t y of gas (1-7) p = gas pressure (28) r = r a d i u s (passim) r ^ T, R 4 t = s t a r t i n g d i s t a n c e of cloud (7) r = d i s t a n c e at which i n i t i a l l y t h i n c l o u d becomes t h i c k (7) r c = r a d i u s of hot c a v i t y (71) r o = c h a r a c t e r i s t i c r a d i u s of OER (Chapters I I I - VI; 22) r = i n n e r r a d i u s of OER (25) R = range of cosmic ray 182 R «wy = P ° ^ n t °f maximum gas d e n s i t y (Chapter V) s(ft) = s l o w l y v a r y i n g p a r t of c r o s s s e c t i o n f o r i o n i z a t i o n (20) (^> = v/c here only) t ^ = dynamical time s c a l e = r e / v (66) t ^ = thermal i n s t a b i l i t y growth time (66) t ^ T = R a y l e i g h - T a y l o r i n s t a b i l i t y growth time (69) T = temperature (passim) u = gas or clo u d v e l o c i t y (passim) u^ = t e r m i n a l c l o u d v e l o c i t y (11) v = cl o u d or gas v e l o c i t y (passim; v - u/c s i n Chapter I I ; 10) v*K\y = hi g h e s t s h e l l v e l o c i t y (Chapter V) v 0 = cosmic ray v e l o c i t y with r e s p e c t to c l o u d (16) v^ , = r e s t frame cosmic ray v e l o c i t y (15) x = f r a c t i o n a l i o n i z a t i o n , n e /n (1) y "'= I H / k T (32b); v o 3 / V 3 = (normalized f r e q u e n c y ) - 3 (1-15) z = r e d s h i f t ; d i s t a n c e penetrated i n s i d e c l o u d (13) Z = normalized column d e n s i t y to cosmic rays (1.12) Zjc ,= normalized column - d e n s i t y t o X-rays (1.15) <>(#^ (T) — recombination c o e f f i c i e n t (1,32) s p e c t r a l index f o r f l u x d e n s i t y (19) . ^= s p e c t r a l index f o r number d e n s i t y (1.8) o(', '^ = numerical 0(1) co n s t a n t s i n F('J) c o n v e r s i o n (22,24) X= a d i a b a t i c exponent 1 = i o n i z a t i o n r a t e per hydrogen atom (19) X= p e r t u r b a t i o n s c a l e i n i n s t a b i l i t y a n a l y s i s (67) /\ (T) = net r a d i a t i v e c o o l i n g (passim) l) = frequency (Hz) <iSTT> = average momentum t r a n s f e r per i o n i z a t i o n (29) 183 ^ = mass d e n s i t y (27) ^ ~^-»<T/T(E) = i o n i z a t i o n or " c o u p l i n g " c r o s s s e c t i o n (1,20) 'V = o p t i c a l depth (3a) / iO = i n s t a b i l i t y growth r a t e (65) ^ "Q., 0-(r) , H(T) = a t t e n u a t i o n of cosmic ray / i o n i z a t i o n r a t e (25a,b) 1 TERMS ARISING FROM THE FINITE DIFFERENCE SCHEME s u p e r s c r i p t s ( f o r i n s t a n c e , u*) i n d i c a t e p o i n t s of the time g r i d ; s u b s c r i p t s (u- v (J i n d i c a t e space g r i d p o i n t s . AMS = = GM c/r 0u 2, (34) C = c o e f f i c i e n t of (<ip/Sp) i n s o l u t i o n a l g o r i t h m (44) ^ = normalized e l e c t r o n i o n i z a t i o n r a t e (34) CFLAM = A / A (T) = (> cr 0/m 2u3 ( 3 4 ) e c = i n t e r n a l energy s c a l e = u 2 (34) g = e r r o r i n c o n t i n u i t y equation (50) G (r) = e x t e r n a l f o r c e per gram (29) \^  = normalized h e a t i n g r a t e (34) A = normalized c o o l i n g r a t e (34) \{^= normalized c e n t r a l mass (34) N = number of space g r i d p o i n t s P 0 = pressure s c a l e •= ,p0u2/Y (34) p = f i r s t s o l u t i o n f o r p^+i (45) q = a r t i f i c a l v i s c o s i t y (40) r 0 = d i s t a n c e s c a l e (34) $>r = s i z e of space g r i d (41) R(p,r) = net energy s o u r c e per gram (31) t Q = t ime s c a l e = r o / u c ( 3 4 ) *6t - s i z e of t i m e s t e p (4 1) u 0 = v e l o c i t y s c a l e (34) ZETA = f = [ r c / u D (34) ZETAX = 5 = ^ . r c / u c ( 5 9 ) ^ - n o r m a l i z e d r e c o m b i n a t i o n r a t e (34) % - n o r m a l i z e d i o n i z a t i o n r a t e (34) © = time c e n t e r i n g r a t i o i n c o n t i n u i t y e q u a t i o n (43) X - a r t i f i c i a l momentum d i f f u s i o n c o e f f i c i e n t (35) = n o r m a l i z e d momentum t r a n s f e r r a t e (34) = d e n s i t y s c a l e (34) T' = a r t i f i c i a l mass d i f f u s i o n c o e f f i c i e n t (40) 0 = time c e n t e r i n q r a t i o i n momentum e q u a t i o n (4 1) 185 APPENDIX I SPATIAL DEPENDENCE OF THE IONIZING FLUX E n e r g e t i c p a r t i c l e s of energy E pas s i n g through a gas l o s e energy by i o n i z a t i o n o f , and by Coulomb i n t e r a c t i o n with, the gas, a t a r a t e p r o p o r t i o n a l to £ - ° - 5 . T h i s means a m o n o t o n i c a l l y d e c r e a s i n g p a r t i c l e spectrum w i l l e v olve t o a lower magnitude slope upon i n t e r a c t i o n with the ambient gas. To e v a l u a t e the i o n i z a t i o n r a t e of a hydrogen atom, then, at each p o i n t of the g r i d at each time s t e p of the numerical c a l c u l a t i o n r e q u i r e s an i n t e r g r a l of j ( E , r ) over energy — see e g u a t i o n (1.10) -- and the i m p l i e d i n t e g r a l over r a d i a l d i s t a n c e i n v o l v e d i n the N (r) term. E v a l u a t i n g the s p a t i a l a t t e n u a t i o n of an X-ray spectrum i n v o l v e s two s i m i l a r i n t e g r a l s . T h e r e f o r e , to save the computer time i n what was a long c a l c u l a t i o n anyway, I i n v e s t i g a t e d whether the s p a t i a l dependences of these i o n i z a t i o n r a t e s c o u l d be r e p r e s e n t e d by some a n a l y t i c f u n c t i o n of the column d e n s i t y , N (r) •= ^  n (r) dr. T h i s i s the important p h y s i c a l q u a n t i t y i n the problem. I t t u r n s out that the n o r m a l i z a t i o n of N (r) i n v o l v e s the low-energy s p e c t r a l c u t o f f . The a n a l y t i c e x p r e s s i o n i n both c a s e s , photons and cosmic ray p a r t i c l e s , was d e r i v e d by numerical e v a l u a t i o n and t r i a l and e r r o r f i t t i n g . Cosmic Rays The energy l o s s of an e n e r g e t i c e l e c t r o n or proton p a s s i n g through a t o t a l l y i o n i z e d gas i s d i s c u s s e d by Arons, McCray and S i l k (1971). They show that the main l o s s e s f o r e l e c t r o n s of energy l e s s than 1 BeV, and f o r protons below 10 BeV, are 186 Coulomb l o s s e s . For a p a r t i a l l y i o n i z e d gas, i o n i z a t i o n l o s s e s must a l s o be c o n s i d e r e d . The t o t a l l o s s e s , Coulomb p l u s i o n i z a t i o n , of a proton with v e l o c i t y v i n a gas with kT'<< m^2/2 can be expressed (sea T s y t o v i c h , 1962, a l s o Goldsmith et a l . , 1969) (1.1) dE UlTe4 [ n w In 4E m e + n e l n j£E_ dt mev L m^, -fiiO where m^ , m^  are the e l e c t r o n and proton masses, i s the plasma frequency, ^ 2 = 4Te 2n e/m e , n £ and n U r are the e l e c t r o n and n e u t r a l hydrogen number d e n s i t i e s , and AE i s the average energy l o s s per i o n i z a t i o n , given by Dalgarno and McCray (1970) as 27 eV i n a gas with n e/n = 0.1. The term i n p a r e n t h e s i s can be evaluated i n terms of InE and x=n c/n (s i n c e ne+na-;=n, and n W T/n=1-x). The equation becomes (cgs u n i t s ) dE = -6. 7x10~ 2 2E-°<- 5 n i t *(19.7 + lnE + x (24.0+1.21n (10"&nx))) % - 1 . 6x10- 2°nE~O's (x + 0. 3) ergs s-» (1-2) the approximate form holds f o r E ^ 1 MeV and n 10 6 cm - 3. Since x has a value between zero and one, equation 1.2 i s r e l a t i v e l y i n s e n s i t i v e to the value of x. T h i s p r o v i d e s an e s t i m a t e of the range o f a cosmic r a y , s i n c e dE/dr = 1/v dE/dt, one has dE ^ 1 . 5 x l 0 ~ 3 2 n (x + 0.3) ergs cm- 1 (1.3) dr " " E o r , f o r the range of the p a r t i c l e , with c o n s t a n t n and assuming (x + 0. 3) * 1, R K, 6 . 7 x 1 0 3 i E 2 / n . For a proton with energy 1 MeV (the lower energy p a r t i c l e s are the most e f f i c i e n t i n heating) the range i s only 50/n pc. T h e r e f o r e d e p l e t i o n of the i o n i z i n g f l u x by energy l o s s e s w i l l 187 be important. \ In order t o e v a l u a t e the change i n the i o n i z a t i o n r a t e due to t h i s a t t r i t i o n , we must assume an i n i t i a l form f o r the spectrum. A power law form, N (E) = n oE~(* , with/4 low-energy c u t o f f , E 0 , resembles the g a l a c t i c cosmic ray spectrum / ( l i n g e n f e l t e r , 1973) , although i t probably d i s t o r t s the lower energy end. T h e . k i n e t i c equation f o r a p a r t i c l e spectrum with energy-dependent l o s s e s , b ( E ) , and no s o u r c e s , i s given by Kaplan (1956): (E) -f ^_(N b (E) ) = 0. (1.4) ^ t ^ £ A handy s o l u t i o n i s o b t a i n e d by analogy with the c o n t i n u i t y e g u a t i o n , 3f + A(f v) = df + p^v = 0 (1.5) which has the "Lagrangian" s o l u t i o n , . ^ r / ^ R = ^ 0 . T h i s i s shown as f o l l o w s . Let R be the i n i t i a l c o o r d i n a t e of the Lagrangian p o i n t , R = r (t = 0). E u l e r ' s theorem shows t h a t , i f J = 3r/^B, dJ = J^y; ' d t *r therefor.e (1.5) becomes, a f t e r m u l t i p l y i n g by J , J dp + pJ>y = J dp + p dJ = 0. At \ or 5x ^ cVt That i s , J = c o n s t a n t . T h e r e f o r e , e q u a t i o n (1.4) has the s o l u t i o n 188 N(E,t) = N(E,0) dE,/dE (1.6) i f E, = E (t=0) i s the Lagrangian c o o r d i n a t e . But dE ( /dE can be found from the approximate form f o r b (E) , namely, (1.2); i n t e g r a t i o n g i v e s E i o S - E i . 5 _ 1.6x10 - 2 0 ^ n dt (1-7) o = E|- 5 - 1. 6x10 - 2 0 N ( t ) N (t) can be thought of as a column d e n s i t y seen by the p a r t i c l e i n time t . The e v o l u t i o n of the p a r t i c l e spectrum, a f t e r combining e v e r y t h i n g , and remembering t h a t N(E,0) = n^E~^, i s given by N(E,t) = n E 0 * 5 ( E i . s + 1.6x10 - 2 0 N (t) )"«20 + i ) / 3 (1.8) o T h i s can be converted to a f l u x by m u l t i p l y i n g by a v e l o c i t y : j ( E , t ) = n o p / n ? r E ( E L S . + 1. 6 x 1 0 - 2 o N ( t ) ) - c 2 p i ) / 3 (1.9) For a p a r t i c l e of energy E, the time s i n c e i t s i n j e c t i o n can be converted to d i s t a n c e , with dt = \J ra^/21 E ~ 0 * 5 d r ; t a k i n g 2-o.s o u t s i d e the i n t e g r a l (which i s not s t r i c t l y a c curate) , the i n t e g r a l i s now the u s u a l column d e n s i t y , N (r) = ^ n (r) dr. Here, r, i s the r a d i u s at which the p a r t i c l e beam f i r s t h i t s the c l o u d (at t=0). The f l u x becomes j ( E , r ) = n0J27mJ! E ^ p * ^ / * d-10) [ E 2 + 1 . 6 x 1 0 - 2 0 ^ / 2 ' N(r) ] C 2 p + o / 3 Then, assuming continuous i n j e c t i o n of p a r t i c l e s at the source, i n t e g r a t i n g t h i s f l u x over energy w i l l g i v e the t o t a l f l u x seen at a d i s t a n c e r from the source. To f i n d the s p a t i a l dependence of the i o n i z a t i o n r a t e , we use the e x p r e s s i o n , • DO 1 (r) = ^ j (E,r) <r (E) dE. 189 Then, u s i n g the form of the i o n i z a t i o n c r o s s - s e c t i o n , "sTfE) =<ToE~1 (equation 20), we get J ( r ) = n oT,p7^ D S ( I . H ) R \ [ E2+z (r) ](2^+i>/3 normalize t h i s , by s e t t i n g e = E/E c» Then \ ( e2_j.7) < 2 ^ ,+ 1 .>/ 3 I where Z = 0.86 x 10-20 E - 2 N (r) , i s a normalized column d e n s i t y , with E e i n MeV. At the sour c e , when Z = 0, r "WR> * \ e - < ? + x / 2 ) d e = V ( ( H 1 / 2 ) ; so the a t t e n u a t i o n of the i o n i z a t i o n r a t e i s given by Vp(r) = (£ + 1/2) T ( r ) \ e < 2 ^ + i ) / 6 d e (1.12) ^ V J ( e 2 + Z ) « 2 f i Of c o u r s e , i n the s p h e r i c a l case, a geometric f a c t o r ( r 2 / r 2 ) must be added. Numerical e v a l u a t i o n of the i n t e g r a l i n (1-12) i n d i c a t e s t h a t f o r a ^ of 1 or 2, the i n t e g r a l i s reasonably approximated over the range Z = 0 to 10 3 by the e x p o n e n t i a l exp (-a Z 0 * 2 5 ) . The value of a depends on the s p e c t r a l index, ^ - The value of a used i n the program i s 0.5, d e r i v e d from a curve with ^>=1.5. As ^ i s i n c r e a s e d to 3.0, the b e s t - f i t value f o r a i n c r e a s e s by a f a c t o r of two, which decreases t'he r a t i o y ^ ( r ) / \ ^ ( r t ) by about one h a l f i n the l a r g e Z r e g i o n . When Z > 100 t h i s r a t i o i s c l o s e enough t o zero t h a t the v a r i a t i o n i n a w i l l not a f f e c t the numerical models. The value ^=1.5 was chosen to re p r e s e n t the expected l e s s steep behavior of the s o f t 190 end of the g a l a c t i c cosmic ray spectrum. The f i n a l form of the i o n i z a t i o n r a t e , then, i s ^ (r) = T t RJr, ) ( r f / c * ) e x p ( a Z°.2S) . ( I . 12a) we note the dependence of Z on the low energy c u t o f f , E 0 : Z i s l a r g e r f o r s m a l l e r E , producing a more r a p i d a t t e n u a t i o n , as would be expected from (1.3). X-rays The a l l - o r - n o t h i n g nature of the energy l o s s of a photon beam p a s s i n g through a gas a l l o w s the a t t e n u a t i o n of the beam to be expressed i n simple o p t i c a l depth form. I f the i n i t i a l spectrum i s a power law, L (v) = above some c u t o f f frequency t ) e , the spectrum a f t e r t r a v e l l i n g a d i s t a n c e r has the form L(v,r) •= L^-*1 exp[-<7 (v) \ n dr ] (1.13) Again, r, i s the d i s t a n c e from the source to the edge of the gas and ^fil)) i s the t o t a l i o n i z a t i o n c r o s s s e c t i o n . T h i s equation i s analogous to (1.10) f o r cosmic r a y s , and i s o b t a i n e d without the energy-space argument. The X-ray i o n i z a t i o n c r o s s s e c t i o n f o r hydrogen has a steep energy dependence, namely, ^ (v) •= ~ y\ The t o t a l i o n i z a t i o n c r o s s s e c t i o n , i n c l u d i n g helium, must be used i n the exponent of (1.13); the helium c r o s s s e c t i o n has a d i f f e r e n t form, T u (t>) = T-t.U'7' Combining t h i s with the photon f l u x , j (Vr r) = L ( u , r ) / h y , g i v e s the i o n i z a t i o n r a t e ad ^ ^ y ( r ) = < I ( L 0 ( D - ( ^ < ) expC-tX-r.v-HYT^V- 2) [ n dr ] dV r \ where X and Y are the f r a c t i o n a l abundances of H and He by number. T h i s h o l d s only i f hu. i s above the i o n i z a t i o n energy of He. T h i s i n t e g r a l i s made more t r a c t a b l e b y / s e t t i n g / / y = V03/v3, and d e f i n i n g a mean normalized column densxty, Z x (Y) = (X % ^ " ^ y + Y ^ x i - 2 y 2 / 3 ) J n dr . d-15) when r = r, , the i n t e g r a l f o r If becomes •<*> d-t/ = (3 + ^ ' ) - i i J - " + o(>, So the f i n a l form f o r the i o n i z a t i o n r a t e i s ^c(r) = (1 + V 3 ) ^ ( r , ) ( YA/3 e - V j ) ay (1.16) o Again, the f a c t o r r 2 / r 2 must m u l t i p l y ^ ( r ) i n a s p h e r i c a l geomet ry. T h i s i n t e g r a l can be approximated by the e x p o n e n t i a l , exp (-a Z ° . S ) f o r Z •= 0 to 10 2. The approximation i s q u i t e h f-i n s e n s i t i v e to the value of ot , and a = 1.0 i s a good r e p r e s e n t a t i o n f o r an o£, of 1 to 3. As i n the cosmic ray case, f o r l a r g e Z y the i n t e g r a l i s very s m a l l and the v a r i a t i o n with r e s p e c t to i s unimportant. The form f o r the X-ray i o n i z a t i o n r a t e used i n the program, then, i s ^ ( r ) = Xx(r, ) ( r f / r 2 ) e x p ( - a Z 0 . s ) (1-17) we note that t h i s decreases f a s t e r with column d e n s i t y than does the cosmic ray i o n i z a t i o n r a t e , equation (I.12a). The dependence on low energy c u t o f f i s r e f l e c t e d i n the n o r m a l i z a t i o n of Z i n equation (1.15). 192 APPENDIX I I NUMERICAL TESTS The system of program equations i n Chapter IV can be reduced to the i s e n t r o p i c case by s e t t i n g the heat sources and c o o l i n g terms to z e r o . Then the r e s u l t s can be compared with numerical s o l u t i o n s e x i s t i n g i n the l i t e r a t u r e , and, f o r a few cases, known a n a l y t i c s o l u t i o n s . The most s e n s i t i v e p a r t s of an E u l e r i a n scheme are the a d v e c t i o n and pressure terms; these t e s t s w i l l i n d i c a t e t h e i r accuracy without i n v o l v i n g the h e a t i n g - f u n c t i o n e f f e c t s . Also i n these t e s t s the c e n t r a l mass and the i o n i z a t i o n are i g n o r e d , and only the i s e n t r o p i c expansion of a monatomic gas under i t s own pressure i s c o n s i d e r e d . The f u l l system of equations does not have known s o l u t i o n s , t h e r e f o r e the numerical r e s u l t s cannot be d i r e c t l y compared with other work. The accuracy of the production runs was e s t i m a t e d , t h e r e f o r e , i n terms of f i d e l i t y to p h y s i c a l c o n s e r v a t i o n laws not e x p l i c i t l y b u i l t i n t o the coding, and by t e s t i n g the e f f e c t of the g r i d s i z e . The i s e n t r o p i c t e s t s w i l l be d e s c r i b e d f i r s t , then the p r o d u c t i o n run checks. S i m i l a r i t y S o l u t i o n The s i m p l i f i e d f l u i d e g u a t i o n s , as d e s c r i b e d above, admit sep a r a b l e s o l u t i o n s of the form • u(t) = r f (t) ( I I . 1) where p and D s a t i s f y 193 dp/dr = -c^ >r ( d f / d t + f 2 ) (II. 2) For the case of c o n s t a n t outer v e l o c i t y , u (t) = u (r/R ( t ) , where R (t) i s the edge of the s p h e r i c a l gas c l o u d , the i s e n t r o p i c d e n s i t y and pressure s o l u t i o n s s a t i s f y i n g (28) are ^ ( r , t ) = DC/R3 ( 1 - r 2 / R 2 ) i/<ir-i > (II.3) P ( r , t ) = p c/R 3 ( 1 - r 2 / R 2 ) Tf/<jr-i ) ("# i s the a d i a b a t i c exponent; p/(» = constant i s i m p l i e d by the i s e n t r o p i c c o n d i t i o n . ) The behavior of the outer edge a l s o f o l l o w s from (28) : ( d R / d t ) 2 = UTS/3(Y-1) p o (1-(R^/R) 3 Y - 3 , (II. 4) (See G u l l , 1973.) The v e l o c i t y of the edge, t h e r e f o r e , approaches the l i m i t i n g value u, = 4/3 c,/(V-1) (H-5) i f c s i s the sound speed. A l s o , from c o n s e r v a t i o n of energy, the t o t a l thermal energy must behave as E ot R - 3 1 >. Using i n i t i a l c o n d i t i o n s c o n s i s t e n t with t h i s s o l u t i o n , the program was run through 140 c y c l e s , a s h o r t run. Both a r t i f i c i a l v i s c o s i t y and mass d i f f u s i o n were turned o f f , to a v o i d c o m p l i c a t i o n s ; the good behaviour of the i n i t i a l v a r i a b l e s should not r e q u i r e "smoothing". The energy s o l u t i o n was converted to temperature by s e t t i n g the v e l o c i t y s c a l e at u c •= 10 km s - 1 . There were i n i t i a l l y 40 g r i d p o i n t s , c o v e r i n g 0.025 < r < 1.00. The r e s u l t s are shown i n F i g u r e s A.1, A.2 and A. 3. A f t e r 140 c y c l e s , the program had reached a time 194 t = 0. 1502t o, and the d e n s i t y d i s t r i b u t i o n f i t the formula ( 1 — r 2 / 1 . 5 2) 2 « 5 to w i t h i n two per cent over the r a n g e 1 , 0.35 < r < 1.15, and w i t h i n f i v e per cent over the l a r g e r range, 0.10 < r < 1.30. The d e v i a t i o n s f o r r < 0.10 a r e , t o be expected, s i n c e the boundary c o n d i t i o n u ( r ( ) = 0 where r , •= 0.025 does not d u p l i c a t e the u(0) = 0 s e l f - s i m i l a r c o n d i t i o n . The e r r o r s at the outer edge are i n the sense t h a t the c a l c u l a t e d d e n s i t y t a i l extends f a r t h e r out, t o very low v a l u e s , than the a n a l y t i c s o l u t i o n does. T h i s may be an a r t i f a c t of the g r i d - e x t e n s i o n method of h a n d l i n g the r a r e f a c t i o n . The other e f f e c t of the outer boundary, a p p a r e n t l y , i s the n e a r l y constant r e g i o n i n temperature (and the r e s u l t a n t v e l o c i t y plateau) d e v e l o p i n g i n the very low d e n s i t y outer r e g i o n s — that i s , where o< 0. 1 ^ 0 f o r r > 1-4 i n the 140th c y c l e - The same i n i t i a l c o n d i t i o n s were a l s o run with an outer boundary c o n d i t i o n , T N = constant; t h i s appeared to i n t r o d u c e i n s t a b i l i t i e s . The v e l o c i t y maintains the expected l i n e a r form except a t the o u t e r edge; the t r a n s i t i o n t o s u p e r s o n i c v e l o c i t y causes no problems. The t o t a l mass i s well conserved over the.computation, changing by 0.16 per cent; the t o t a l energy (thermal plus k i n e t i c ) i s conserved to three decimal p l a c e s . However the thermal energy i t s e l f decreases f a s t e r than R— 1• 2, which the s i m i l a r i t y s o l u t i o n p r e d i c t s i f the a d i a b a t i c exponent, '^ =1.4. The value of ^ was chosen a r b i t r a r i l y f o r the t e s t s . I t i s 1 The value R = 1.5 i s not n e c e s s a r i l y a best f i t to the d e n s i t y curve, but was used as a good round-number e s t i m a t e . 195 / / F i g u r e s A.1, A.2 and A.3 d i s p l a y the s i m i l a r i t y s o l u t i o n used t o t e s t the numerical code. F i g u r e A. 1 shows the d e n s i t y at fo u r epochs. The squares r e p r e s e n t the a n a l y t i c form the d e n s i t y should f o l l o w at the l a s t epoch c a l c u l a t e d . The agreement i s good except at the i n n e r boundary, where the computational boundary c o n d i t i o n d i f f e r s from the a n a l y t i c one. F i g u r e A.2 shows the v e l o c i t y c u r v e s , which a n a l y t i c a l l y must be l i n e a r at a l l times. End e f f e c t s appear i n curve (b). F i g u r e A.3 shows the temperature s o l u t i o n , which again agrees w e l l with the a n a l y t i c form except f o r end e f f e c t s . 196 °-25 0.50 0.75 1.00 1.25 1.50 . r / r 0 F i q u r e A.1 S i m i l a r i t y S o l u t i o n - D e n s i t y 197 198 1 9 9 b u i l t i n t o the c a l c u l a t i o n s only as a s c a l i n g parameter, and i t appears i n the i n i t i a l c o n d i t i o n s of t h i s t e s t , p(r)«< ^ (r) . The e f f e c t i v e a d i a b a t i c exponent was c a l c u l a t e d at v a r i o u s epochs using ) f ^ = l n (p/p c) /In (^/(3,) . The average over the g r i d of t h i s q u a n t i t y maintained i t s i n i t i a l value to w i t h i n one per cent . The i n d i v i d u a l p o i n t s near the outer boundary had l a r q e r e r r o r s , up to ten per cent, r e f l e c t i n g the u n c e r t a i n t y i n the g r i d - e x t e n s i o n method. As r e q u i r e d by the i t e r a t i o n r o u t i n e , the equation of s t a t e i s a l s o s a t i s f i e d t o one per cent. I n i t i a l l y Homogeneous Sphere A case which w i l l approach a s e l f - s i m i l a r s o l u t i o n a t l a r g e times, and which i n i t i a l l y has l a r g e d i s c o n t i n u i t i e s a l l o w i n g t e s t i n g of the mass d i f f u s i o n term, i s a sphere a t r e s t with o and p constant i n s i d e a r a d i u s R 0, and zero o u t s i d e R 0. At t = 0 t h i s sphere i s allowed to expand a d i a b a t i c a l l y . T h i s example i s d i s c u s s e d i n d e t a i l by Z e l ' d o v i c h and Hai z e r (1966), and numerical c a l c u l a t i o n s using the f r e e - m o l e c u l a r approximation (a c o l l i s i o n l e s s Maxwellian gas) were done by Molmud (1960) and by Narasiraha (1962). As the gas s t a r t s expanding, a r a r e f a c t i o n wave moves i n toward the c e n t e r , and the outer r e g i o n s expand with the maximum v e l o c i t y , 2c s/("£-1) (as demonstrated, f o r i n s t a n c e , by Thompson, 1972). Narasimha d i s c u s s e s the inward-moving r e g i o n of r a p i d d e n s i t y change, which he d e f i n e d as 6: a r e g i o n of width shows t h a t , with the f r e e molecular e q u a t i o n s , t h i s r e g i o n c e n t e r e d at the point He 200 expands at a c o n s t a n t r a t e ; the width &(t) = '\. Me^t/Ji^. Molmud compared the f r e e molecular s o l u t i o n s with a numerical s o l u t i o n y to the f l u i d a q u a t i o n s with p/c? = c o n s t a n t , and found good agreement. A f t e r very long times, when the gas has expanded s u f f i c i e n t l y t h a t Ro i s no l o n g e r an important parameter, one would expect the s o l u t i o n to approach s e l f - s i m i l a r i t y . As d i s c u s s e d above, t h i s would have uo<r/t, ^ = P t,(r/R)/R 3, and E <X. R-3 ( V - i >. S t a r t i n g with the homogeneous c o n d i t i o n s ^ = 1, e = 2, u = 0 w i t h i n R 0 = 1, and s t a r t i n g on the same g r i d as i n the p r e v i o u s example, 300 c y c l e s were run. T h i s advanced the s o l u t i o n to t - 0.3262t o, which i s 0.45 of the i n i t i a l sound-wave c r o s s i n g time. The mass d i f f u s i o n term was found n e c e s s a r y . With T = 0 s p a t i a l o s c i l l a t i o n s grew very q u i c k l y . For a g r i d s i z e of 0. 0 2 5 r p , a value T - 10 ( S r ) a = 0.25o\r was found to best s t a b i l i z e the s o l u t i o n . The r e s u l t s are shown i n F i g u r e s A.4, A-5 and A.6. The d e n s i t y e v o l u t i o n appears to agree with r e s u l t s given by Molmud. A r e g i o n to compare with o(t) can be d e f i n e d , say, by the d i s t a n c e over which () changes from 0.75^(0) to 0.25 p(0) — a f t e r the f i r s t s t a g e s of the expansion, t h i s r e g i o n s e t t l e s down to growth at a c o n s t a n t r a t e , i n agreement with Narasimha's model. The v e l o c i t y i s approaching a l i n e a r form i n the r a r e f a c t i o n r e g i o n , as a s e l f - s i m i l a r s o l u t i o n would. The same boundary problems — an extended l o w - d e n s i t y r e g i o n of roughly constant temperature, and a r e g i o n of d e c r e a s i n g du/dr — appear a g a i n . The e f f e c t i v e a d i a b a t i c 201 exponent (defined above) has a value of 1.6 when averaged over the r e g i o n between ^= 1.0 and the p o i n t of temperature t u r n o v e r . Mass i s conserved to t h r e e decimals, and the t o t a l energy grows by 4.7 per cent over the 300 c y c l e s . T h i s energy gain i s probably due to the mesh e x t e n s i o n procedure. G u l l (1973) n o t i c e d t h a t r e z o n i n g w i l l a f f e c t the net energy. Since so l i t t l e mass i s i n v o l v e d i n these outer r e g i o n s , the e r r o r s here are probably w i t h i n reasonable t o l e r a n c e s . (In comparison. G u l l quotes 10 per cent accuracy f o r a Lagrangian scheme; Larson, 1969, quotes 20 per cent f o r an E u l e r i a n scheme.) Accuracy of Computed R e s u l t s A standard t e s t of numerical codes i s the behavior of the r e s u l t s as a f u n c t i o n of the g r i d s i z e used; these r e s u l t s should be i n s e n s i t i v e to &r. T h i s was t e s t e d a f t e r the f i r s t p r o d u c t i o n runs of the model were completed, by r e p e a t i n g the e a r l y c y c l e s of the l o w - f l u x , c o o l mode model with a g r i d s p a c i n g h a l f as l a r g e . The in n e r h a l f of Model 1 (NP=40; c f . Table V I I , Chapter V) was d u p l i c a t e d with a 40 poin t g r i d using cSr/2, and l e t evolve u n t i l the s h e l l had neared the outer edge of the g r i d . F i g u r e A.7 compares r e p r e s e n t a t i v e times i n the v e l o c i t y and d e n s i t y e v o l u t i o n of the two models. The o r i g i n a l c a l c u l a t i o n i s shown by s o l i d l i n e s and the r e p e t i t i o n by dashed l i n e s . The two models agree reasonably w e l l with each other; the d i f f e r e n c e s between them are that the peak (in both d e n s i t y and v e l o c i t y ) i s sharper and moves more s l o w l y , i n the case of the 202 F i g u r e s A. 4, A.5 and A.6. The homogeneous sphere c a l c u l a t i o n i s shown. The i n i t i a l c o n f i g u r a t i o n i s a cons t a n t d e n s i t y sphere out to a r a d i u s r - R c, as shown i n F i g u r e A.4. The subsequent expansion compares w e l l with n u m e r i c a l s o l u t i o n s computed by other methods. F i g u r e A.5 shows the v e l o c i t y development, which s t a r t e d from the s t a t i c case. F i g u r e A.6 shows the temperature s o l u t i o n . 203 F i g u r e A-4 Homogeneous Sphere - Densi t y 204 205 0.25 0.50 0.75 1.00 1.25 , 1.75 F i g u r e A.6 Homogeneous Sphere - V e l o c i t y 206 narrower g r i d s p a c i n g . Both of these r e s u l t s are known e f f e c t s of the a r t i f i c i a l v i s c o s i t y term — the widening of near d i s c o n t i n u i t i e s i n the flow to s e v e r a l g r i d s p a c i n g s as the v i s c o s i t y i s i n c r e a s e d , and the v a r i a t i o n of speed of p r opagation of t h i s d i s c o n t i n u i t y with the s i z e of the v i s c o s i t y parameter. The dependence of the v i s c o s i t y and mass d i f f u s i o n c o e f f i c i e n t s on Gr, g i v e n by equations (53) and (54) of Chapter IV, i s c o n s i s t e n t with these c a l c u l a t e d e f f e c t s . S i nce the computed d i f f u s i o n c o e f f i c i e n t s are necessary f o r the s t a b i l i t y of the program, t h i s u n c e r t a i n t y i s unavoidable. The c a l c u l a t e d v a r i a t i o n i n the s t r e n g t h of the v e l o c i t y maximum about ten per cent -- i n d i c a t e s that the s h e l l v e l o c i t i e s d e r i v e d f o r the c o o l mode models can be s p e c i f i e d o n l y to t h i s a ccuracy. A l s o , the broadening of the d i s c o n t i n u i t y means d e r i v e d s h e l l widths are upper l i m i t s to the t r u e widths. C o n s e r v a t i o n Laws The mass of the gas i s computed at each c y c l e of the c a l c u l a t i o n as a check. Over each c a l c u l a t i o n , even the l o n g e s t (about 30,000 c y c l e s ) , the mass was conserved to w i t h i n a few per c e n t . The d i s c r e p a n c y seems to have two s o u r c e s . The r e z o n i n g sometimes necessary, f o r i n s t a n c e i n the hot mode models as they expand, i s the most s e r i o u s c o n t r i b u t i o n to t h i s ; e a r l y experiments with the value of the s t a b i l i t y parameters showed t h a t a s l i g h t mass l o s s r e s u l t s from the optimum v a l u e s . Momentum and e n e r g y . t r a n s f e r were not monitored dur i n g the c a l c u l a t i o n s but were checked a f t e r w a r d . For the Model 1, Case 207 F i g u r e A.7 compares s e l e c t e d epochs of the low f l u x (Model 1) c a l c u l a t i o n using two d i f f e r e n t g r i d s i z e s , one which i s h a l f of the ot h e r . Density and v e l o c i t y p l o t s are shown. The c a l c u l a t i o n should be s t r i c t l y independent of the g r i d s i z e . The d i s c r e p a n c i e s , namely the width and p o s i t i o n of the s h e l l , are known e f f e c t s of the d i f f u s i o n c o e f f i c i e n t s which the E u l e r i a n code r e g u i r e s f o r s t a b i l i t y . 208 F i q u r e A-7 Half G r i d T e s t s 209 B run 1000 c y c l e s were used to estimate the t o t a l energy and momentum budget, and to check c o n s e r v a t i o n of those q u a n t i t i e s . The momentum q a i n from the cosmic rays goes mainly i n t o work a g a i n s t g r a v i t y . The gas pressure provided a s m a l l e r outward f o r c e (28 per cent of the cosmic ray fo r c e ) and the momentum of the gas changed by a s m a l l amount (29 per cent of the g r a v i t a t i o n a l work) d u r i n g the p e r i o d being monitored. The net f o r c e balance was i n e r r o r by only s i x per cen t . T h i s f i g u r e c o u l d r e p r e s e n t e r r o r s i n the nume r i c a l code, or i t cou l d r e p r e s e n t i n a c c u r a c y of the dis c r e t e - s u m r e p r e s e n t a t i o n of the space- and t i m e - i n t e g r a l c o n s e r v a t i o n checks. The energy budget r e f l e c t s the near balance of the cosmic ray h e a t i n g and the r a d i a t i v e c o o l i n g which was apparent i n the temperature e g u i l i b r i u m a t t a i n e d by the models i n Chapter V. The net gains i n k i n e t i c , thermal and g r a v i t a t i o n a l p o t e n t i a l energy together amounted to only a few per cent of the r a d i a t i v e l o s s e s . The energy g a i n from the cosmic rays balanced the net l o s s e s to w i t h i n f o u r per cent over the p e r i o d being monitored. As b e f o r e , t h i s c o u l d r e p r e s e n t i n a c c u r a c y i n the code or i n the d i s c r e t i z a t i o n procedure used to check i t . 210 APPENDIX I I I PROGRAM LISTING c c v e r s i o n l x x v i i of wind program, with p o t e n t i a l l y v a r i a b l e c g r i d s i z e S s p a c i n g ; c c a l c u l a t i n g advanced-time d e n s i t y and v e l o c i t y ; c with donor c e l l o p t i o n t o r f l u x c a l c u l a t i o n , c c and readi n g i n i t i a l a r r a y s from u n i t 3. c c non-conservative e x p l i c i t energy and v e l o c i t y s o l u t i o n s ; c i t e r a t i o n r o u t i n e s on c o n t i n u i t y and s t a t e e g u a t i o n s . c i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l long c pr = check whether to p r i n t or not to p r i n t ; c i f pr = . t r u e , every njump'th c y c l e i s p r i n t e d . c long = check whether to p r i n t f u l l e r r o r d i a g n o s t i c o/p l o g i c a l pr l o g i c a l * 1 tpar(20) l o g i c a l equc i n t e g e r * 2 f l a g (100) dimension re (12) dimension y (200) dimension px (1 00) , p i (100) common/grid/dn (10 0) ,u(101) ,p(100) ,e(100) ,x(100) ,r(100) , Sr2 (101) * common/hum/bt (100), bt 2(100) ,c2(100),g(9 8 ) ,rr(99) common/source/sree (100) common/gt/dnt (101) ,ut(101) ,pbar (101),xt (100),et (1 00) common/para/ams,xi,xe,delr,delt,tht,phi,gamma,yi,alam common/parb/tau,arec,zeta,opac (100) ,ckb,umu c o m m o n / p a r c / h t i , h t e , c f l a m , u o 2 , t c f , c o l l , e h y , c o o l f , e h c common/parx/zetax , h t h , h t 1 , h t 2 , b i g x , b i g y , a r e c 1 , a r e c 2 , x i h , S x i 1 , x i 2 common/part/t (1 00) , y 1 (1 00) , y 2 (10 0) , t t (1 00) ,y11 (1 00) , .Syt2(100) common/out/lcng,pr c o m m o n / o f f / t o l , f l a g , i c o n s t na melist/par/zeta,zetax,uo2,ams,cflam,ncyc 5, t a u , a l a m , t o l , c o o l f , n j u m p , n s t 1 , n s t 2 6, a c o e f f , i c o n s t , d n m i n , e n s e a l e x t e r n a l d e r l e x t e r n a l der2 e x t e r n a l derp e x t e r n a l dersg c c i n i t i a l i z e parameters. c ams = mass c o e f f i c i e n t c x i •= momentum c o e f f i c i e n t c based on mom (cr) = 1.0d-9 * energy c or momentum (xr) = 3.d- 11 *energy c p hi = s p l i t t i n g of pressure space d e r i v a t i v e c t h t = s p l i t t i n g of d e n s i t y space d e r i v a t i v e 211 c c gamma = r a t i o of s p e c i f i c heats c d e l r , d e l t = space 5 time g r i d s c alam = a r t i f i c i a l v i s c o s i t y ( v e l o c i t y d i f f u s i o n ) c tau = a r t i f i c i a l mass d i f f u s i o n c uo2 = square of r e f e r e n c e v e l o c i t y i n r e a l u n i t s c ht = hea t i n q c o e f f i c i e n t -- i o n i z a t i o n p l u s Coulomb c cflam = s c a l i n g f o r r a d i a t i v e energy l o s s c t c f * uo2 = c o n v e r s i o n of e ( i ) t o r e a l temperature c c o n v e r s i o n = 8592 degrees i f uo2=1.d12 c a r e c , a r e d arec2 = hydrogen, helium ( i i S i i i ) recombination c c o e f f i c i e n t s , see Gould & Thakur — low t approximation use c so recombination i s thereby overestimated f o r t > 13.6 ev. c z e t a = (cosmic ray) i o n i z a t i o n r a t e c zetax = (X-ray) hydrogen i o n i z a t i o n r a t e c c o o l f = zero to tu r n o f f r a d i a t i v e c o o l i n g c n s t 1 , n s t 2 = range of energy p o i n t s to s k i p i n t2 ac c o u n t i n g c bigx = f r a c t i o n a l hydrogen abundance by number c b i g y = f r a c t i o n a l helium abundance by number c umu = mean atomic wt. per atom, see Mihalas c ucsmc, t t o p = c u t o f f s i n h e a t i n g e f f i c i e n c y l o n g = . f a l s e . c a l l par (tpar,nt,20,510,&10) 10 continue i f ( e g u c ( ' l ' , t p a r ( 1 ) )) long=.true. i f ( equc(•1',tpar(6)) ) long=.true. write(6,151) t p a r 151 format(1x,20a1) gamma=1.4 n j u m p= 1 acoeff=0.5 enscal=0. c e n s e a l = 5.0 f o r 0.1 mev; c =1.0 f o r i . 0 mev; c = 0.56 f o r 10. mev. iconst=0 tau=10. tau=0. . alam=10. alam=0. coolf=1.0 tol=1.d-2 zeta=0. zetax=0. nst1=0 nst2=1 dnmin=1.Od-6 c read (5, par, end= 10 0) c tpast=0. t h t =0.5 phi = 0.5 pi=3.1415926536d 0 bohr=,529172d-08 elm=9.1084d-28 212 ckb=1.38046d-16 q=dsqrt(2.*ckb/elm) c o l l = 4 . 4 * d s q r t ( p i ) * b o h r * b o h r * q ehy=2.17953d-11/ckb prm = 1.66033d-24 e0=1.60207d-6 eO=eO/ (prm*uo2) ratms=gamma*prm/elm , ehc=ehy*ckb/(prm*uo2) ( ehei=3.94d-11/(prm*uo2) / eheii=8.70d-11/(prm*uo2) 1 tcf=2.*prm/ (3.*ckb) hti=1.0d-11*zeta/ (prm*uo2) hte=5.0d-10*zeta/ (prm*uo2) hte = h t e * e n s c a l h t i = h t i * e n s c a l hth=2.9 8d-10*zetax/(prm*uo2) ht1=2.8 4d-10*zetax/ (prm*uo2) ht2 = 2.34d-10*zetax/(prm*uo2) x i = 1 . 0 d - 9 * h t i * d s q r t (uo2) xe=1.0d-9*hte*dsgrt (uo2) xih=3.0d-11*hth*dsqrt (uo2) x i 1=3.0d-11*ht 1 * d s q r t (uo2) xi2=3„0d-1 1*ht2*dsqrt (uo2) ucsmc=1.0d 9/dsqrt(uo2) ttop=1.d 7 arec=3.0d-11*cflam*uo2*prm arec1=7.6d-11*cflam*uo2*prm arec2=9-2d-11*cflam*uo2*prm bigx=0.9d 0 bigy=0.1d 0 i f (zetax.eg.0.) bigx=1.0d 0 i f (zetax.eq. 0. ) bigy -= 0. bigx=1. bigy=0. umu = bigx+4.*bigy read (3,20) . n p , d e l r , t p , t p a s t , z a i , z a j , z a k , n y i f (nst1.eq.0) nst1=np 20 format(i5,6g10.3,i5) if(ny.eq.O) ny=2*np npm=np-1 npp=np+1 np0=ny/2 w r i t e (6,42) w r i t e (6, par) do 651 i=1,np 651 f l a g ( i ) = 0 c c space g r i d s r.(1) =0.5 do 65 i=1,npm r (i+ 1) =r (i) +delr r2 (i)=r (i) - 0 . 5 * d e l r 65 c o n t i n u e r2 (np) =r2 (npm) +delr 213 r2 (npp) ~r2 (up) + d e l r c c marker p o s i t i o n s read(3,21) (y(i) , i - 1, ny) yl=y (ny-1) c c d e n s i t y , dn read(3,21) {dn (i) ,i= 1,np) c c p r e s s u r e , p. read(3,21) (p (i) , i= 1, np) 21 format(5z16) c c v e l o c i t y , u. nota, u(1) and u (npp) are " m i r r o r " q u a n t i t i e s read(3,21) (u (i) , i= 1, np) u(npp)=2.*u (np)-u (npa) dnuO=dn (1) * (u (1) +u (2) ) c c i n t e r n a l enerqy, e read{3,21) (e (i) , i= 1, np) c c f r a c t i o n a l i o n i z a t i o n , x c x = n ( h i i ) / n (hi + h i i ) read (3, 21) (x (i) - i= 1, np) c y1 = n ( n e i i ) / n (he) c y2 = n ( n e i i i ) / n (he) read (3,21) (y 1 (i) ,i= 1,np) read (3,21) (y2 (i) , i= 1, np) c CJ- n ( e l e c ) / n (h + he, t o t a l ) •= bigx*x + faiqy*y1 +2*bigy*y2 c pr=.true. i f ( t p a s t . n e . 0.) qo to 201 dnbar=0. opac (1) = 1. 0 i f (t (1) . gt. ttop) opac (1) =opac (1) * (t (1) / t t o p ) ** (- 1.5) dnt(1)=dn(1) t (1) =2.*e (1) * t c f * u o 2 / 3 . opac (2) = 1. 0 i f (t (2) . g t . t t o p ) cpac (2) =opac (2) * (t (2)/ttop) ** (-1.5) dnt(2)=dn(2) t (2) =2. *e (2) *tcf*uo2/3-do 202 i=3,np t s c l = 1 . i f (t (i-1) . g t . t t o p ) t s c l = (t (i-1) /ttop) ** (-1.5) dn b a r = d n b a r + d n ( i - 1 ) * t s c l * d e l r aco=5.0d-23*cflam*dnbar r z - r (i) - r (1) rz=1. rz= (aco*rz)**0.25 opac (i) =dexp (-acoef f *rz) t (i) =2. *e (i) * t c f * u o 2 / 3 . i f (t (i) . g t . ttop) opac (i) =opac (i) * (t (i) / t t o p ) ** ( - 1. 5) 202 dnt (i) =dn (i) dnbar=0. i f (zetax.eg.0.) go to 205 do 204 i=2,np dnbar=dnbar+dn(i-1)*delr*(1.-x(i-1) ) aco=5.3d-25*cflam*dnbar rz=1. r z = d s g r t ( a c o * r z ) 204 opac (i) = dexp (-acoef f *rz) 20 5 continue c a l l i o n (np, 1) 203 con t i n u e i f ( t p a s t . n e . 0 . ) go to 201 c a l l ionhe(np,1) do 200 i=1,np e (i) =e (i) +ehc*xt (i) *bigx/umu+ (ehei*yt1 (i) + e h e i i * y t 2 (i)) o*bigy/umu y1 (i)=yt1 (i) y 2 ( i ) = y t 2 ( i ) 200 x (i)=xt (i) 201 continue c a l c u l a t e temperature do 600 i=1,np dnm=ehc*x ( i ) * b i g x + (ehei*y 1 ( i ) + e h e i i * y 2 (i) ) * b i g y ddn = (1.+bigx*x (i) +bigy*(y1 (i) +2.*y2 (i) ) ) t ( i ) = (e (i) -dnm/umu) / (ddn/umu) 600 t (i ) = t (i) * t c f *uo2 601 co n t i n u e do 101 kn=1,ncyc mean d e n s i t y , mean o p a c i t y dnbar=0. dnm=0. opac (1) =1. 0 i f (t (1) . gt. ttop) opac (1) =opac (1) * (t (1) /ttop) **(-1.5) i f (u (1) .gt.ucsmc) opac(1)=opac(1) *(e0/(u (1) *u (1) 6 + ratms*p (1)/dn (1) ) ) **2. opac (2) =1.0 i f ( t 2 . g t . ttop) opac (2) =opac (2) * (t (2)/ttop) ** (-1. 5) i f (u (2) . g t . ucsmc) opac (2) =opac (2) * (e0/ (u (2) *u (2) S + ratms*p (2) /dn (2) )) **2. do 211 i=3,np t s c l = 1 . i f (t (i-1) . g t . ttop) t s c l = (t (i-1) / t top) ** (-1.5) i f (dn (i-1) .eq.0.) go to 212 i f ( u (i-1) .gt. ucsmc) t s c l = (e0/(u ( i - 1 ) * u (i-1) + Sratms*p ( i - 1 ) / d n (i-1) ) ) **2. 212 continue dnbar=dnbar+ dn ( i - 1 ) * t s c l * d o l r aco=5.0d-23*cflam*dnbar r z = r ( i ) - r ( 1 ) rz=1. rz= (aco*rz)**0.25 opac (i) =dexp (-acoef f * r z ) i f (t (i) . g t . ttop) opac (i) =opac (i) * (t (i) / t t o p ) ** (- 1.5) i f (dn (i) .eg. 0.) go to 215 215 i f (u (i) . g t . ucsmc) opac (i) =opac (i) * (eO/(u (i) *u (i) + 5 r a t m s * p ( i ) / d n ( i ) ) ) * * 2 . 215 c o n t i n u e dnm=dmax1(dnm,dn(i)) 211 c o n t i n u e dnm=dnmin*dnm dnbar=0. i f (zetax.eg-0.) go to 213 do 214 i=2,np dnbar=dnbar+dn (i-1) * d e l r * (1.-x (i-1) ) aco=5.3d-25*cflam*dnbar rz=1-0 rz = d s g r t ( a c o * r z ) 214 opac (i) =daxp (-acoef f * r z ) 213 c o n t i n u e i f ( k n . e q . l ) write(6,39) ( ( i , r (i) , opac (1) ) , i=1, np) c c choose time step c p r = . f a l s e . if(mod(kn,njump).eg.0) pr=.true. kcnt=1 sm= 1. i f (dn (1) .eg. 0.) go t o 711 sm=p(1) /dn (1) 711 c o n t i n u e um=1. tm=1. do 70 i=2,npm * i f (dn (i) . I t . dnm) go to 70 utest=dabs (u (i) ) umt=dmax1(um,utest) um=umt stm=p (i) /dn (i) sm = dmax1 (sm,stm) if(z e t a . n e . 0 . . a n d . c o o l f . e g . 0 . . a n d . z e t a x . e g . 0 . ) go to 70 i f ( i . gt. nst 1. and. i . l t . nst2) go t o 70 i f ( k n . g t . l ) go to 705 72 continue t n = h t i * (1.-x (i) ) +hte*x (i) tn = tn + hth* (1.-x (i) ) *bigx + (ht 1 * (1.-y 1 (i) -y2 (i) ) 6 + ht2*y1 ( i ) ) * b i g y tn=tn*opac (i) ec=t (i) c t f n=cool (x (i) ,ec) t n = t n / ( r (i) * r (i) ) - c o o l f * c t f n * d n (i) tnn=-0. 25*dn (i) * (u (i) +u (i+1) ) *der2 ( f l a g ( i - 1) , (i-1) ,e) tnn=tnn-p (i) *der1 ( f l a g (i),i,u)/gamma tn=tn+tnn/(delr*dn (i) ) go to 706 705 tn = s r c e ( i ) / d n (i) 706 continue tn=0. 15*e ( i ) / d a b s (tn) tm=dmin1 (tm,tn) 70 c o n t i n u e s=dsgrt(qamma*sm) 216 delt=0.1*delr/(s+um) d e l t = 3 . * d e l t delt=0.5*dmin1 (delt,tm) d e l t = 2 . * d e l t d e l t = 0 - 5 * d e l t w r i t e (6,20) np,ucsmc,tm,s,um,delt 105 c o n t i n u e t d = d e l t / d e l r c c hold your b r e a t h , d i v e i n . . . s o l v e i m p l i c i t p r e ssure c eguation . . i f (kcnt. ne. 1) go t o 163 i f ( ( . n o t . long) . and. pr) write(6,35) a m s , x i , a l a m , d e l r , d e l t , &tpast 163 c o n t i n u e npq=np-2 na=4*~(np-2) c a l l imp(pbar,np,npm,npp,na,npg) c c use advanced p r e s s u r e s o l u t i o n t o get advanced d e n s i t y . . . do 80 i=2,np dnt (i) = dn (i) +c2 (i) * (pbar (i) -p (i) ) i f (dnt (i) . q t . 0. . and-pbar (i) . gt. 0. ) go to 80 dnt (i)=0. pbar (i) =0. 8 0 c o n t i n u e dnt (1) =2.*dnt (2) - dnt (3) i f (dnt (1) . gt. 0. ) go t o 801 dnt (1) -0. pbar(1)=0. 80 1 continue i f (p (np).eq.0.) go to 106 dnt (np) =dn (np) + dn (np- 1) *u(np) * ( (r (np-1) / r (np) ) **2.) * d e l t pbar (np) =p (np) + (dnt (np) -dn (np) ) /c2 (np) 106 c o n t i n u e i f (kcnt.gt.7) go to 100 kcnt=kcnt+1 c c . . . and advanced v e l o c i t y , a t i n t e r i o r p o i n t s , do 8 1 i=2,npm tut=0. tutq=0. i f (dn (i) . eq. 0. and. dn (i+1) . eq. 0. ) go to 82 tutq=- 1. * d e l t * a l a m * (dersq (f l a j (i) , i , u,r 2) / (r (i) * r (i) ) & - dersq ( f l a g ( i + 1) , (i+1) , u , r 2 ) / ( r ( i + 1) * r (i+1) ) )/gamma tutq=tutq*0. 5* (dn (i) +dn ( i + 1) ) avopac=0. 5* (opac (i) +opac (i+1) ) t u t = - 0 . 5 * t d * u ( i + 1 ) * d e r 2 ( f l a g ( i ) , i , u ) & + d e l t * (avopac* {xi* (1.-x (i) ) + xe*x (i) +biqx*xih* (1.-x (i) ) + & b i g y * ( x i 1 * {'..-y 1 (1) -y2 (i) ) + xi2*y 1 (i) ) ) -ams) / 5 (r2 ( i + 1) *r2 (i+1) ) p l ( i ) = t u t t u t - t u t - a . * t d * ( p h i * d s r 1 ( f l a g (i) ,i,pbar) + 6 (1.-phi) * d e r 1 ( f l a g (i) , i , p ) ) / 5 (gamma* (dnt (i+ 1) +dat (i) +dn (i+ 1)+dn (i) ) ) px (i) = t u t - p l (i) 217 82 continue ut ( i +1) =u (i+1)+tut + t u t q \ 81 continue ut (1)=2.*ut (2) -ut (3) ut(1)=0. ut(2)=0. go to 325 i f (equc {» z' ,tpar (1) ) . or.equc (• z' ,tpar (6) ) ) 7 u t (1) =-ut (2) i f (iconst.eq.0) go to 325 / dnt (1)=dnu0/{ut (1)+ut (2) ) 325 continue ./ ut (npp)=2.*ut(np) -ut (npm) c t e s t : can c o n t i n u i t y equation get s a t i s f a c t i o n ? c a l l ch (np , delr,delt,tau,yl,gamma,5 106) i f (pr. and, long) write(6,39) ( ( i , dnt (i) , ut (i) ) ,i= 1, np) i f (pr.and. long) write(6,39) ( ( i , p i (i) ,px (i) ) , i= 1, npm) kctt=0 c c use these to get i o n i z a t i o n s o l u t i o n c a l 1 ion (np,1) 107 continue kctt=kctt+1 i f (kott-gt.7) go to 100 i f (zetax. ne. 0. . and-bigy. ne. 0. ) c a l l ionhe (np, 1) c c f i n a l l y , get advanced temperature ( i . e . , i n t e r n a l energy c per gram) c a l l gnome (np,npm,npp,nst1,nst2,5105,S106,5107) go to 108 j- pbar (1) =gamma*dnt (1) * t t (1) * (1. +bigx*xt (1) +bigy* (yt1 (1) + 52. *yt2 (1) ) ) pbar (1)=pbar (1) / (1. 5 * t c f *uo2*umu) 108 continue c c advance program time and w r i t e t h i n g s out i f pr = . t r u e , c t p a s t = t p a s t + d e l t i f (pr.and.long)write(6,35) a m s , x i , a l a m , d e l r , d e I t , t p a s t 35 format(1h1,' mass =',g10.3,' mom c o e f f =',g10.3, 5* v i s c o s i t y c o e f f =', 5 g10.3,' d e l r =«,g10.3,« d e l t =«,g10.3,' time =»,g10.3) i f ( p r ) write(6,34) 39 format (1h0, 20 (/5 ( i 3 , 2g11. '4) ) ) umach=0. do 9 0 i=1,np i f ( p b a r ( i ) . e g . 0 . . o r . d n t ( i ) . e q . 0 . ) go to 75 souud2=gamma*pbar ( i ) / d n t (i) umach=ut (i) /dsqr t (sound 2) 75 continue i f (pr) write (6, 36) r2 (i) , u (i) , ut (i) , umach p l l - p ( i ) dnl-dn (i) temp=t (i) pbl=pbar (i) dntl=dnt (i) 218 tempt=tt (i) f=0.5*dn (i) * r (i) * r (i) * (u (i) + u (i+1) ) f 2=0. 5*dnt (i) * r (i) * r (i) * (ut (i) +ut (i+ 1)) -••='«-•' write (6,37) i , r (i) , p l l , d n l , tern p,opac (i) , p b l , dn t l , 90 17 36 42 34 33 37 i f (pr)Stempt,f2 continue c o n t i n u e i f ( p r ) w r ite (6,36) r2 (npp) , u (npp) , ut (npp) format (1x,f7.2,g12.3,48x,g10.3,3 6x,.f7.3) format(1h1) format (1h0) format (5x,f 8.4,g1 1. 4) i f (zetax.ne. 0. ) c a l l ionhe(np,1) format(1x,i3,f6.2,10x,2g10.3,q13.6,g10.3,15x,2g10.3 , &g13.6,g10.3) c c c c c c c check c o n s e r v a t i o n p r o p e r t i e s and s t a b i l i t y c o e f f i c i e n t s c a l l cons(np) c a l l d i f f u s (np,ttau,tlam) i f (tau.ne.0.) tau = t t a u i f (alam.ne.0.) alam = 1.5*tlam advance markers i f ( p r ) w r i t e (6, 41) c a l l y mark(y,u,ut,np,np0,delr,delt,tht) yl=y (ny-1) i f ( y l . g t . r 2 (npp)) c a l l mac(up,npm,npp,ny,delr,yl,ehc) "7 2 0 continue 41 format(1h0, 1 markers') now, f o r the next time . i f ( . n o t . p r ) go to 712 tp=tcf*uo2 w r i t e (7,20 write (7,21 write (7,21 write (7,21 wr i t e (7,21 w r i t e (7,21 write (7,21 wr i t e (7,2 1 w r i t e (7,21 712 continue n p , d e l r , t p , t p a s t , z e t a , c f l a m , a m s , n y (y (i) ,i=1,ny) (dnt (i) ,i= 1, np) (pbar (i) , i= 1, np) (ut (i) ,i=1,np) (et (i) ,i=1,np) (xt (i) ,i=1,np) (yt1 (i) , i ' (yt2 (i) , i : = 1»np) \ ,np) 71 do 71 i=1, np u ( i ) = u t ( i ) dn (i) =dnt (i) e (i)=et (i) t (i) =tt (i) p (i) =pbar (i) x (i) =xt (i) y1 (i)=yt1 (i) y2 (i) =yt2 (i) c ontinue 219 u (npp) =ut (npp) 101 c o n t i n u e 100 continue i f ( k c n t . g t . 5) write (6,900) 900 format(1h0,'energy equation has blown up; stop') stop end 220 c c s u b r o u t i n e imp(pbar,np,npm,npp,na,npq) i m p l i c i t r e a l * 8 (a-h,o-z) i n t e g e r * 2 f l a g (100) l o g i c a l l o n g l o g i c a l pr dimension pd (100) ,rp (100) dimension a (400) dimension b (93) ,pbar (100) dimension t1 (100) ,t2 (100) ,t3 (100) ,t4 (100) , t r (100) , t r 2 (101) dimension i p (98) , bb (98) , res (98) common/grid/dn (100) , u (10 1) ,p(100),e(100),x(100),r(100), 6r2 (101) common/para/ams,xi,xe,delr,delt,tht,phi,gamma,yl,alam common/parb/tau,arec,zeta,opac(100) ,ckb,umu common/hum/bt (100),bt2 (100) ,c2 (100) , g (9 3) , r r (99) coamon/uarx/zetax,hth,ht1,ht2,bigx,bigy,arec1,arec2,xih, S x i 1,xi2 common/par t / t (100),y1(10 0 ) , y 2 ( 1 0 0 ) , t t (100), yt 1(10 0), Syt2(100) c c c c c common/tst/ptest (100) comjnon/out/long, pr c o m m o n / o f f / t o l , f l a g , i c o n s t s u b r o u t i n e t o s o l v e i m p l i c i t equation f o r advanced s o l u t i o n to pseudo p r e s s u r e , pbar. f i n d s s o l u t i o n by s o l v i n g band-matrix e g u a t i o n . c~ s t a r t : s e t up a r r a y f o r r.h.s of e g u a t i o n . c c r e a t e s t o r a g e a r r a y s f o r o f t - r e p e a t e d terms do 10 i-= 1. nD c i-1> t r 2 (i) =r2 ( i j *r2 (i) 10 t r (i) =r (i) * r (i) t r 2 (npp) =r2 (npp) *r2 (npp) do 12 i=1,np t1 ( i ) = t r 2 ( i + 1) *u ( i + 1 ) / t r ( i ) t3 (i) = t r 2 (i) *u (i) / t r (i) t2 ( i ) = t r (i) *dn (i) *u (i) *u (i+1) c f i n i t e d i f f e r e n c e form f o r momentum f l u x : c donor c e l l , u (i-1/2) **2 or u ( i + 1/2)**2 ; c or z i P , u (i+1/2) *u (i-1/2) t 2 ( i ) =tr (i) *dn (i) *u (i) *u (i) i f (u (i+ 1) - l t . 0 . ) t2 (i) =tr (i) *dn (i) *u ( i + 1) *u (i+1) 12 co n t i n u e c c move ..on t o r r ( i ) a r r a y ; d e f i n e r r f o r i = 1,np-1 do 15 i=1,npm dn2=0.5* (un*(i)+dn ( i + 1) ) r r ( i ) = (opac (i) * ( x i * (1.-x ( i ) ) +xe*x ( i ) +bigx*x i h * (1.-x ( i ) ) + b i g y * ( x i 1 * (1.-y 1 ( i ) - y 2 ( i ) ) +xi2*y 1 ( i ) ) ) -ams) *dn2 r r ( i ) = r r ( i ) - d e r 1 ( f l a g (i) , i . t P W d P i r-5 •' i • - _ y ' i r> - y^ r 11) • r r i =rr (r) -der 1 (f l a g (i) , i , t2) / d e l r r r (i) = r r ( i ) / t r 2 (i+1) t r y v a r i a b l e alam . . . a3 = alam*dn (i) 221 a1=alain*dn ( i + 1) t 4 ( i ) = a l a m * ( d e r p ( f l a q { i ) , i , dn, 13) -derp ( f l a g (i) , i , d n , 11) ) r r (i) =rr ( i ) - t 4 (i) /gamma 15 co n t i n u e 130 format(1h0/1h0) i f (pr.and.long) w r i t e (6,100) (rr (i) ,i= 1,npm) c c r e d e f i n e temporary a r r a y s do 21 i=1,np r t 2 = r2 (i) / r (i) t r 2 (i) = r t 2 * r t 2 r t = r2 (i+1)/r (i) t r ( i ) = r t * r t 21 c o n t i n u e do 22 i=2,npra t1 (i) =0. 5*u (i) * (dn (i-1) + dn (i) ) pd (i)=der1 ( f l a g (i-1) , (i-1) rp) pd (i+1)=der1 ( f l a g (i) # i , p ) t2 (i)=dersg ( f l a g (i) ,i,pd,r2) / (r (i) * r (i) ) 22 continue 11 (np) =0. 5*u (np) * (dn (npm) +dn (np) ) + t h t * r r (npm) * d e l t c c c r e a t e sound-speed-sguared a r r a y , c2. cc=gamma cc- 1.5*cc do 26 i-1,npm c2 (i) =cc i f (dn (i) .eq.0.) go to 26 c2 (i) =cc*p (i) /dn (i) c2 (i) =1. 0/c2 (i) 26 continue c2 (np) =c2 (npm) c c c r e a t e g f o r i = 1,np-2 d l = d e l t / d e l r r p(2)=rr(1) dl2 = dl*dl*tht*(1.0-phi)/gamma do '25 i-2,npm rp (i+ 1) =rr (i) g ( i - 1 ) = - d l * d e r 1 ( f l a g (i) , i,t1) - d e l t * d n (i) * (u (i) +u (i+ 1) ) / & r (i) + d l 2 * t 2 (i) g ( i - 1) =g ( i - 1) - d l * d e l t * t h t * d e r s g ( f l a g (i) , i , r p , r 2 ) / 5 (r (i) * r (i) ) c a r t i f i c i a l mass d i f f u s i o n p d ( i ) = d e r l ( f l a g (i-1) , (i-1) ,dn) pd(i+1)= der 1 (f l a g (i) , i , dn) 9" (i-1) =9 +de l t * t a u * d e r s q ( f l a g (i) , i , pd, r2) / (r (i) * r (i) ) g (i-1) = -g (i-1) - p (i) *c2 (i) b ( i - 1 ) = g (i-1) 25 continue i f ( p r . a n d . l o n g ) w r i t e (6,130) i f (pr.and. long) w r i t e (6, 100) (g (i) ,i= 1 ,npq) c c now c r e a t e l . h . s . of matrix equation c the b i g a r r a y : pseudo pressure c o e f f i c i e n t s . put i t , c column by column, i n t o a (4*npq) 222 c c a l s o s e t up c o e f f i c i e n t s f o r s u b r o u t i n e impress, bt and bt2 beta = tht*phi*dl*dl/gamma a(1) = 0. a(2) = 0. bt{1)=0. bt(np)=0-bt2 (npm) =0. bt2 (np) =0. bt (2)=beta*tr (1) bt2 (1) =beta*tr2 (2) a (3) =-beta*tr (2) -c2 (2) a (4) = b e t a * t r 2 (3) bt2(2)=a(4) jg = np-4 do 29 j=1,jq i=j+1 a(4*j+1) = 0. jo=4*j+2 a (jo) =beta-*tr (i) bt (i+ 1) =a (jo) i=i+1 a ( 4 * j + 3) = - b e t a * ( t r (i) +tr2 (i) ) - c2 (i) i=i+1 jo=4*j+4 a (jo) =beta*tr2 (i) bt2 (i-1) =a (jo) 29 continue c ^ **** i n s e r t i n g p(1) = l i n e a r e x t r a p o l a t i o n c o n d i t i o n *** a(na-3)=0. a (na-2) =beta*tr (npq) a (3) =a (3) -bt2 (1) *2. a (6) =a (6) +bt2 (1) bt (npm) =a (na-2) c c ,**note: a(na-1) i s d e r i v e d from the boundary c o n d i t i o n c f o r the r e a l p r e s s u r e , which i s then a p p l i e d s t r a i g h t to the c pseudo p r e s s u r e . * * a (na- 1) =-beta* ( t r (npm) +tr2 (npm) ) - c2 (npm) sgnp=p (np) /p (npm) c c o u t e r c o n d i t i o n when p (yl) = 0.; p (np) < 0. i f (sgnp. I t . 0. ) a (na- 1) -a (na- 1) +beta*tr (npm) * (r (np) -y 1) / & ( y l - r (npm) ) c c outer c o n d i t i o n with f i n i t e p r e ssure past p(np); p (np) > 0. i f (sgnp. gt. 0.) b (np-2) = g (np-2) - b e t a * t r (npm) * (p (np) * & • (dn (npm) *u (np) * ( d e l t / d e l r ) * (r (npm)/r (np) ) **2.)/c2 (np) ) a(na)~0. 100 format(1x,8g10.3) dl2= (1.-phi)*dl/gamma dl=phi*di/gamma al=a(na-1) c a l l impres (al,dl2,dl,delt,pq,np,npm,npq) c c go ahead and s o l v e the matrix pg=p (1) pq=0. ud l = d e l t * (opac (1) *xe*x (1) -ams)/ (r (2) * r (2) ) 6 d e l t * (0. 5*u (2) * (u (3)-u (1) ) + & 2.*(p (2)-p(1) ) / (dn (2) +dn (1) ) ) / d e l r i f (iconst.eg-1) pg = p (1) *u (1) / (u (1) +udl) b(1)=b(1) -bt2 (1) *pq go to 31 31 continue ! c a l l dgband(a,b,npq,1,1,1,ip,det,ncn) / pbar (1) =pq do 30 i=1 rnpq 30 pbar(i+1) = b ( i ) pbar (1) =2. *pbar (2) -pbar (3) i f (pbar (1)-gt.O.) go to 850 pbar(1) = pbar (2) *p (1)/p (2) 850 c o n t i n u e pbar (np) - pbar (npm)* ( y l - r (np) ) / ( y l - r (npm) ) i f (p (np) , eq. 0.) pbar (np) =0. r e t u r n end s u b r o u t i n e p r e s s l (d!2,dl,dell,a,p1,p2) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l long l o g i c a l l a n g l o g i c a l pr dimension a (1) common/grid/dn (1 0 0) , u (10 1) , p (100) , e (1 00) , x (10 0) , r (1 00) , &r2 (101) common/out/lang,pr common/para/ams,xg,xe,delr,delt,tht,phi,gamma,yl,alam s u b r o u t i n e t o set up inner-boundary z e r o - v e l c c i t y c o n d i t i o n using e x t r a p o l a t i o n of known values, e x p r e s s i n g u i n term of p b a r ( 1 ) . r e s u l t s i n q u a d r a t i c equation f o r p b a r ( 1 ) . long=.true. f1 = a(9)/a(5) f2-a (7) /a (5) f 3=a(10)/a (6) f 4=a (8) /a (6) pd4=4. * p h i * d e l t / (delr*gamma) N-gk1 = u (2) +delt* (xq* (1. + x (1) ) +xe*x (1) -ams) / (r 2 (2) *r2 (2) ) £ - d e l t * u (2) * (u (3)-u (1) ) / (2. * d e l r ) - (alam/qamma) & * (r2 (3) *r2 (3) *u (3) -r2 (2) *r2 (2) *u (2) ) / (r (2) * r (2) ) 5 + (alam/gamma) * (r2 (2) *r2 (2) *u (2)-r2 (1) *r2 (1) *u (1)) 6 / ( r ( 1 ) * r ( 1 ) ) a1 = pd4*(1.-f1) a2 = -pd4* (f2-(1--phi) * (p(1)-p (2) )/phi) a3 = a (3) + f 1*a (4) a4 = a (4) * f 2-a (3) *p (1)-a (4) *p (2)+2. *dn (1) <-2.*dn (2) gk2=u(3) + d e l t * (xq* (1.+x (2) ) +xe*x (2)-ams) / (r2 (3) *r2(3) ) 5 - d e l t * u (3) * (u (4)-u (2) ) / (2. * d e l r ) - (alam/gamma) 6 * (r2 (4) *r2 (4) *u (4) -r2 (3) *r2 (3) *u (3) ) / (r (3) * r (3) ) & + (alam/gamma) * (r2 (3) *r2 (3) *u (3) -r2 (2) *r2 (2) *u (2)) & / ( r ( 2 ) * r (2)) b1 = pd4* (1.-f 3) * f 1 b2 = pd4* (f 2* (1.-f 3) - f 4 & +(1--phi) * ( p ( 2 ) - P ( 3 ) )/phi) b3 = f 1* (a (11) * f 3 + a (4) ) b4 = a (4) * f 2 + a (11) *f4 + a (11) * f 2*f 3-a (4) *p (2)-a (11) *p (3) 5 + 2.*dn (2)+2. *dn (3) combine these i n t o q u a d r a t i c c o e f f i c i e n t s pk r e f l e c t s the boundary c o n d i t i o n : pk = 1 f o r u (1) = u (2) pk = 2 f o r u(1)=0 ( l i n e a r l y e x t r a p o l a t e d ) pk = 3 f o r u(1)=-u(2) ( i . e . , u(1-1/2) = 0.) pk=1. i f (u (2) . l t . 0 . ) pk=3. qk1=qk1-qk2/pk qk2=0. pa = pk* (b3*gk1*a3+b3*a1)-a3*b1-a3*qk2*b3 pb = pk* (b3*a2 + gk 1* (b3*a4 + b'4*a3) +a1 *b4) 6 -a3*b2-qk2* (a3*b 4 + a4*b3) -b1*a4 pc = pk* (a2*b4+qk1*a4*b4•-b2*a4-qk2*b4*a4 i f (pr.and.long)write(6,30) a 30 format (1h0,10g10.3) i f (pr.and.long)write(6,30) f 1 , f 2 , f 3 , f 4 i f ( p r . a n d . l o n g ) w r i t e (6,30) a1,a2,a3,a4 i f (pr.and.long) write (6,30) b1,b2,b3,b4 q = pb*pb-4.*pa*pc i f (pr.and.long) w r i t e (5, 3 0) pa,pb,pc,q i f ( q . lt.O.) go to 51 eps=1.0e-06 xi=-pc/pb 5 xo=xi xi= -pc/pb - pa*xo*xo/pb i f ( (xi-xo).gt.eps) go to 5 p1 = x i p2=xi go t o 50 51 stop 50 r e t u r n end 226 \ s u b r o u t i n e impres (al,d!2,dl,delt,pbar,np,npm,npg) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l long l o g i c a l pr dimension gp (98) , d (98) , a r r (11) common/husi/bt (1 00) , bt2 (100) ,c2 (100) , g (9 8) , r r (99) common/tst/ptest (100) / common/out/long,pr ' subr. t o s o l v e f o r advanced pressure using r e c u r s i o n r e l a t i o n s p b a r ( i ) = f (pbar (i-1)) and the inner-boundary s o l u t i o n f o r p b a r ( 1 ) . pbar-0. go to 50 d (npq) =al d (npq) =-al d (npq) =-al gp (npg) =g (npg) gp (npg) =-gp (npq) jk=npq-1 do 10 i=1,jk ji=npm-i g p ( j i - l ) = g ( j i - 1 ) -bt (ji+1) *gp ( j i ) /d ( j i ) gp ( j i - 1 ) =-gp ( j i - 1 ) d ( j i - 1 ) = -c2 ( j i ) -bt2 ( j i ) *bt ( j i + 1 ) / d ( j i ) - b t ( j i + 1 ) - b t 2 ( j i - 1 ) d ( j i - 1 ) = c 2 ( j i ) - b t 2 ( j i ) * b t ( j i + 1 ) / d ( j i ) + b t ( j i + 1 ) +bt2 ( j i - 1 ) 10 continue p i c k parameters f o r pbar inner-boundary s o l u t i o n do 20 j=1,2 a r r ( j ) = r r ( j ) a r r (j + 4) =d {j) a r r (j + 2)=c2(j) a r r (j+6)-gp (j) a r r (j + 8) =bt2 (j) 20 continue a r r (11)=c2 (3) s o l v e f o r pbar(1) c a l l p r e s s l ( d l 2 , d l , d e l t , a r r , p 1,p2) next: s e t up sweeping pbar s o l u t i o n , then r e t u r n . 35 format (1h0,8g10.3) i f ( p r . a n d . l o n g ) w r i t e (6,35) p1,p2 pbar = p1 pg=pt pbar=p2 pq=p2 37 format (1h0) 36 format(8x,i2,g12.3) 50 continue r e t u r n end 227 c subroutine ch (np,delr,delt,tau,yl,gamma,*) c s u b r o u t i n e t o improve the d e n s i t y s o l u t i o n , using a newtonian-c s t y l e r e l a x a t i o n matrix based on the c o n t i n u i t y equation-i m p l i c i t r e a l * 8 (a-h,o-z) i n t e g e r * 2 f l a g (100) dimension aa (40 0) ,b (100) ,go (100) , i p (100) ,c2 (100) dimension td(100) common/grid/dn (100) , u (10 1) , p (100) , e (1 00) , x (10 0) , & r (100) ,r2 (101) common/gt/dnt (101) , ut (101) ,pbar (101) ,xt (1 00) ,et (100) c o m m o n / o f f / t o l , f l a g , i c o n s t npm=np-1 g t = t o l d t = d e l t / d e l r gsum=0. npq-0 c c c a l c u l a t e e r r o r i n the c o n t i n u i t y equation c do 60 i=2,npm rx=r ( i - 1) / r (i) rx=rx**2. c4=dnt (i) -dn (i) td (i)=ut (i) * (dnt (i) +dnt (i-1) ) +u (i) * (dn (i) +dn (i-1) ) td (i+ 1) = ut (i+1) * (dnt (i) +dnt ( i + 1) ) +u ( i + 1) * (dn (i) +dn ( i + 1) ) c1 = -de r s q ( f l a g (i) , i , t d , r2) * d t / ( 2 . * r (i) * r (i) ) c1=c1/2. td (i)=der1 ( f l a g (i-1) , (i-1) , dn) td (i+ 1) =der1 ( f l a g (i) ,i,dn) c.3 = t a u * d e l t * d e r s q (f l a g (i) , i , td,r2) / (r (i) * r (i) ) cs=c4-c1-c3 c2 (i) =gamma i f (dn (i) -eg. 0.) go to 105 i f (dnt (i) .eq-0. ) go to 105 c2 (i) =gamma*p ( i ) / d n (i) c2 (i)=1.5*'c2 (i) go (i) =cs goabs=dabs(cs/dn ( i ) ) gsum=gsum+goabs npg=npg+1 write(8,30) i , c s , dn (i) , gsura 105 continue 60 continue g3Um=gsum/npg gt=dmax1 (gt,gsum) 30 format (1x,i5,5g 10.3) 40 format (1x,*cont- equation o f f by •,g10.3,'; i t e r a t e ' ) 85 format (1h0, 10 (/1x, 6g10. 3) ) c c i f e r r o r too b i g , d e r i v e c o r r e c t i o n s to the d e n s i t y c i f (gt.eq. t o l ) go to 100 wri t e (6,40) gt a1 = 0. do 61 i=2,npm 228 c d.g ( i - 1 ) / d . d n t (i) c i f ( i . e g . 2) go to 71 a1 = 0. 5*dt*ut (i) * (r2 (i) / r (i-1) ) **2. 71 continue aa (4*i-7) =0. aa (4*i-6) =a1 c c d.g ( i ) / d . d n t (i) c a2 = 1.+-0. 5*dt* (ut (i+1) *r2 (i+1) *r2 (i+1) -& u t ( i ) * r 2 ( i ) * r 2 ( i ) ) / ( r ( i ) * r ( i ) ) aa(4*i-5)=a2 c c d. g (i+1)/d. dnt (i) c if(i.eg.npm) go to 62 a3 = -0. 5*dt*ut ( i + 1) * (r2 (i+1)/r ( i + 1) ) **2-go t o 63 62 a3=0. 63 aa (4*i-4) =a3 b(i-1)=-go (i) 50 format (i5) 6 1 c o n t i n u e c anp= (ut (np) *dt/4. - t a u * d e l t ) * (r2 (np) / r (npra) ) **2. anp=0. na=4*npm-5 * aa (na) =aa (na) +anp* ( y l - r (np) ) / ( y l - r (npm) ) c a l l dgband (aa,b, (np-2) , 1, 1, 1,ip rdet,ncn) c do 70 1=2,npm dnt (i) =dnt (i) + 1. 0*b (i-1) pbar (i) =pbar (i)+1 ,0*b (i-1) *c2 (i) i f (dnt (i) . ge. 0. ) go to 70 dnt (i) =0. pbar (i) =0. 70 continue w r i t e (6,86) ( (dnt (i) , b (i) ) ,i= 1, npm) 86 format (1x,10g10.3) r e t u r n 1 100 continue r e t u r n end 229 c c \. sub r o u t i n e cons (np) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l pr common/grid/dn (10 0) ,u(101) , p (100) ,e (100) ,x (100) , r (1 00) , Sr2(101) common/gt/dnt(100) ,ut (101) , pbar (1 01) , xt (1 00) , et (1 01) co mm on/para/a m s , x i , x e , d e l r , d e l t , tht,phi,ga'mma,yl,alam common/parb/tau,arec,zeta,opac (100) ,ckb /u /inu common/out/long,pr 1 c c to check i n t e g r a l c o n s e r v a t i o n p r o p e r t i e s of s o l u t i o n c sumdn=0. sa = 0. sb=0. sumom=0. sump=0. sumpb=0. \ sumg=0. smass=0. sen=0. sent=0. c c s e t up the i n t e g r a l s npm=np-1 do 50 i=2,npm rs=r (i) * r (i) * smass=smass+dn(i)*rs sumdn = sumdn+ (dnt (i) -dn (i) ) * r s sumom=sumonn- (dn t (i) * (ut ( i + 1) + ut (i) ) -dn (i) * (u (i+ 1) +u (i)) ) * r s sumg=sumg+ ( x i * ( 1 . - x (i))-ams) *dn (i) sumpb=sumpb + pbar (i) * (r2 (i)+0. 5*delr) sump=sump + p ( i ) * ( r 2 ( i ) + 0.5*delr) sen=sen + dn ( i ) * e ( i ) * r s sent=sent + dn (i) * ( e ( i ) + 0 . 5 * u ( i ) * u ( i + 1)) * r s 50 co n t i n u e c ends=r2 (2) **2. * (phi*pbar (2) + (1.-phi) *p (2) ) +r2 (2) **2. S *dn (2) *u (2) *u (2) ends=ends-r2 (np)**2.*(phi*pbar(np) + (1.-phi)*p (np) ) S - r (np) **2. *dn (np) *u (np) *u (np) c c ** note donor c e l l form ** c sumdn=sumdn*delr sumom~sumom*0.5*delr sumg=sumg*delr*delt sump=sump*2.*delt*delr* (1.-phi) sumpb-sumpb*2.*delr*delt*phi sa = r2 (2) **2. * ( t h t * u t (2) * (dnt (1) +dnt (2) ) + (1.-tht) 5 * (dn (2) +dn (1) ) *u (2) ) sa=sa-r2 (np) **2. * ( t h t * (dnt (np) +dnt (np-1) ) *ut (np) 6 + (1.-tht) * (dn (np) +dn (np- 1) ) *u (np) j sa=sa*delt*0„5 230 sb=r2 (2) **2. * (da (1) -dn (2) ) - r2 (np) **2. *.(dn (np- 1) -dn (np)) s b = s b * t a u * d e l t / d e l r ends=ends*delt smass=smass*delr sen=sen*delr s e n t = s e n t * d e l r smuf=sa+sb-sumdn c c w r i t e e v e r y t h i n g out i f ( p r ) w r ite (6,79) smass i f (pr) write (6,80) srauf i f (pr) write (6,88) sen i f ( p r ) w r ite (6,39) sent i f (pr) write (6,81) sumom i f ( p r ) w r i t e (6, 85) suing i f (pr) write{6,82) sumpb i f (pr) w r i t e (6, 33) sump i f (pr) write (6,78) ends 80 format(1hO,•mass unaccounted f o r =',g10.3) 81 format (1h0,'momentum gain =',g10.3) 82 format(1h0,'pbar * r sum =»,g10-3) 83 format(1h0,•p * r sum =«,g10.3) 85 format (1h0,•net mom source =',g10.3) 78 format (1h0,'mom ands =',g10.3) 79 format ( 1 h 0 , ' t o t a l mass =',g10.3) 88 f o r m a t ( 1 h O , ' t o t a l thermal energy =',g10.3) 89 format ( 1 h 0 , ' t o t a l energy =',g10-3) r e t u r n end s u b r o u t i n e d i f f u s { n p , t t a u l , t l a m l ) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l pr common/grid/dn (10 0) ,u (101) ,p(100) ,e (100) ,x (100) , r (100), Sr2 (101) common/gt/dnt (10 1) , ut (10 1) , pbar (101) , xt (1 00) , et (1 00) common/para/ams,xi,xe,delr,delt,t1,t2,gamma,t3,t4 common/out/long,pr t d = d e l t / ( d e l r * d e l r ) do 60 i=1,np i f (dnt (i) .eq.0. ) go to 60 c2 gamma*pbar (3.) /dnt (i) t t a u = 0.5* (ut (i)+ut (i+1) ) / r (i) + (ut ( i +1)-ut (i) ) / d e l r t t a u = 0.5*ttau + t d * (c2 + ut (i) *ut (i) ) tlam = (dnt (i+1)-dnt (i) ) / ( d e l r * (dnt (i+1)+dnt (i) ) ) & + 1./r2(i+1) tlam = u t ( i + 1)*tlam + t d * (c2 + 3. *ut (i) *ut (i) ) i f (i.eg.1) go to 65 ttau=dmax1(ttau,ttaul) tlam=dmax1 (tlam,tlaml) 65 t t a u l = t t a u tlaml=tlam 6 0 continue i f (pr) write (6,30) t t a u l , t l a m l 30 format (1hO,•tau max =',g10.3,' alam max = ',g10.3) t h i s s u b r o u t i n e e v a l u a t e s , approximately, the mass- and momentum-diffusion c o e f f i c i e n t s needed f o r s t a o i l i t y at each s t e p , and p r i n t s the l a r g e s t . r e t u r n end 232 c \ s u b r o u t i n e ymar k ( y , u , u t , n p , n p O , d e l r , d e l t , t h t ) c c c to advance marker p a r t i c l e s by one time s t e p c / i m p l i c i t r e a l * 8 (a-h,o-z) / l o g i c a l pr / dimension k (200) / dimension y(1) , u(1) ,ut (1) ' dimension r2(101) common/out/long,pr r2 (1) =0.5-0- 5* d e l r do 7 0 i=1,np 70 r2 (i+ 1) =r2 (i) +delr rn2 = r2 (np) +2. * d e l r n2=2*np0-1 do 60 i=1,n2 im= (y (i) -r2 (1) ) / d e l r + 1 udel= (y (i) -r2 (ira) ) / d e l r uy=u (im) + (u (im+ 1) -u (im) ) *udel uyt = ut (im) + (ut (im+1) - ut (im) ) *udel y ( i ) = y ( i ) + t h t * d e l t * u y t + ( 1 - - t h t ) * d e l t * u y i f (y (i) . gt. rn2) y (i) =rn2 k(i)=im 6 0 continue i f (y (n2) . gt.rn2) y (n2) =rn2 i f ( p r ) w r i te (6, 31) i f ( p r ) write(6,30) ( (k (i) ,y (i) ) , i = 1, n2) 31 format(1h0) 30 format (5x, 10 ( i 4 , f 8. 4) ) r e t u r n end 233 c c \ s u b r o u t i n e gnome(np,npm,npp,n1,n2,*,*,*) i m p l i c i t r e a l * 8 (a-h,o-z) i n t e g e r * 2 f l a g (100) l o g i c a l long l o g i c a l pr dimension vt(101) ,vtt(100) ,tg(100) / dimension ek (10 1) ,ekt (100) / dimension t1(100),t2(100),tp(100),tpr(100),q(100) common/grid/dn (100) ,u (101) , p (100) ,e (100) ,x (100) , r (100) , £rh(101) common/gt/dnt (1 0 1) , ut (1 0 1) , pbar (101) , xt (100) ,et (100) commcn/source/srce (100) common/par a/am s . x i , x e , d e l r , d e l t , t h t , p h i , g a m m a , y l , a l a m c o m m o n / p a r c / h t i , h t e , c f l a m , u o 2 , t c f , c o l l , e h y , c o o l f , e h c common/parb/tau,arec,zeta,opac(100),ckb,utnu common/par K / z e t a x , h t h , h t 1 , h t 2 , b i g x , b i g y , a r e c 1 , a r e c 2 , x i h , S x i 1 , x i 2 common/part/t (1 00) ,y 1 (10 0) , y2 (10 0) , 11 {1 00) , y11 (100) , Syt2 (100) common/out/long,pr c o m m o n / o f f / t o l , f l a g , i c o n s t c c c s u b r o u t i n e to s o l v e e x p l i c i t equation f o r advanced energy c s o l u t i o n u s i n g a backwards g r a d i e n t e v a l u a t i o n f o r the l a s t c g r i d p o i n t and a s u b f u n c t i o n to e v a l u a t e r a d i a t i v e energy c l o s s , e v e r y t h i n g being e v a l u a t e d at advanced times; c -* i n c l u d i n g a p o t e n t i a l i t e r a t i o n based i n equation of s t a t e c r e l a x a t i o n . c prra=1.66033d-24 do 51 i=1,np c non c o n s e r v a t i v e f . d . e . i s s t a t e d i n terms of i n t e r n a l energy c per gram. e k ( i ) = e ( i ) vt (i) =x (i) v t t ( i ) - x t ( i ) 51 continue c c b u i l d up source terms c n o t e * * * * l i n e a r i n t e r p o l a t i o n f o r outer g r a d i e n t s ; c * * * * t r e a t . i n n e r boundary using v e l = z e r o , c dsum=0. ek (npp) =2*ek (np) -ek (npm) dnt (npp) =2*dnt (np) -dnt (npm) pbar (npp) = 2.*pbar (np) -pbar (npm) vt (npp) =ut (np) *0. 5* (ut (np) +ut (npp)) * r (np) * r (np) do 65 i=2,npm r2=r (i) * r (i) c d i r e c t coulomb + i o n i z a t i o n h eating t2 (i) = ( h t i * (1.-xt (i) ) +hte*xt ( i ) ) *dnt (i) /r2 t2 (i) = t2 (i) + (hth*bigx* (1. -xt (i) ) +ht1*bigy 5 * (1.-yt1 ( i ) - y t 2 (i) ) + ht2*.bigy*yt 1 (i) ) *dnt ( i ) / r 2 234 t2 (i) =t2 (i) *opa.c (i) ax1 = 0. 5* (x (i)+xt (i) ) ax1=ax1*bigx+0.5*bigy*(y1 (i) + yt1 (i) +2.*y2 (i) +2.*yt2 ( i ) ) ax2=t(i) c t f = c o o l f * c o o l ( a x 1, ax 2) t2 (i) = t2 (i) - c t f *dnt (i) *dnt (i) c c pressure plus a r t i f i c i a l v i s c o s i t y r3=rh (i+1) *rh ( i + 1) r1 = rh (i) *rh (i) a1=alam* (dnt ( i - 1) +dnt (i) ) /2. a1=a1*delr*delr a2=alaui*dnt (i) a2=a2*delr*delr a3=alam* (dnt (i) +dnt (i+1) )/2. a3=a3*delr*delr tp (i) =pbar (i) -a2*dersq ( f l a g (i) , i , ut, rh) / (r (i) * r (i) * d e l r ) g (i) =tp (i) -pbar (i) c * * * n o n - c o n s e r v a t i v e form f o r energy c o n v e c t i o n t l (i)=-0. 25*dnt (i) * (ut (i) +ut (i+ 1) ) G *der2 (f l a g ( i - 1) , ( i - 1) , ek) / d e l r c e v a l u a t e de/dr i f d e n s i t y = 0 behind i f (dnt (i-1) . ne. 0.) go to 115 t n d l = -ek (i) -ek ( i - 2 ) / 3 . +4. *ek ( i + 1) /3. i f ( i . eg. 3) t n d l = 4. *ek (4)-3. *ek (3)-ek (5) t 1 ( i ) = -0. 25*dnt (i) * (ut (i)+ut ( i + 1) ) * t n d 1 / d e l r 115 c o n t i n u e c e v a l u a t e de/dr i f d e n s i t y = 0 ahead i f (dnt (i+1).ne.0.) go to 116 * t n d l = ek (i) +ek (i+2)/3.-4.*e'k (i-1) /3. i f ( i . e g . (npm-1) ) t n d l = 3. *ek (npm-1) -4. *ek (npm-2) 5 +ek (npm-3) t1 (i) •= -0. 25*dnt (i) * (ut (i)+ut (i+1) ) * t n d 1 / d e l r 116 continue t p r (i) =-tp (i) * (dar 1 ( f l a g (i) , i , u t ) ) / (delr*gamma) tp (i) =-tp (i) * (ut (i) +ut ( i + 1) ) / (r (i) *gaiama) '50 con t i n u e c t g ( i ) = 0 . temp=t (i) i f (pr.and. long) w r i t e ( 6 , 100) temp,ctf,t1 (i) r t 2 (i) ,tp (i) , 6 t p r ( i ) , t g ( i ) 60 c o n t i n u e i f ( i . g t . n l - a n d . i . It.n2) t2 (i) =0. s r c e (i) = t2 (i) +t 1 (i) + tu (i) +tpr (i) i f (dnt ( i ) . g t . 0 . ) go to 105 e k t ( i ) = a k ( i ) e t (i) =ekt (i) go to 6 5 105 continue 100 format (1x,•temp =',g12.5,' degrees; c o o l i n g =',g10.3, & » / »,5g12.U) ekt (i) =ek (i) + 2. 0*srce (i) * d e l t / (dnt (i) +dn (i) ) et (i) =ekt (i) tnm = ehc*xt (i) *bigx + ( e h e i * y t 1 ( i ) + e h e i i * y t 2 (i) ) *bigy tdn= (1. +bigx*xt (i) +bigy* (yt 1 (i) +2. *yt2 (i) ) ) 235 t t (i) = (et (i) -tnra/umu) / (tdn/umu) t t ( i ) = t t ( i ) * t c f * u o 2 i f ( t t ( i ) - I t . 100.) t t ( i ) = 1 0 0 . delp=pbar (i)-2.*gamma*dnt (i) * t t (i) * (1.+bigx*xt (i) +bigy* S (yt1 (i) +2. *yt2 (i) ; ) / (3. * t c f *uo2*umu) delp = d a b s ( d e l p / p b a r ( i ) ) dsum=dsura+delp 30 format(1x,i5,2g10.3) dele=et (i) -e (i) erat=dabs (dele) /e (i) i f (erat.Ie..25) go to 65 i f (pr) write (6,35) i , d e l t d e l t = d e l t * 0 . 2 0 / e r a t r e t u r n 1 65 c o n t i n u e s r c e (np) = s r c e (npm) c npmm-npm-1 do 106 i=3,npmm i f (dnt (i) .gt.0. ) go to 106 e t (i) =0.5* (et (i-1)+et (i+1) ) t t ( i ) = e t ( i ) * t c f * u o 2 106 c o n t i n u e i f (dnt (2) .gt.0. ) go t o 107 et (2) =et (3) t t (2) =et (2) * t c f *uo2 107 i f (dnt (npm) . g t . 0. ) go to 108 et (npm) =et (npm- 1) t t (npm) =2. *et (npm) * t c f * u o 2 / 3 . -"108 c o n t i n u e c et(1)=et(2) et (np) =et (npm) t t ( 1 ) = t t ( 2 ) t t (np) =tt (npm) c a l l i on (np,-1) xsum=0. do 200 i=1,np xsum=xsum + xt (i) - v t t (i) 200 continue xsum=xsum/float (np) i f (dabs (xsum) . g t . 0. 005) write (6,301) xsum xsum=dabs (xsum) 301 format(5x,'ion.n o f f by', g10.3) if(xsum.It.0.005) go to 210 do 201 i=1,np xt (i) =0. 5* (xt ( i ) + v t t (i) ) xt (i) =0. 25*xt (i) +0. 75*vtt (i) 201 continue r e t u r n 3 210 do 202 i=1,npp 202 xt (i) =vtt (i) c check, average s a t i s f a c t i o n of equation of s t a t e . c i f i t can't be s a t i s f i e d , d e f i n e a new pbar and do i t aqain dsum=dsum/ (npm-1) i f (dsum.qt.tol) go to 25 236 35 f o r m a t ( 1 x f ' s t e p • , i 5 , * r e j e c t e d . d e l t -',g10.3) go to 27 approximations i n pbar i t e r a t i o n : dn = dnt ** and ** g << pbar 25 do 26 i=2,npm i f (dnt (i) .eq.O.) go to 26 pdb=ehc*bigx*xt (i) + ( e h e i * y t 1 (i) +eheii*y t2 (i) ) * b i g y pdb= (e (i)-pdb/umu) *dnt (i) + (t2 (i) + t 1 (i) ) * d e l t ux= (ut (i)+ut (i+1) )/r (i) + der 1 ( f l a g (i) , i , u t ) / d e l r pdb=pdb-ux*q(i)*delt/gamma pd = 3./(2. *gamma) +2. *dnt (i) * d e l t * u x / (dnt (i) +dn (i) ) pbar (i) =0. 75*pdb/pd + 0.25*pbar(i) i f (pbar (i) .lt.O.) " p b a r ( i ) = 200. *gamma*dnt (i) 6 * (1. +bigx*xt (i) + b i g y * (yt1 (i) +2. *yt2 (i) ) ) / (3. * t c f *uo2*um 26 c o n t i n u e pbar (1) =2. *pbar (2) - pbar (3) i f (pbar (1) .gt. 0.) go to 850 pbar(1) = pbar (2) *p(1)/p(2) 850 continue w r i t e (6,300) dsum 300 format(1x,'pbar t o l o f f by »,g10.3) r e t u r n 2 27 c o n t i n u e r e t u r n end 237 c c s u b r o u t i n e i o n (np,nstart) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l pr dimension r a t (100) common/gt/dnt (101) , ut (101) ,pbar (101) ,xt (100) ,et (101) common/grid/dn(10 0),u(101),p(100),e(100),x(100),r(100), 5r2(101) / common/parb/tau,arec,zeta, opac (100) , cXb, uiau comraon/parc/hti,hte,cflam, uo2, t c f , c o l l , eh'y, c o o l f , ehcc common/parx/zetax,hth,ht1,ht2,bigx,bigy,ared,arec2,xih, 5 x i 1 , x i 2 common/par t / t (1 00) , y 1 (10 0) , y 2 (10 0) , t t {1 00) , y 11 {10 0) , &yt2 (100) common/out/long, pr s o l v e s t e a d y - s t a t e e g u a t i o n f o r f r a c t i o n a l i o n i z a t i o n of hydrogen; c.ray i o n i z a t i o n (x) + s l p r t r n n w nyarogen; c.ray i o n i z a t i o n (x)  e l e c t r o n c o l l i s i o n a l c i o n i z a t i o n (dn,x) = recombination (x,dn) c c note, temp = tp*e (i) ; recombination time = s q r t ( t e m p ) / c (arec*dnt) ; c o l l . i o n ' n formula from MacAlpine. c tp=tcf*uo2 dp=dsqrt(tp) c1=coll*uo2 ex1=-ehy c2=arec ehc=ehcc N nzero=0 i f (nstar t. g t . nzero) ehc=0. nst=iabs ( n s t a r t ) do 6 0 i=nst,np tx = t t (i) i f (ehc. eg. 0. ) t x = t ( i ) r a t (i) =0. xp=0. i f ( d n t ( i ) . l e . 0 . ) go t o 95 i f (e (i) .eg. 0.) go to 95 i f ( t x . l e . 0 . ) go to 36 de=dsqrt(tx) b 1= (zeta + zetax) *opac (i) / (r (i) *r ( i ) ) b2=c1*dnt (i) *de*dexp (ex1/tx) b3=c2*dnt ( i ) / d e i f (bl.eg.0.) go to 96 ra t ( i ) = b 2 / b 1 f1=b1-b2 f2=b3+b2 g=f1*f1+U.*b1*f2 i f ( q . l t . 0 . ) go t o 36 q=dsqrt (q) xp=-0. 5* (g + f 1)/f 2 if(xp.le.1..and.xp.ge.0.) go t o 95 xp=0. 5* (g-f 1)/f 2 i f (xp.le.1..and.xp.ge.0.) go to 95 238 36 c o n t i n u e i f ( p r ) write (6,35) i,.f 1, f 2,q, xp 35 format(1x,'stopped at ,,i5,4g11.3) stop 96 continue xpn=1.+arec/(c 1*tx) xpn = 1.+o3/b2 xp = 1./xpn 95 x t ( i ) = x p j 60 c o n t i n u e / i f ( n s t . e q . l ) xt(1)=xt(2) ' i f (pr) w r i t e (6,30) ( ( i , r a t (i) , xt (i) ) , i=nst,np) 30 format (5 ( i 4 , 2g10. 3) ) 31 format (1x, ' ! I ') 32 format(1x,4g10.3) r e t u r n end 239 c c s u b r o u t i n e i o n h e ( n p , n s t a r t ) i m p l i c i t r e a l * 8 (a-h,o-z) l o g i c a l pr dimension s t a r t (3) , i p t (4, 4) ,isub (4) ,coe (4,4) , tmp (4) ,pt (4) common/fkpar/cfa,cfb,cfc,bx,by common/gt/dnt (1 01) , ut (101) , pbar (101) ,xt (1 01) ,et (1 01) common/grid/dn (1 00) , u (10 1) , p (100) ,e (1 00) , x {100) , r {100) , &r2 (101) common/parb/tau , are c , z e t a , opac (100) ,ckb,umu common/parx/zetax,hth,ht1,ht2,bigx,bigy,arec1,arec2,xih, S x i 1 , x i 2 commori/part/t (100) ,y1 (100) ,y2(100) , t t (100) ,yt 1 (100) , &yt2 (100) common/out/long, pr bx=bigx by=bigy c c t h i s c a l c u l a t e s the f r a c t i o n a l helium i o n i z a t i o n s , c u sing the s t a r t i n g assumption t h a t he i s decoupled from c h, due to lower abundances, t h e r e f o r e the hydrogen e l e c t r o n s c determine the recombination; then i t e r a t i n g . c n s t a r t = i a b s (nstart) do 60 i = n s t a r t , n p yt1 (i)=0. y t 2 ( i ) = 0 . i f ( d n t ( i ) . e g . 0 . o r . o p a c ( i ) - eg.0.) go to 70 * i f (zetax.eq.0.) go to 70 i f (bigy.eq.0.) go to 70 a r e c i = a r e d / d s q r t (t (i) ) a r e c i i = a r e c 2 / d s g r t (t ( i ) ) ddn = 1.0 + a r e c i * x (i) *dnt (i) / (1 4. *zetax*opac (i) ) ddn = ddn + 13. *zetax*opac (i) / ( a r e c i i * d n t (i) *x (i) ) yt1 (i) = 1.0/ddn y t 2 ( i ) = yt1 (i) *1 4. *zeta.x*opac (i) / ( a r e c i i * d n t (i) *x (i) ) c max=20 s t a r t (1) =xt (i) s t a r t (2) =yt 1 (i) s t a r t (3) =yt2 (i) c f a=zetax/ (arec*dnt (i) / d s q r t (t (i) ) ) c f b = 1 3 . * z e t a x / ( a r e c i * d n t (i) ) c f c = 1 4 . * z e t a x / ( a r e c i i * d n t ( i ) ) c a l l n o n l i n ( 4 , m a x , 4 , i s i n g , s t a r t r i p t , i s u b , c o e , t m p , p t ) i f (max.ne.20) go to 6 1 i f ( i s i n g . n e . 0 ) go to 61 w r i t e (6,35) i , max, i s i n g , xt (i) ,yt1 (i) ,yt2 (i) , s t a r t 35 forraat(1hO,'no convergeance at p o i n t ' , i5,2i6,6g10.3) stop 61 c o n t i n u e xt ( i ) = s t a r t (1) yt1 (i) = s t a r t (2) yt2 (i) = s t a r t (3) i f (pr) write (6, 33) ( ( i , xt (i) ) , i=nstart,np) 33 format(5x,8(i4,g10. 3)) 70 c o n t i n u e 60 c o n t i n u e i f ( p r ) write(6,30) ( ( i , y t 1 (i) r y t 2 ( i ) ) , i = n s t a r t , 30 format (5 ( i 4 , 2g10. 3) ) r e t u r n end 241 c c \ s u b r o u t i n e fk(n,x,y,k) dimension x(n) common/fkpar/cfa,cfb,cfc,bigx,bigy go to (10,20,30),k 10 y=bigx*x (1) *x (1) +bigy*x (1) * (x (2) +2. *x (3) ) +cfa* (x (1) -1.) r e t u r n / 30 y = b i g x * x ( 1 ) * x ( 3 ) + b i g y * x ( 3 ) * ( x ( 2 ) + 2 . * x ( 3 ) ) - c f c * x ( 2 ) 20 y=bigx*x (1) *x (2) +tigy*x (2) * (x (2) +2.*x (3) ) 6 + c f b * (x (2)+x (3)-1.) r e t u r n r e t u r n end 242 c f u n c t i o n c o o l (x,e) i m p l i c i t r e a l * 8 (a-h,o-z) dxmension t h o t ( 3 5 ) , t l o ( 2 5 ) , a (3) C c c c c common/parc/hti,hte,cflam,uo2,tcf,coll,ehy,coolf,ehc common/rad/px1 (25) , px2 (25) , px3 (2 5) , ph (2 9) ,phe (21) coramon/rad2/pc (21) , po(2 1) , p s i (2 1) , ps (21) , pns (2 1) ,prag (21) common/radnew/plo (25),panp (33) using r a d i a t i o n l o s s e s c a l c u l a t e d f o r c o l l i s i o n a l e x c i t a t i o n only, f o r d i f f e r e n t abundances, s c a l e p's a c c o r d i n g l y . c c c c c xl=-3. t=e cool=0. i f (t.eg.O.) go to 200 tl=dlog10 (t) i f (x.eq.O.) go t o 65 xl=dlog10 (x) 65 continue do 50 i=1,25 50 t l o (i) = 1.0+(i-1) *0. 125 do 51 i=1,35 51 t h o t (i)=4.0 + (i-1) *0. 125 i f ( t l . g t . 4 . ) go to 100 cool=0. i f ( t l . l e . 2 . ) go to 200 c a l c u l a t e (log of) t o t a l c o o l i n g f u n c t i o n f o r t under 10,000 degrees p2=daint(25,tlo,px2,tl,3,g) go to 150 i f ( x l . g t . - 2 . ) go to 120 p 3 = d a i n t ( 2 5 , t l o , p x 3 , t l , 3,g) i f ( x l . l t . - 3 . ) x l = -3, px=p3+ ( x l + 3.)*(p2-p3) go to 160 120 continue p1=daint(2 5,tlo,px1,tl,3,g) px=p2+(xl+2.)*(p1-p2) i f ( x l.gt.-1) px-p1 + 1.+xl 160 continue 31 format(5x,3g10.3) cool=x* (1. -x) *1 0. ** (-px) go to 200 new c o o l i n g f u n c t i o n from Aldrovandi S Pequinot, low tempi 150 continue px=daint ( 2 5 , t l o , p l o , t l , 3 , q ) c o o l = x*10.**(-px) go t o 200 100 continue i f ( t l . g t . 8 . ) go t o 152 243 c i f t above 10,000 f i n d (log of) a s i m p l i f i e d , smooth c t o t a l c o o l i n g c go t o 151 p1=daint(29,thot,ph,tl,3,g) p 1 = 10. ** (-pi) cool=p1*x/.3 c higher t r e g i o n : c o o l i n g goes as t**(1/2) 152 c o o l = x*10.** (-panp (33)) c o o l = c o o l * 1 - 0 d - 4 * d s g r t (t) go to 200 c c new c o o l i n g f u n c t i o n , high temperatures 151 co n t i n u e px=daint (33,thot,panp,tl,3,g) c o o l = x*10.**(-px) c 200 c o o l = c f l a m * c o o l 30 format (1x,'temp =',g12.5,' degrees; c o o l i n g =',g10.3) r e t u r n end 244 i m p l i c i t r e a l * 8 (a-h,o-z) block data common/rad/px1 (25) , px2 {/5) , px 3 (25) , ph (2 9) ,phe (21) common/rad2/pc(21),po ( 2 1 ) , p s i (21) ,ps (21),pne (21) ,pmg (21) common/radnew/plo (25) ,panp (33) c data px1/2.800d1,2.70 2d1,2.631d1,2.586d1,2.55 3d1, 6 2. 531d1,2.51 3d 1,2.5 31d1,2.49Od1,2. 4 80d1,2.471d 1, 2. 46 3d1, & 2.457d1,2.4 51d 1, 2. 44 6d1, 2. 44 2d 1,2. 4 38d1 ,2. 4 3 3d1, 2. 428d1, 5 2.421d1,2.419d1, 2. 422d1,2. 421d1,2. 413d 1, 2. 385d 1/ c data PX2/2. 900d1,2. 322d 1, 2. 7 38d 1 , 2. 686d 1 , 2. 653d 1 , 6 2.631d1, 3. 608d1,2.591d1, 2.582d1,2. 568d1, 2. 558d1, 2. 552d1, 5 2.54 7d1,2.541d1,2.534d1, 2. 530d1 ,2. 526d1,2. 5 2 2d1, 2. 520d1, 6 2.518d1,2.516d1,2.513d1,2.51Od1,2. 506d 1,2. 485d1/ c data px3/2. 900d1,2. 900d1 , 2. 900d1,2. 8 2 3d1,2.74 2d 1, & 2.695d1,2.66 7d1,2. 64 9d1,2. 636d1 , 2. 62 5d1,2.613d1, 2. 604d1, C 2.596d1,2. 53 7d1,2. 57 9d1, 2. 572d1,2.56&d1,2.56 2d 1, 2. 55 8d1, 6 2.554d1,2.55 3d 1,2.551d1, 2. 548d1,2.541a1,2.538/ c data ph/2. 500d1, 2. 309d1,2.216d1,2.217d1,2.215d 1, 5 2. 20 8d1,2.19 4d1,2.136d1,2-182d 1,2. 176d1,2. 166d1, 6 2.160d1,2. 178d1,2. 190d1,2.2OOd1 ,2. 237d1,2.271d1, G 2.28 2d1,2.2 35d1,2.290d1,2.290d1,2.300d1,2.30 5d1, & 2.310d1,2.305d1,2.305d1,2.300d1,2.293d1,2.285d1/ c c--" cosmic-ray i o n i z e d A S P c o o l i n g f u n c t i o n data plo/2.720d1,2.650d1,2.592d1,2.555d1, 5 2.52 9d1 , 2. 505a1,2.49Od 1, 2. 47 0d1, 2.4 6 0d 1,2. 44 8d1, 6 2.4 34d1 , 2. 425d1,2.417d1,2. 408d1 ,2.404d 1,2.400d 1, & 2. 4OOd 1 ,2. 400d 1,2. 4OOd 1,2. 400d1, 2. 40Od 1, 2. 396d 1, & 2.392d1,2.375d1,2.300d1/ c data panp/2.300d1,2.258d1,2.243d1,2.240d1, & 2.222d1,2. 191d1,2. 160d1,2. 132d1,2. 130d1,2. 119d1, 5 2.110d1,2. 110d1,2.120d1,2. 17 8a1,2.195d1,2.215d 1, 8 2.255d1,2.287d1,2.298d1,2.307d1,2.310d1,2.311d1, 5 2.31 2d 1,2.311d1,2. 306a1,2.30 2d 1, 2. 30 0d1,2. 29 7d 1, 6 2.29 1d1,2. 286a1,2.280a1,2.273a1,2.268a 1/ end \ • s u b r o u t i n e mac(np,npm,npp,ny,delr,yl,ehc) i m p l i c i t r e a l * 8 (a-h,o-z) dimension a (3,5) common/grid/dn (100) ,u (101) , p (100) ,e (1 00) ,x (100) , r (1 00) 6r2{101) common/gt/dnt(101),ut (101) ,pbar (101),xt (100) ,et (100) common/part/t (100) ,y1 (100) ,y2 (100) , t t (100)',yt1 (100) , Syt2(100) j s u b r o u t i n e to add anothar g r i d p o i n t to outer edge i f l a s t marker has escaped. r2 (np+2) = r2 (npp) + d e l r r (np + 1) =r (np)-t-delr 31 format(1x,•!!!') i n t e r p o l a t e l a s t - b u t - o n e d e n s i t y p o i n t ; s e t dens(yl) = 0. et (npp) =et (np) e(npp)=e(np) i f (et (np) - ne. e (np) ) go to 50 do 5 i-1,3 a ( i , 1) =r (np-i) * r (np-i) a ( i , 2 ) - r (np-i) a ( i , 3 ) = 1 . a (i,5)=e (np-i) a ( i , 4) =et (np-i) 5 c o n t i n u e a (3, 1) =r (np + 1) * r (np+ 1) a (3,2)=r (np+1) a (3,5)=e (np) a (3,4) =et (np) c a l l dsolvk (a,2,3,3,det) dsn=a (1,4) * r (np) * r (np) +a (2, 4) * r (np) +a (3,4) dsn2=a (1, 5) * r (np), * r (np) +a (2, 5) * r (np) +a (3, 5) et (np) =dsn et (np) =0. 5* (et (npp) +et (npm) ) i f (et (np) . l e . 0 . ) et (np) =0. 1*et (npp) e (np) =dsn2 e (np) =0.5* (e (npp)+e (npm) ) i f (e (np) .Ie. 0. ) e (up) = 0. 1*e (npp) 50 co n t i n u e dnt (np) =0. 375*dnt (np-1) dn (np) =0. 37 5*dn (np-1) rat= ( y l - r (npp) ) / ( y l - r (np) ) dnt (npp) =0. dn (npp) =0. f i n d v e l o c i t y using g u a d r a t i c i n t e r p o l a t i o n ut (npp) = ut (np) do 6 i= 1, 2 a ( i , 1) = r2 (np-i) *r2 (np-i) a ( i , 2) =r2 (np-i) a ( i , 3 ) = 1. 6 a ( i , 4) =ut (np-i) 246 a(3,J) = 1. a (3,4)=ut (npp) a (3, 1) =r2(npp) *r2 (npp) a(3,2)=r2 (npp) c a l l dsolvk (a,1,3,3,det) ut (np) =a (1 ,4) *r2 (np) *r2 {np)+a (2,4) *r2 (np)+a (3,4) ut(np + 2)=2.*ut (upp) - ut(np) i f (et (npp) . eg. e (npp)) go to 21 c c e x t r a p o l a t e energy d i r e c t l y to avoid c n e g a t i v e pressure problems. et (npp) =2. *et (np) -et (npm) e (npp) =2. *e (np) -e (upm) et (npm) =0. 5* (et (np) +et (np-2) ) t t (npp) =2. * t t (np) - t t (npm) t (npp) =2. * t (np)-t (npm) t t (npm) =0. 5* ( t t (np) +tt (np-2) ) go to 21 21 continue c c c a l l i o n i z a t i o n s u b r o u t i n e f o r l a s t point3 c e v a l u a t e pressure d i r e c t l y c a l l i o n (npp,np) pbar (np) =2. *gamma* (et (np) -ehc*xt (np) ) *dnt (np) /3. pbar (np) =et (np) *dnt (np) * (1. +xt (np) ) pbar (npm) =2. *g am ma* (et (npm) -ehc*xt (npm) ) * dnt (npm) /3. pbar (npm) =et (npm) *dnt (npm) * (1. + xt (npm) ) pbar (npp) =0. c c " i n c r e a s e i n d i c e s np-np+1 npm=np-1 npp=np+1 wri t e (6,30) np, dnt (npm) , pbar (npm) ,et (np) ,xt (np) , ut (np) ,.. Sut (np+ 1) 30 format (1x,i5,6g10.3) c r e t u r n c c end 247 f u n c t i o n der 1 ( i l a g , i , p) i m p l i c i t r e a l * 3 (a-h,o-z) dimension p (100) der 1=p ( i + 1) -p (i) r e t u r n end f u n c t i o n der2 ( i l a g , i , p) i m p l i c i t r e a l * 8 (a-h,o-z) dimension p(100) dar2=p ( i + 2) -p (i) r e t u r n end f u n c t i o n derp ( i l a g , i , p , t ) i m p l i c i t r e a l * 8 (a-h,o-z) dimension p (100) , t {100) derp=p (i+1) * t ( i + 1) -p (1) * t (i) r e t u r n end f u n c t i o n d e r s q ( i l a g , i , p , r ) i m p l i c i t r e a l * 8 (a-h,o-z) dimension p (100) , r (10 1) dersg^p ( i + 1) * r ( i + 1) * r ( i + 1) -p (i) * r (i) * r (i) r e t u r n end 

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