Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A fast, robust algorithm for the solution of the equation of state for late-type stellar atmospheres Bennett, Philip Desmond 1983

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1983_A6_7 B46.pdf [ 3.28MB ]
Metadata
JSON: 831-1.0085777.json
JSON-LD: 831-1.0085777-ld.json
RDF/XML (Pretty): 831-1.0085777-rdf.xml
RDF/JSON: 831-1.0085777-rdf.json
Turtle: 831-1.0085777-turtle.txt
N-Triples: 831-1.0085777-rdf-ntriples.txt
Original Record: 831-1.0085777-source.json
Full Text
831-1.0085777-fulltext.txt
Citation
831-1.0085777.ris

Full Text

A  FAST, OF  ROBUST  THE  ALGORITHM  EQUATION  OF  STELLAR  FOR  STATE  THE  FOR  SOLUTION  LATE-TYPE  ATMOSPHERES  by  PHILIP DESMOND BENNETT B.Sc,  Simon Fraser U n i v e r s i t y ,  A THESIS SUBMITTED  IN PARTIAL  1975  FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  •in  THE FACULTY OF GRADUATE STUDIES (Department o f Geophysics and Astronomy)  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1983 ©  P h i l i p Desmond  Bennett, 1983  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by  department or by h i s or her  the head of  representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department Of  Geophysics and  The  U n i v e r s i t y of B r i t i s h  1956  Main M a l l  Vancouver, Canada V6T 1Y3 Date  October 12.  1983  Astronomy  Columbia  written  i i  ABSTRACT A fast,  but a c c u r a t e  for late-type s t e l l a r realistic  procedure to s o l v e the e q u a t i o n o f  atmospheres i s an e s s e n t i a l  model atmosphere c o d e .  the chemical  state  component o f a  T h i s r e q u i r e s the d e t e r m i n a t i o n o f  e q u i l i b r i u m o f a gas c o n t a i n i n g s i g n i f i c a n t amounts o f  perhaps one to two hundred s p e c i e s , over a wide range o f  temperature,  p r e s s u r e and c o m p o s i t i o n . A general method o f s o l u t i o n , based on a l i n e a r i z a t i o n a p p r o a c h , is derived f i r s t .  T h i s i s a c c u r a t e but has the disadvantage o f  the s o l u t i o n o f a l i n e a r  system o f equations o f order N, where N i s  number o f elements c o n s i d e r e d i n the e q u i l i b r i u m , f o r each I  then show t h a t the o r d e r o f t h i s  the  iteration.  system can be reduced to 8 by the  i n t r o d u c t i o n o f a " f i c t i t i o u s " metal s o l u t i o n t i m i n g without  requiring  element, thereby t r i p l i n g  s i g n i f i c a n t loss of accuracy.  and economized a l g o r i t h m s make no assumptions as to the s p e c i e s to be c o n s i d e r e d i n the e q u i l i b r i u m ; a l l  the  Both the  general  particular  such i n f o r m a t i o n  is  read from a s p e c i f i c a t i o n f i l e at e x e c u t i o n t i m e . Finally,  the e q u i l i b r i u m abundances o f s i g n i f i c a n t s p e c i e s  are  d i s p l a y e d g r a p h i c a l l y over a range o f t e m p e r a t u r e , p r e s s u r e and comp o s i t i o n (C/0 r a t i o ) , with these r e s u l t s temperature and C/0 r a t i o .  p l o t t e d both as a f u n c t i o n o f  The importance o f o b t a i n i n g  accurate  e q u i l i b r i u m abundances o f the v a r i o u s o p a c i t y s o u r c e s , and the i m p l i c a t i o n s f o r model atmosphere c o n s t r u c t i o n are d i s c u s s e d .  iii  TABLE  OF  CONTENTS  Page  ABSTRACT  ii  LIST OF TABLES  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  ix  CHAPTER ONE THE EQUATION OF STATE FOR LATE-TYPE STELLAR ATMOSPHERES  1  1.  Introduction  ^  2.  Historical  3  Development  CHAPTER TWO A GENERAL SOLUTION OF THE EQUATION OF STATE 1.  Chemical E q u i l i b r i u m o f a Gas o f Composition  2.  E v a l u a t i o n o f E q u i l i b r i u m Constants  9  Arbitrary  9 16  CHAPTER THREE AN OPTIMIZED SOLUTION OF THE EQUATION OF STATE PROBLEM 1.  2.  Initial  Estimates' o f P a r t i a l  Pressures  20 20  A;  S i m p l i f y i n g Assumptions  20  B.  Estimates o f E l e c t r o n Pressure  22  C.  L i n e a r i z a t i o n o f Charge N e u t r a l i t y  D.  Estimates o f Atomic P a r t i a l  Equation  Pressures  Economized S o l u t i o n o f Chemical E q u i l i b r i u m  26 29 33  iv  Page CHAPTER FOUR DISCUSSION OF RESULTS .  40  1.  D e t a i l s o f Results  40  2.  Astrophysical Implications  44  REFERENCES  65  LIST  OF  TABLES  TABLE I.  Species  c o n s i d e r e d i n Equation o f S t a t e  vi  LIST  OF  FIGURES Page  FIGURE 1.  2.  3.  4.  5.  6.  7.  Chemical  E q u i l i b r i u m o f Hydrogen:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  47  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  47  Chemical  E q u i l i b r i u m o f Carbon:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  48  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  48  Chemical  Equilibrium of Nitrogen:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  49  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  49  Chemical  E q u i l i b r i u m o f Oxygen:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  50  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  50  Chemical  Equilibrium of S i l i c o n :  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  51  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  51  Chemical  Equilibrium of  Sulfur:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  52  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  52  Chemical  E q u i l i b r i u m o f Aluminium:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  53  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  53  vii  Page FIGURE 8.  9.  10.  11.  12.  13.  14.  15.  Chemical  E q u i l i b r i u m o f Magnesium:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  54  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  54  Chemical  Equilibrium of  Chlorine:  a)  C/0 = 0.58 ( s o l a r ) , l o g P=2  55  b)  C/0 = 0.58 ( s o l a r ) , l o g P=5  55  Chemical  E q u i l i b r i u m o f Hydrogen:  a)  C/0 = 1,  b)  C/0 = 1 . 5 ,  Chemical  C/0 = 1 ,  b)  C/)  = 1.5,  56  l o g P=3  56  l o g P=3  ' •  l o g P=3  57 57  Equilibrium of Nitrogen:  a)  C/0 = 1,  b)  C/0 = 1 . 5 ,  Chemical  .  E q u i l i b r i u m o f Carbon:  a)  Chemical  l o g P=3  l o g P=3 l o g R=3  58 58  E q u i l i b r i u m o f Oxygen  a)  C/0 = 1 , l o g P=3  b)  C/0 = 1 . 5 ,  l o g P=3  Abundance o f Important  59 59  Species:  a)  T = 2000K,  l o g P=3  60  b)  T = 2000K,  l o g P=3  60  Abundance o f Important  Species:  a)  T = 2500K, l o g p=3  61  b)  T = 2500K, l o g P=3  61  vi i i  Paje FIGURE 16.  17.  18.  Abundance o f Important  Species:  a)  T = 3000K, l o g  P=3  62  b)  T = 3000K, l o g  P=3  62  Abundance o f Important  Species:  a)  T = 3500K, l o g  R=3  63  b)  T = 3500K, l o g  P=3  63  Abundance o f Important  Species:  a)  T = 4000K, l o g  P=3  64  b)  T = 4000K, l o g  P=3  64  ix  ACKNOWLEDGEMENTS  I would l i k e  to express my thanks to my s u p e r v i s o r ,  D r . Jason Auman f o r i n t r o d u c i n g me to the realm o f  late-type  s t a r s , and i n p a r t i c u l a r ,  f o r h i s a d v i c e and encouragement  throughout t h i s  Many thanks a l s o go to D r .  project.  Richer and Dr. Hoi l i s  Johnson f o r t h e i r  c o n c e r n i n g the atmospheres o f l a t e - t y p e  Harvey  stimulating discussions stars.  I am indebted to  Dr. C o r r i e Kost f o r h i s c o n t i n u i n g support o f my d e c i s i o n to to graduate school under r a t h e r must express my g r a t i t u d e typing of this  trying circumstances.  to Carmen de S i l v a  I  both f o r her  return  also beautiful  t h e s i s , and her perserverance with the pages o f  equations c o n t a i n e d w i t h i n .  Finally,  I wish to d e d i c a t e  this  work to my c h i l d r e n , Max and A t h e n a , i n the hope t h a t they too may someday f e e l  the sense o f wonder about the u n i v e r s e t h a t has  i n s p i r e d me over the  years.  1  CHAPTER ONE  THE  EQUATION  OF  STELLAR  1.  STATE  FOR  LATE-TYPE  ATMOSPHERES  Introduction The b a s i c s t e l l a r atmospheres problem i s to determine the run  o f t e m p e r a t u r e , p r e s s u r e and r a d i a t i o n f i e l d s through the atmosphere, which i s c o n s i d e r e d to be the o u t e r r e g i o n o f the s t a r from which photons may escape d i r e c t l y to space without or s c a t t e r i n g . local  If  the assumption t h a t the s t e l l a r m a t e r i a l  thermodynamic e q u i l i b r i u m (LTE)  temperature o f the gas e x i s t s  (e.g.  is  in  i s made, then a w e l l - d e f i n e d  the k i n e t i c temperature = e x c i t a -  t i o n temperature = i o n i z a t i o n temperature) gas w i l l  intervening absorption  and the c o m p o s i t i o n o f  be w e l l d e s c r i b e d by e q u i l i b r i u m p r o c e s s e s .  It  this  must be  n o t e d , however, t h a t the temperature o f the r a d i a t i o n f i e l d i n  the  o u t e r atmosphere g e n e r a l l y d i f f e r s from t h i s gas t e m p e r a t u r e , and i n fact  i s o f t e n s u f f i c i e n t l y non-Planckian t h a t an unique r a d i a t i o n  temperature cannot be d e f i n e d . The e s s e n t i a l arises  c o m p l i c a t i o n o f the s t e l l a r  atmosphere problem  from the c o u p l i n g between the r a d i a t i o n f i e l d and the  material  v i a the o p a c i t y .  the case o f l a t e - t y p e  stars  stellar  T h i s s i t u a t i o n i s f u r t h e r aggravated s i n c e t h e i r atmospheric temperatures  for are  2  low enough to permit m o l e c u l a r a s s o c i a t i o n to o c c u r . molecules e x h i b i t s t r o n g e l e c t r o n i c spectral  transitions  Many o f  i n the  r e g i o n , or s t r o n g r o t a t i o n a l - v i b r a t i o n a l  M stars  visible  transitions in  i n f r a r e d and so are v e r y s i g n i f i c a n t o p a c i t y s o u r c e s . the c l a s s i c a l  the  For example,  MK c l a s s i f i c a t i o n scheme (Keenan and M c N e i l , 1976)  for  i s based p r i m a r i l y on the s t r e n g t h o f TiO bands, which  c o m p l e t e l y dominate the o p t i c a l o f T i O , f o r example, a f f e c t s  spectra of late M s t a r s .  f l u x i n t o the "windows"  The presence  the atmosphere i n two b a s i c ways.  i s the l i n e b l a n k e t i n g a f f e c t which serves to channel  the  between the a b s o r p t i o n f e a t u r e s .  i s the backwarming o f the photosphere t h a t r e s u l t s  One  emergent The o t h e r  from the a b s o r p t i o n  and subsequent t h e r m a l i z a t i o n o f photons i n the TiO bands. typical  these  Since  energy o f the photons absorbed by these bands ( p r i m a r i l y  the blue and v i s u a l  spectral  regions)  exceeds the mean thermal  the in  energy  o f the gas p a r t i c l e s , the net a f f e c t  i s to t r a n s f e r  radiation field  i n a warming o f the p h o t o s p h e r i c  layers.  to the gas r e s u l t i n g  Furthermore, the abundance o f TiO i n the atmosphere (which  determines the magnitude o f the above e f f e c t s ) f u n c t i o n o f temperature  (and to a l e s s e r  T h i s example i l l u s t r a t e s  well  is  extent,  itself  necessary to have an a c c u r a t e , y e t determine the chemical  the b a s i c d i f f i c u l t i e s  gas f o r temperatures  reasonably f a s t  equilibrium ( i . e .  a strong  pressure).  s t r u c t i n g model atmospheres f o r l a t e - t y p e s t a r s .  of con-  As a b a s i s , i t  ranging from 1500-5000K. is  It  is this  of a  problem  intended to be the  s t e p i n the development o f a model atmosphere code f o r  type g i a n t s t a r s with extended atmospheres.  is  procedure to  the e q u a t i o n o f s t a t e )  t h a t I address i n t h i s work, and as such i t first  energy from the  late-  3  2.  Historical  Development  The c a l c u l a t i o n o f the chemical  e q u i l i b r i u m i n cool  stellar  atmospheres remained a b a s i c o b s t a c l e f o r many years to the f o r m u l a - , t i o n o f model atmospheres f o r these s t a r s . of Russell  (1934) y i e l d e d q u a l i t a t i v e l y  The p i o n e e r i n g  c o r r e c t r e s u l t s , but remained  l i m i t e d by the l a c k o f b a s i c thermochemical and by c o n s t r a i n t s Russell  o f manual  effort  and s p e c t r o s c o p i c d a t a ,  calculation.  determined the chemical  e q u i l i b r i u m o f a gas at a  s p e c i f i e d temperature and p r e s s u r e through the use o f e q u i l i b r i u m c o n s t a n t s to express mass balance among the s p e c i e s i n terms o f the p a r t i a l  p r e s s u r e s o f the f r e e atomic s p e c i e s .  While the problem  can a l s o be approached through the m i n i m i z a t i o n o f the t o t a l energy o f the system (White e t . chemical  full until  a l . , 1 9 5 8 ) , most i n v e s t i g a t i o n s  e q u i l i b r i u m i n the s t e l l a r  the c l a s s i c a l  Russell e q u a t i o n s .  free of  atmospheres c o n t e x t have used  The numerical  complexity of  the  s o l u t i o n o f the problem, however, d i s c o u r a g e d f u r t h e r work the advent o f high speed e l e c t r o n i c computers rendered  calculations The f i r s t  the  tractable. investigator  to perform a d e t a i l e d a n a l y s i s  m o l e c u l a r e q u i l i b r i u m problem coupled with a model s t e l l a r  of  the  at-  mosphere was Vardya ( 1 9 6 6 ) , who computed e q u i l i b r i u m c o n c e n t r a t i o n s o f one hundred chemical p o s i t i v e and n e g a t i v e  species  ions)  ( i n c l u d i n g neutral  formed from f i f t e e n  temperatures and p r e s s u r e s r e p r e s e n t a t i v e  atoms, m o l e c u l e s ,  elements,  for  o f l a t e K and M s t a r s .  4  Vardya  used the  inverse  temperature  G = 5 0 4 0 . 3 9 / T and  the  * fictitious  partial  variables.  p r e s s u r e o f h y d r o g e n p^,  The f i c t i t i o u s p a r t i a l  the r e s u l t i n g p a r t i a l  ions into free neutral  =  P  +  H  P + H  the atmospheres  +  PH  +  of these  estimate o f the e l e c t r o n  and I.,  I I ^ / d i  is  A -> A  i,  +  H  2 p  recombinahydrogen  pressure  H , +  of  2  approximation that  late-type  stars  H , N a , K,  pressure  + P ),  p  g  where i  e  Si  all  the  arise  ,  I. =  e  from the  in  first an  by  = H , N a , K, S i , Mg  P + ( P /PA ) A  electrons  and M g , V a r d y a o b t a i n e d  given  i.e.,  + e"  the  i n the forms o f H,  the e q u i l i b r i u m constant d e s c r i b i n g the  of element  assuming  to  be:  i o n i z a t i o n of the elements  e  state  refers  For e x a m p l e , i f  i n t h e gas o n l y  A s s u m i n g f o r an i n i t i a l  p  atoms.  then the r e s u l t i n g ' f i c t i t i o u s p a r t i a l  * h y d r o g e n p^ w o u l d  PH  species  m o l e c u l e s and t h e c o m p l e t e  i s c o n s i d e r e d t o be p r e s e n t H~ and W^,  independent  p r e s s u r e o f an e l e m e n t  pressure of the atomic  complete d i s s o c i a t i o n of a l l tion of a l l  as h i s  first  ionization  5 Vardya s i m p l i f i e s the m o l e c u l a r e q u i l i b r i u m equations by o n l y the more abundant s p e c i e s ( f o r  approximately solar composition),  y i e l d i n g simple equations which can be s o l v e d d i r e c t l y estimates  of  , p , c  p ,  ,  Q  simultaneous equations  retaining  . . . etc.  to g i v e  The complete s e t o f  initial  fifteen  (one equation f o r each element c o n s i d e r e d )  were  then s o l v e d e x a c t l y by a p p l y i n g the Newton-Raphson method to t h i s and i t e r a t i n g  W^, CO, H 2 O ,  to convergence.  These c a l c u l a t i o n s  OH, Ng. S i 0 and HC1  g i a n t atmospheres.  indicated  are the most abundant molecules i n M  o f s t a t e under the assumption o f a grey atmosphere, and the p u b l i s h e d i n a s e r i e s o f very u s e f u l  supergiant s t a r s .  graphs i l l u s t r a t i n g  equation  results  the march o f m o l e -  depth f o r l a t e K and M dwarf, g i a n t and  T h i s work was a s i g n i f i c a n t advance over the o l d s i n g l e -  l a y e r m o d e l s , s i n c e temperature and p r e s s u r e v a r i a t i o n are now accounted f o r .  Still,  i n the atmosphere  the assumption o f a grey atmosphere  presents a c o n s i d e r a b l e source o f e r r o r s i n c e the stars  that  Model atmospheres were c a l c u l a t e d u s i n g t h i s  c u l a r abundances w i t h o p t i c a l  system,  opacity  in  re-  late-type  i s h i g h l y non-grey. A related  study by Vardya  (1967)  concerned the importance o f  negative  ions i n cool s t a r s .  Another s e r i e s o f graphs were presented showing the  variation of partial  p r e s s u r e o f these ions with o p t i c a l  depth f o r  K and M s t a r s , again under the assumption o f a grey atmosphere. prevalent  i o n i c s p e c i e s were found to be H~, C l ~ , 0~, and S~,  the l a r g e abundance o f C I " these models.  A CI  i s an a r t i f a c t  The most  although  o f the high c h l o r i n e c o n t e n t  in  l o g number d e n s i t y of 7.2 on a s c a l e with l o g n^ = 12  was assumed; a more reasonable c u r r e n t v a l u e i s Noyes, 1972).  late  Even s o , the p a r t i a l  log n ^  pressure of C l ~ w i l l  comparable to t h a t o f H~ i n the o u t e r l a y e r s  o f an M6 III  = 5.4  (Hall  still star.  be  and  6 Goon and Auman (1970) molecules above an o p t i c a l characteristic  spectral  depth T^= 3 f o r wavelengths  features  atmospheres o f Auman (1969) temperatures  c a l c u l a t e d column number d e n s i t i e s o f  o f t h a t molecule o c c u r .  for late-type  s t a r s with  The non-grey  effective  between 2000K and 40Q0K, t h a t i n c l u d e d the a t o m i c , mole-  c u l a r and i o n i c s p e c i e s c o n s i d e r e d by Vardya study.  a t which the  (1966), were used i n t h i s  The c o n c l u s i o n reached by the authors was t h a t HC1, HS, S i S ,  NH^, HCO, H^S and COr, should be p o t e n t i a l l y d e t e c t a b l e  in  stellar  spectra. Tsuji  (1973)  p u b l i s h e d a more e x t e n s i v e a n a l y s i s  atmosphere chemical a total  e q u i l i b r i u m problem, c o n s i d e r i n g 36 elements and  o f 232 s p e c i e s over a range o f temperatures and p r e s s u r e s  a p p r o p r i a t e to l a t e - t y p e coefficients of quartic  stars.  T h i s paper c o n t a i n s a t a b u l a t i o n o f  polynomial a p p r o x i m a t i o n s t o l o g  t i o n o f the i n v e r s e temperature 0 , where  c o n s t i t u e n t atoms.  equilibrium calculation  chemical  I have used these values in t h i s research.  the c h e m i s t r y o f d i f f e r e n t  as a f u n c -  i s the e q u i l i b r i u m c o n s t a n t  d e s c r i b i n g the d i s s o c i a t i o n o f the p a r t i c u l a r its  o f the s t e l l a r  species i  into  f o r my m o l e c u l a r  While T s u j i  investigated  temperature and p r e s s u r e regimes i n c o n -  s i d e r a b l e d e t a i l , no attempt was made to i n c o r p o r a t e these  results  i n t o a model atmosphere. Irwin (1978, 1981) presented an even more complete t a b u l a t i o n o f polynomial approximations to the p a r t i t i o n molecular species. chemical  The i n t e r n a l  partition  f u n c t i o n s f o r 344 atomic and f u n c t i o n o f a gaseous  s p e c i e s i s d e f i n e d by  Q = I g. expC-E./kT) 1=o • where the summation 1 1s over the bound e l e c t r o n i c  states,  each with  7  energy E.  above the ground s t a t e and s t a t i s t i c a l  summation i s is  t r u n c a t e d at  weight  g^.  The  s t a t e n w i t h a c u t - o f f energy A E , where AE  the d e p r e s s i o n o f the continuum due to the i n t e r a c t i o n with n e i g h -  bouring p a r t i c l e s the p a r t i t i o n  i n the g a s .  This truncation  f u n c t i o n from d i v e r g i n g to  As shown by Irwin ( 1 9 7 8 ) , l o g a r i t h m o f the p a r t i t i o n  In Q =  5  I  a.(lnT)  .  i s n e c e s s a r y to  prevent  infinity.  given a polynomial approximation to  function,  the  i.e.  ,  1  i=o then the l o g a r i t h m o f the e q u i l i b r i u m c o n s t a n t f o r t h i s  s p e c i e s can be  w r i t t e n as  In  K =  5  I. b . ( l n T ) i =o  where the c o e f f i c i e n t s d i s s o c i a t i o n energy D.  + b /T  1  g  ,  b. can be c a l c u l a t e d  T h i s procedure ensures t h a t the p a r t i t i o n  (required  f o r the o p a c i t y c a l c u l a t i o n s )  (required  f o r the e q u a t i o n o f s t a t e )  fortunately,  from the v a l u e s a^ and the  and the e q u i l i b r i u m c o n s t a n t  are i n t e r n a l l y  consistent.  s i n c e T s u j i d i d not p u b l i s h h i s p a r t i t i o n  (which I  adopted i n t h i s work) and i n d e p e n d e n t l y d e r i v e d o p a c i t i e s , are an e x h a u s t i v e  1600 s p e c i e s formed from a l l  study of the chemical  inhave  inevitable.  equilibrium of  nearly  o f the 92 elements H through U was  performed by Johnson and Sauval weighted column d e n s i t i e s  Un-  functions,  c o n s i s t e n c i e s between h i s m o l e c u l a r e q u i l i b r i u m c o n s t a n t s  Recently,  function  (1982).  The a u t h o r s c a l c u l a t e d  f o r these s p e c i e s based on a g r i d o f red  model atmospheres o f s o l a r c o m p o s i t i o n c a l c u l a t e d by Johnson,  giant  Bernat,  8  and Krupp (1980).  These column d e n s i t i e s , which i n c l u d e d a f l u x  w e i g h t i n g f u n c t i o n , were e v a l u a t e d at s p e c i f i c  wavelengths  c o r r e s p o n d i n g to the p o s i t i o n o f p r i n c i p a l m o l e c u l a r a b s o r p t i o n f e a t u r e s , as well as f o r the standard wavelength results reaffirmed e a r l i e r  o f 1 ym.  The  s t u d i e s r e g a r d i n g the dominance o f  molecules such as H^, CO, OH, N , H 0 , S i O , HC1, HF, HS, SiH i n 2  atmospheres o f l a t e - t y p e  giants.  2  It  the  a l s o demonstrates t h a t most o f  the remaining p o t e n t i a l l y d e t e c t a b l e ,  but c u r r e n t l y  molecules i n T a t e M g i a n t s are e i t h e r  i o n i z e d metal monohydrides  + + or i o n i z e d monoxides, e . g . CaH and TiO .  undetected  9  CHAPTER TWO  A GENERAL  SOLUTION OF  1.  Chemical In t h i s  STATE  OF T H E PROBLEM  E q u i l i b r i u m o f a Gas o f A r b i t r a r y section  I will  e q u i l i b r i u m o f a gas o f  c o m p o s i t i o n , f o r which an a r b i t r a r y be c o n s i d e r e d i n the e q u i l i b r i u m . species  In the f o l l o w i n g , may r e f e r  index k, k = 1...K H-  Let  P  o f chemical  refers  species  present  to the n e u t r a l  f r e e atomic s p e c i e s o f  p r e s s u r e o f s p e c i e s ny  PL. =  partial  pressure o f neutral  k  P p  E  f r e e atomic s p e c i e s  k.  f i c t i t i o u s p a r t i a l p r e s s u r e o f element k, i . e . the r e s u l t i n g p a r t i a l p r e s s u r e o f the f r e e atomic s p e c i e s k i f a l l molecules i n the gas were d i s s o c i a t e d .  =  electron  =  total  * _ P =  formed from K elements.  i n the g a s , w h i l e the  partial  * p 5  s p e c i e s a r e to  Assume the gas c o n s i s t s o f N  =  N  arbitrary  i s to be assumed t h a t the index n, n = l . . . , N  to any a r b i t r a r y  element  list  (atoms, m o l e c u l e s , and i o n s ) it  Composition  d e s c r i b e the development o f an  a l g o r i t h m to s o l v e the chemical  chemical  EQUATION  pressure.  gas p r e s s u r e , i n c l u d i n g e l e c t r o n  pressure.  f i c t i t i o u s total pressure, i . e . resulting partial p r e s s u r e o f the f r e e atoms (not i n c l u d i n g the e l e c t r o n s ) i f a l l molecules were d i s s o c i a t e d .  10 q  =  n  charge ( i o n i z a t i o n s t a t e )  of species  n.  N ^ =  number o f atoms o f element k i n s p e c i e s  N  =  E N ^ = total  =  fractional  =  i o n i z a t i o n e q u i l i b r i u m c o n s t a n t r e l a t i n g the  n  I  n.  number o f atoms i n s p e c i e s n.  abundance o f element k,  by number, with £ a^=  p r e s s u r e o f i o n i z e d s p e c i e s n with charge q n e u t r a l parent s p e c i e s . K  =  n  1.  partial to t h a t o f  n  its  m o l e c u l a r e q u i l i b r i u m c o n s t a n t r e l a t i n g the p a r t i a l p r e s s u r e o f the n e u t r a l m o l e c u l a r s p e c i e s n to the p a r t i a l pressures o f i t s c o n s t i t u e n t n e u t r a l f r e e atoms.  Also, T will  be used throughout t h i s chapter to denote the system  temperature (assumed to be i n thermodynamic e q u i l i b r i u m ) , and h and k refer  to the Planck and Boltzmann c o n s t a n t s r e s p e c t i v e l y .  between these q u a n t i t i e s  P  *  Pk p  K  =  n  =  ak  K  N  n  n  * Pk  y I  p*  E N p n n n  N .p nk^n H  E E N . p tr „• n  n  i s the n e u t r a l  i o n i z a t i o n equation  n  1  k  —  * n'  n k Pn  END n  *• 2  If  immediately.  Pe + n £ Pn  =  K  follow  Some r e l a t i o n s  n*n  n  E p E N , n k nk n  —  n  n n K  E p N *n n  n  n  —  -j  n n r  parent s p e c i e s o f s p e c i e s n with charge q , n  is  n + q e" n M  and the c o r r e s p o n d i n g i o n i z a t i o n e q u i l i b r i u m c o n s t a n t I  is  the  11 I  = P P n e  n  H  /P  H  For a n e u t r a l  /  p  n  or  species, q  Similarly,  if  P  n  = I  q n P'/P n V e H  = 0 and thus  n is a neutral  1 = 1 .  m o l e c u l e , then the d i s s o c i a t i o n equation  can be w r i t t e n  where n  k  indexes the k-th element present i n s p e c i e s n.  d i s s o c i a t i o n e q u i l i b r i u m constant K  The c o r r e s p o n d i n g  i s then g i v e n by  or  For a f r e e neutral I  atomic s p e c i e s n t h i s  s p e c i e s to the p a r t i a l  pressures of i t s  pressure of  pressure of  parent and the e l e c t r o n p r e s s u r e , and s i m i l a r l y ,  e x p r e s s i o n r e l a t i n g the p a r t i a l partial  = 1.  have now o b t a i n e d an e x p r e s s i o n r e l a t i n g the p a r t i a l  an i o n i z e d ( p o s s i b l y m o l e c u l a r ) neutral  reduces to K  I  have an  p r e s s u r e o f a n e u t r a l m o l e c u l e to  s p e c i e s and the p a r t i a l  pressures of i t s  atoms, along with the e l e c t r o n p r e s s u r e .  n —-—  „ _ nk n p  K P "  k »k  q  n  e  the  f r e e atomic c o n s t i t u e n t s . ' T h e s e expressions may be  combined to y i e l d a r e l a t i o n between the e q u i l i b r i u m p a r t i a l an a r b i t r a r y  its  pressure of  constituent  free  T h e r e f o r e , f o r any s p e c i e s  n,  (2-1)  12 The equations o f mass-balance among the s p e c i e s  i n chemical  e q u i l i b r i u m can how be imposed i n terms o f the s p e c i f i e d abundances.  T h e r e f o r e , f o r each o f the k elements,  Pt  I n  k  J  N  a.k = — + =p*  nk n p  . Y N p k t n  n n K  I(a N  or  elemental  k  n  n  - Y N ,p = 0 nk n L  p  - N ) n k  P  n  = 0  ,  k=2,...K  (2-2)  Note t h a t the complete s e t o f K equations are not l i n e a r l y s i n c e Y a. k K-l.  = 1 and so any one equation may be d e r i v e d from the chosen the mass balance equations f o r  2 , . . . , K - 1 , o m i t t i n g the equation f o r K=l Charge n e u t r a l i t y  o f the gas y i e l d s  c o n s i d e r as equation 1 o f my s e t .  n  It  P A  - P  e  (normally  atoms, and so t h e r e f o r e  total  hydrogen).  This is  (2-3)  pressures p  p r e s s u r e s p^ o f the f r e e  there e x i s t a total  k  Hence one a d d i t i o n a l  r e q u i r e d to c l o s e the system, and t h i s  n  (p ,k=l,...K k  k = l , . . . , K and the  equation  is just  i s to be  constituent  o f K+l unknowns  to be d e t e r m i n e d , g i v e n the c o m p o s i t i o n a ,  gas p r e s s u r e p.  shall  simply expressed by  = 0  w r i t t e n i n terms o f the p a r t i a l  e  elements  another r e l a t i o n , which I  should be r e c a l l e d t h a t each o f the p a r t i a l  and p )  remaining  K  I have t h e r e f o r e  I  independent,  (K+l  the t o t a l  o f my set) i s  pressure  relation.  13 P  +  e  I n I  n  I  P  n  PA  = P  n  IP  +  n  =  p  n  P (q n  n  + 1) = P  (2-4)  T h i s system o f K+l equations must be s o l v e d f o r the K+l pressures to determine the chemical One model o f approach i s available. chemical this  e q u i l i b r i u m o f the gas.  to assume t h a t an approximate s o l u t i o n  The h i g h l y n o n l i n e a r system o f equations d e s c r i b i n g  e q u i l i b r i u m o f the gas i s  resulting linear  s o l u t i o n was s u f f i c i e n t l y straightforward  the  estimate.  to convergence provided the  good.  is  then l i n e a r i z e d , and the s o l u t i o n o f  system used to improve the o r i g i n a l  procedure can then be i t e r a t e d  relatively  partial  I will  show l a t e r  that i t  to o b t a i n s u i t a b l e s t a r t i n g  This  starting is  indeed  solutions.  Let us proceed with the l i n e a r i z a t i o n , b e g i n n i n g with the mass balance  equations,  n kn  T a. N t  I  nk')*np' = 0  - N ,  assume t h a t the p  p^ with P  n  =  P  n  +  n  SP n  available T  n  e  i s o n l y an e s t i m a t e o f the exact  objective  i s to s o l v e f o r the  solution  correction  term 6p .  n  N  N  n k  )(P  nk> n 6p  + 6p ) = 0  n  n  =  K nk N  n  "  <W  P  n  (2-5)  14 T h i s expresses the l i n e a r i z e d mass balance equation i n terms o f changes i n the p a r t i a l  pressures 6 p  v a r i a b l e s , however, are the p a r t i a l and the e l e c t r o n p r e s s u r e p  of a l l  p  „  n  -  I  o n l y , and so the c o r r e c t i o n 6 p  g  k  n  T h i s i s accomplished by  nk k  n  + <5p = "  -  g-  Kn ( p ' e+ ' S o  }"  'n  .  e  0  In K  n  N  %  k  &  n  V  e  k ^  6  P  nk + ? —  (1 + E  N  nk  K P^  k  P  \  6p  - q n q  6p - q Pn  %  k  ^/ifVU' nk  k  X  n  „  , k  e  N  P  Pn  K  ^  p  ^  6pe ,  k  n  (1 - q —^}(1  N  k  nk \ N (1 + -)' '  Nnk. /,  N . p + p (E — Pn Pn ^ p k  >n  k  qnk v  P  p  n  n '  6 p  +  r  p  +  E p  n e  n e  n  Q  KP>  K  Pn  k  K P Ni ^) n  (  11  q  I  5  must be  n  (2-1)  TT  "  and  f r e e atoms p^  N . n  K p n e  p  Our independent  p r e s s u r e s o f the n e u t r a l  expressed i n terms o f these q u a n t i t i e s . l i n e a r i z i n g equation  the s p e c i e s .  the  ^Pe, e  Sp  K  k  "fe  Nnk, P  \ r  n K  \  )  e —- ) p 6 p  e  6p  —) p '  '  (2  "  6)  15  where i t i.e.  Sp  has been assumed t h a t the q u a n t i t i e s 6p /p.. k k  n  «  n  expansions  1 , 6p /p e  are s m a l l ,  k  «  e  , <Sp  n e 1 i n o r d e r t h a t the v a r i o u s f i r s t  order  performed above be v a l i d .  Then, s u b s t i t u t i n g back i n t o the l i n e a r i z e d equation o f mass balance,  (2-5),  nk E (a. N - N , )p (Z — 6p • - q k n nk n i< p 'n. ^n n k N  v  n  yt  v  k  nk Z (a. N - N , )p E — Sp k n nk n p n N  / H  n  k  H  n  k  1 - p g  MASS  BALANCE  6 p  = E N . - a. N ) p k nk k n n K  e  S  n  ( a  k n N  "  N  nk  ) p  n n q  a, N )p  5 nk  =  LINEARIZED  —pe )'  (N  (2-7)  EQUATIONS  T h i s i s the l i n e a r i z e d equation o f mass b a l a n c e . Next c o n s i d e r the charge n e u t r a l i t y E p'q n n K  n  - p 'e  =0  1  M  where a g a i n , the primed q u a n t i t i e s r e f e r  p  n  =  p  n  p; = P •"•  E  n  I  ( p  n q  n  +  6 p  +  e  6  6 p  n  n "  6 p  to the exact  solution  n  + 5p P >  equation.  q  e  n "  e  =  ( p  P  e  +  e " \  6 p  e> p  =  0  n n q  and s u b s t i t u t i n g i n the e x p r e s s i o n f o r 6 p  n  (Equation  2-6),  16 N  nk  Z p q  (Z  Z p q  N. nk Sp Z -  LINEARIZED  CHARGE  Sp  P —-) 6  - q  -  - Sp  P  = p  1 •+ — Z p q P n nvn  Z p'(q  the t o t a l  + 1)  - Z p q , *n^n n  (2-8)  EQUATION  y i e l d s the l i n e a r i z e d equation o f charge Finally,  Z p q  5p„ = p *e *e  2  e  NEUTRALITY  -  neutrality.  pressure e q u a t i o n i s l i n e a r i z e d ,  =p  s(p + « p ) ( q + D = P n  n  n  2(q+-D 6p = P - Z p (q n  -  n  n  n  n  n  "  n  + 1)  and t h e r e f o r e  I  PnK  + 1  N , nk « > p n. k n. k v f  p  K  N Z Kp V(q + 1) J — H n  LINEARIZED  2.  £ P  n n  TOTAL  n  Sp  ^  PRESSURE  Sp e p e K  q  n  - -  K  = p - Z p (q + 1) n n n ' K  X M  Z H p q H(q + ;1)  p [n fi  nV n  J  0f  Sp  Z p (q + 1) *n ^n '  e  n  (2-9)  EQUATION  E v a l u a t i o n o f E q u i l i b r i u m Constants Consider the  reaction  AB -»- A + B r e f l e c t i n g chemical T h i s could r e f e r atoms A and B,  e q u i l i b r i u m between the r e a c t a n t s  to the d i s s o c i a t i o n o f a diatomic molecule AB i n t o  or e q u a l l y w e l l ,  to the i o n i z a t i o n o f a n e u t r a l  i n t o a s i n g l y charged ion A and a f r e e e l e c t r o n B. (1966),  and the p r o d u c t s .  i f N^, Ng, N^g are the t o t a l  the  atom AB  As given by Tatum  number o f the r e s p e c t i v e  species,  17 and Z . , Z , Z n  A  D  are t h e i r  n D  respective  total  partition  functions,  then  AD  f o r a system i n e q u i l i b r i u m ,  ¥ B  N AB  where E states  Q  . V B  P  Z -AB  V  K  T  i s t h e energy d i f f e r e n c e between the r e a c t a n t s  and the products i n t h e i r  ground s t a t e s .  in their  ground  T h i s would correspond  to the d i s s o c i a t i o n o r i o n i z a t i o n energy f o r t h e two examples mentioned previously. The t o t a l  partition  f u n c t i o n Z o f a s p e c i e s c o n t a i n e d i n a volume  V i s the product o f i t s i n t e r n a l partition  f u n c t i o n Q, and i t s t r a n s l a t i o n a l  2TrmkT\  3 / 2  /W\  AB  "I  vibrational  partition  electronic)  f u n c t i o n given by  h  3/2  Then  " o E  V  /  k  T  ^AB  2  where m i s now t h e "reduced mass" M^Mg/M^g.  Using pV = NkT, t h e  d e s i r e d e x p r e s s i o n f o r the e q u i l i b r i u m c o n s t a n t equation)  and  v  where m i s t h e s p e c i e s mass.  N  (rotational,  (the g e n e r a l i z e d Sana  i s obtained.  K„CT) - VB =(l™hlY T W - V * i /2  K  AB  \ h  Z  /  E  Q  A B  {  18  For the s p e c i f i c case o f i o n i z a t i o n o f a s p e c i e s X : X i d e n t i f i c a t i o n s AB=X, A=X (the  i o n i z a t i o n energy),  +  X  and B=e~ can be made.* A l s o , E  K^g = 1^, m = m  g  and Q  g  = 2.  + e~, the  +  = x  Q  With these  r e d u c t i o n s , the standard Saha equation i s o b t a i n e d ,  (2-11)  where m  e  is  log I  x  the e l e c t r o n mass, or  numerically,  = 2.5 l o g T - 0.48 + l o g I —^— I - P  For atomic s p e c i e s , at low state s t a t i s t i c a l  temperatures,  0 3  ^-  Jx  9  normally Q  x  (2-12)  - g ^ , the ground x  weight.  An analogous equation can be w r i t t e n f o r m o l e c u l a r d i s s o c i a t i o n , but now the e v a l u a t i o n o f the m o l e c u l a r p a r t i t i o n difficult.  This c a l c u l a t i o n  vibrational  and e l e c t r o n i c  involves  f u n c t i o n i s much more  summing over the v a r i o u s  rotational,  l e v e l s o f the m o l e c u l e ; the i n t e r e s t e d  i s r e f e r r e d to the e x c e l l e n t a r t i c l e by Tatum (1966) on t h i s The most c o n s i s t e n t approach  to s o l v i n g the chemical  reader  subject.  equilibrium  problem would be to express the v a r i o u s e q u i l i b r i u m c o n s t a n t s d i r e c t l y terms o f the polynomial equation  fits  to the p a r t i t i o n  ( 2 - 1 0 ) , as suggested by  mentioned advantage t h a t the same  Irwin  functions required  (1981).  partition  T h i s has the  in  in  previously  f u n c t i o n s may be used i n  the o p a c i t y c a l c u l a t i o n s , e n s u r i n g c o n s i s t e n c y between the o p a c i t y and the equation o f s t a t e .  It  a l s o decouples  the p a r t i t i o n  f u n c t i o n from the  d i s s o c i a t i o n energy i n the e x p r e s s i o n f o r the e q u i l i b r i u m c o n s t a n t .  This  r e a d i l y a l l o w s f o r r e v i s i o n o f the o f t e n p o o r l y determined d i s s o c i a t i o n  energy ( u s u a l l y constant)  the l a r g e s t  without  source o f e r r o r  the n e c e s s i t y o f f i t t i n g  to these e q u i l i b r i u m c o n s t a n t s . compilation of p a r t i t i o n until  i n the new  Unfortunately,  equilibrium  paratnetrizations Irwin's  (1981)  f u n c t i o n approximations was not  available  t h i s work was a l r e a d y underway and so t h i s approach was not  followed.  Instead,  evaluated d i r e c t l y partition  the i o n i z a t i o n e q u i l i b r i u m c o n s t a n t s were from the Saha equation (2-12) u s i n g  f u n c t i o n s f o r 9 = 1.0  (Allen,  1976).  tabulated  The m o l e c u l a r  e q u i l i b r i u m c o n s t a n t s , f o r now, were c a l c u l a t e d d i r e c t l y polynomial approximation o f T s u j i  (1973).  be expressed i n terms o f I r w i n ' s polynomial coefficients  for future  work.  These w i l l , partition  from the  however, function  20  CHAPTER THREE AN O P T I M I Z E D  1.  Initial A.  SOLUTION  Estimates  OF THE  of Partial  EQUATION  OF  STATE  PROBLEM  Pressures  S i m p l i f y i n g assumptions  The 1 i n e a r i z a t i o n method o f s o l u t i o n o f the chemical described e a r l i e r initial  i n the previous c h a p t e r r e q u i r e s r e a s o n a b l y good  estimates o f the s o l u t i o n to converge r e l i a b l y .  since this  equilibrium  technique i s e q u i v a l e n t  Furthermore,  to g e n e r a l i z e d Newton-Raphson method  and thus e x h i b i t s q u a d r a t i c convergence near a s o l u t i o n , i t d e s i r a b l e to achieve as good an i n i t i a l the number o f i t e r a t i o n s such i n i t i a l  is  e s t i m a t e as p o s s i b l e to reduce  i n the l i n e a r i z a t i o n l o o p .  As I w i l l  now show  estimates are r e a d i l y o b t a i n e d , but at the p r i c e o f  ducing assumptions about the c o m p o s i t i o n o f the g a s .  intro-  The b a s i c  assumptions needed to render the problem c o m p u t a t i o n a l l y t r a c t a b l e  i)  p  «  p  :  H  where n r e p r e s e n t s any s p e c i e s except H, H , Hr, or He. +  all i  i  }  P C N ' ^  PC  are:  this  This i s c e r t a i n l y  valid  for  "normal" s t e l l a r compositions, favours M-stars (with n^ < n^) over  C - s t a r s , although the s t a r t i n g actually  estimates  obtained remain useable even  the C-star  case.  for  21  iii)  p<-.Q «  :  PQ  this  is certainly  v a l i d for C-stars,  somewhat l e s s so i n the M-star c a s e . good res-timates are o b t a i n e d f o r iv)  molecule formation among  0 , S,  N,  where p  Si obeys p  n  «  1c  1c  1c  p ,  p ,  p ,  p ,  M-stars.  Q  N  $  that m o l e c u l a r  G  ,C  1c  p. $  ,  pressure o f any m o l e c u l a r  i n v o l v i n g such an element This states  Still,  the elements other than H , H 1c c  i s the p a r t i a l  n  but  species  with any o f C, N , 0 , S or  Si.  a s s o c i a t i o n o f H , C, N , 0 , S,  atoms w i t h o t h e r s does not s i g n i f i c a n t l y d e p l e t e  Si  these  elements. With these a s s u m p t i o n s , the f i c t i t i o u s  total  pressure can be  expressed s i m p l y .  P* * P  +  H  H  2 P  V  +  2  PHe  +  and s i n c e P P*  PH  K  88  P  PH  +  p  +  +  2  - P  H  P  H  +  Pe  +  (3-D  e  The v a r i o u s f i c t i t i o u s as  PHe  +  partial  pressures  follows:  P  H "  p  He"  p  P  C  P  P  K  P  H  H  2 p  +  2  PH+  +  P  H-  He  C  * " 0 p  +  +  +  PH C  PH 0  +  +  P  C 0  PH 0 2  P  +  +  C  +  PQ C  +  P  0  +  can now be approximated  22  ~-  PN  *  P  Si  *  Ps  Pci  *  PTi  *  Pv  P  *  p  a  p  For a l l PK  B.  ci Ti  *  p  v  H  K  P  PHS  +  P  Y  *  s  +  +  PHCI  +  T  p  Y  +  P iO  +  vo H  Zr  +  Psio  +  *  *  P  Si  s  PNH  +  +  +  YO p  Zr  0  P N  2  P  +  2  SiS  +  P  P  H S  +  +  2  N+  P  P  SiH  SiS  +  +  p  P  S  Si+ +  Pci + p  T i +  P+ v  P  Y02  +  p  P+ Y  +  ZrO  ?  + Zr+ p  o t h e r elements assume PK  =  +  P  The Estimates  K  +  of Electron  The most d i f f i c u l t pressure p ,  Pressure  quantity  s i n c e f o r the outer r e g i o n s o f cool  g  e l e c t r o n s are s u p p l i e d by a l a r g e low abundance. temperature  to estimate a c c u r a t e l y  number o f metals each o f  dominates  relatively  rising  l a r g e abundance.  the metals as an  upon t e m p e r a t u r e .  estimate o f p  g  electron  The equation o f s t a t e code f o r a  cool model atmosphere must be a b l e to handle the f u l l e  the  hydrogen begins to c o n t r i b u t e to the e l e c t r o n p r e s s u r e , and  donor because o f i t s  p  electron  s t a r s , most o f  Deeper i n the atmosphere, however, with  at s u f f i c i e n t l y warm temperatures  of  i s the  range of dependence  The procedure I have used to o b t a i n an  i s o u t l i n e d below.  d e r i v e d f o r the more r e a d i l y  Basically,  treated  estimates o f p  g  initial  are  extremes o f high t e m p e r a t u r e ,  23  where e s s e n t i a l l y so p  all  o f the e l e c t r o n s  - p + , and o f low temperature,  g  Fe along with C.  from i o n i z a t i o n o f H and  ./Where most o f the e l e c t r o n s  H  due i o n i z a t i o n o f the r e l a t i v e l y  arise  abundant metals Na, Mg, A l , S i ,  A s i m p l e , though - c r u d e , estimate o f p  range o f temperature  i s s u b s e q u e n t l y improved by a  e  very good estimate i)  (usually  electron  estimated i n the  s u p p l i e s most o f the e l e c t r o n s .  a  or  H  H  P  + p^-  limit,  a (P H  +  H  P*  a (p  In t h i s  P  e  p  - P) e  p )  u  «  = P "  2  PH  +  P  +  2  H  +  PH VP  = p  2  R  + 2p  in this  therefore. p  e  H  H +  high temperature l i m i t where H  The abundance o f H, by number i s :  P-  +  H  e  H 2  +  p  H +  + p H  p _ « u  p, so  H  + (  .  total  + P+  PH Pe H+  limit, = '- p  to the  p and can be n e g l e c t e d , and (always)  / r  +  where the e q u i l i b r i u m r e l a t i o n Also,  value),  : e  pressure i s  estimate  l i n e a r i z a t i o n method to y i e l d a  the c o n t r i b u t i o n from H» denoted by p ^ ,  \ _  over a wide  T h i s crude  w i t h i n ^ 2% o f the c o r r e c t  The high temperature l i m i t  First,  K, C a ,  i s then o b t a i n e d by merely t a k i n g the maximum o f  these two l i m i t i n g v a l u e s o f the e l e c t r o n p r e s s u r e . of p  g  are  1  p^ = p^+(p /I^+) has been s u b s t i t u t e d .  essentially This  }  yields  e  all  o f the e l e c t r o n s come from H and  24 a (p  - p )  - P (P /I  + 1-)  I Pe H+ V e " AHVP=0 H  e  e  +  0  r  P  e * H  ii)  e  1( +  1 L" V  ( 1  H  +  p  +  H  T  i  The low temperature  + ( 1  involved.  so d i r e c t l y ,  I approached t h i s  i o n i z e d metals  H  ]  2  problem by t r e a t i n g a l l K, C a , Fe)  for Z is rather  estimate of p  the above m e t a l s ) , noted t h a t  (the  arbitrary  g  b e a r i n g on the f i n a l provided t h a t t h i s  p  i s o n l y an i n i t i a l The p a r t i c u l a r  is P A  H  a, l  -  ( p  P  H  +  H  P*  +  P  = | a^,  where  "effective"  =  7.3  eV to  2  P  H " Pe  }  P H P  +  =  P  H  PH+  +  2  H  2  +  -  2 p  will  yield  my work.  It  should be  estimate to supply the value o f x^ adopted has no  the l i n e a r i z a t i o n  step,  s t a r t i n g e s t i m a t e o b t a i n e d was s u f f i c i e n t l y c l o s e  the l i n e a r i z a t i o n procedure to converge. H  species Z  e l e c t r o n p r e s s u r e c o n t r i b u t i o n from  value o b t a i n e d a f t e r  g  easily  s i n c e the best v a l u e  found x^  is  M  subsequent l i n e a r i z a t i o n s t e p .  for  as one f i c t i t i o u s  and t h i s v a l u e was used i n a l l  t h i s value o f p  electrons  the r e l e v a n t  The c h o i c e o f an  v a r y with temperature and c o m p o s i t i o n , but I  e  o f the  and abundance  the summation runs over the above m e t a l s .  a satisfactory  all  s i n c e now more than one e l e c t r o n donor  ionization potential  ionization potential  (3-2)  +  supply e s s e n t i a l l y  ( C , Na, Mg, A l , S i ,  with an e f f e c t i v e  a  limit:  T h i s case i n which the metals cannot be t r e a t e d  +  P  H„  +  P  H -  E  +  P  H+  +  PH"  for  A g a i n , the abundance equation  25  In the cool temperature the e l e c t r o n  limit,  pressure p  arises  g  abundant than hydrogen. but  H  (  P  Since  P H  +  2  }  P  =  = P^/p^  H  ,  +  P  from i o n i z a t i o n o f m e t a l s , a l l  Therefore, p  formation may be very  A  hydrogen i o n i z a t i o n i s n e g l i g i b l e , and  2  P  H  ,  p^_,  p  g  «  p and can be i g n o r e d ,  significant.  2  = ^/K p  H  H +  much l e s s  h  H  and t h i s  equation may be expressed  in  terms o f , and s o l v e d f o r p^ .  a (p H  + p  This y i e l d s  p  ) =  2 "2  the q u a d r a t i c  + 2p  equation  ,2_2 (2 - a ) ' p ^ - [ 2 a ( 2 - a ) p + K^] p ^ + a H  with  H  H  H  H  H  H  f o r the f i c t i t i o u s  Pz  Z  P  p*  Z  +  metal  2  =0 .2  H  2 a ( 2 - a ) p + Kj.  2(2-a )' Also,  p  '2a (2-a )p  2a (2-a ) p + H  2  (3-3)  H  2J  Z,  Pz+  Here the assumption i s made t h a t Z does not take p a r t i n molecule and i s  present  states. 1^+,  i n the gas o n l y i n the n e u t r a l  Then, i n v o k i n g a f i c t i t i o u s  atomic or s i n g l y  ionization equilibrium  ionized  constant  which can be expressed in terms o f the p r e v i o u s l y mentioned  potential  x  z  by means o f the Saha r e l a t i o n  formation  (2-11), t h i s abundance  ionization equation  26 f o r Z can be w r i t t e n as  P  a  z * z+^ p  =p  In the cool  ^(p+p^)  or  p  This  yields  + I  I p  e  with p  a n d  p* - p + p  g  2  2  z  p  z i— =  and  l  e  Z+  p+ - p 7  L  6  I  ( p + p^)  z +  + 4a I  Z  P +P  e  z  z +  =0  (p + p  =  H  ) (3-4)  i-  given by the estimate o f e q u a t i o n  u  H  +  z  +^I  limit, P - )  =  e  - a I  -  =  7  Z +  z+ — p ~  sincer  +  e  e  z  r" '  = p (l p  z+  P +P  %  temperature  So,  2  +  e  follows  A u s e a b l e , though c r u d e , estimate over a wide range o f temperatures  p  g o  (3-3). o f the e l e c t r o n  pressure  can now be o b t a i n e d simply by u s i n g  electron  p r e s s u r e a r i s i n g from the dominant source (hydrogen or  i .e.  = max (p  p e  C.  o  e  Linearization A very  , p Z  e  o f charge n e u t r a l i t y  as a s t a r t i n g and i t e r a t i n g  e  =  I nV p  pressure can now be o b t a i n e d , g u e s s , by l i n e a r i z i n g until  convergence.  e l e c t r o n d o n o r s , ( H , He, C, Na, Mg, A l , S i ,  i n c l u d e d i n the l i n e a r i z a t i o n .  p  equation  o f the e l e c t r o n i c  equation o f charge n e u t r a l i t y  neutrality  metals),  )  u s i n g the above rough estimate  are  the  H  good estimate  o n l y the major  valid  Again,  K, C a ,  S t a r t with the e q u a t i o n o f  the  Fe)  charge  27  Assume, as b e f o r e , t h a t the e l e c t r o n s a r i s e  from s i n g l y i o n i z e d  donors t h a t are not s i g n i f i c a n t l y d e p l e t e d by m o l e c u l a r a s s o c i a t i o n . T h i s i s a very  good approximation s i n c e hydrogen a s s o c i a t e s o n l y  temperatures at which i t the charge n e u t r a l i t y relevant  p  e  i s unimportant as an e l e c t r o n donor.  equation can be w r i t t e n ,  at  Therefore,  summing over o n l y  the  species,  I k+ p  =  k  since  q =1for  s i n  9  l y  0 for  By the above a s s u m p t i o n ,  k  p  ionized species neutrals.  V  +  ) p  *  k+  and 1 + P /I e  k +  The e q u a t i o n o f charge n e u t r a l i t y  P  e -k =E  P  k  = p +  *I  kV kV e a  r  H  and s i n c e the t o t a l vary w i t h p . g  -  (3-5)  + p  which I now proceed to l i n e a r i z e  P* * P + P  then becomes  P  in p . g  First,  however,  note  that  e  pressure p i s  Therefore, I  define  f i x e d at a s p e c i f i e d v a l u e ,  p* w i l l  P  1  =  P  P^  +  > which w i l l  p* = p'  - p  remain c o n s t a n t as p  g  g  P  (3-5)  (P  =  P e  I  T  e  +  +  (3-6)  I assume p  g  P  e  i s an approximate s o l u t i o n o f t h i s such t h a t p  The assumption i s made here t h a t 6pg  H  +  e  6p H  = (p*  e  e  , ,y  6  - p  a  T  Vk  e' fe I  + v  +~n~  y ^  k+  a  "  p  5  P  e  e  + 6pg  g  p .  i s an exact s o l u t i o n .  .  +  g  I k >Vv eV k +  eI k k+ e  ~ P ' " Pe  T  equation  Then  g  6pe 6p p kV  p  n  «  .  + p  v  kk V V J,e  or  6p)  -  ^e  K W  Cp'-P )I  neutrality  +~n~  k k I  and so seek a c o r r e c t i o n 6pg  P  The charge  now becomes  " PJ  To l i n e a r i z e ,  So I now have  £  y i e l d i n g the c o r r e c t dependence o f p* upon p . equation  varies.  (  6 0  6  6p  u  5  p  11  }  p  Vk  +  + -  n  -  )  2  ; a  k k T  +  e ^ ' - P e i ^ -  P^  * ^kV_ 7  P  *P< 1  +  &h  +~TT "  ? T  k l. + + k  +  P  Pp  with p* = p'  - p.  e e  Pe  An e x c e l l e n t e s t i m a t e o f the e l e c t r o n p r e s s u r e i s o b t a i n e d i n iterations  (3-7)  several  by t h i s method s t a r t i n g with the p r e v i o u s l y d e r i v e d rough  29  estimate,  f o r a range o f temperatures  from 1500K to 30000K, and  pressures from l o g P = 0 to 10.  D.  Estimates i)  o f Atomic P a r t i a l  Pressures  Hydrogen  Knowing the e l e c t r o n the p a r t i a l  pressure p , an a c c u r a t e g  estimate  of  pressure o f atomic hydrogen (and a b e t t e r value  p^,  f o r p,, ) M  can be o b t a i n e d . a  a  H  A g a i n , the mass balance equation f o r H i s :  p* P* _ P  H P (  +  + 2 pP H "+ Pe P  H  PH "  H 2  +  Pe  2  Using the v a r i o u s to t h a t o f p^,  }  =  p _ H  2  PH  +  2  PH  2  +  P+ H  +  p  H-  equilibrium expressions r e l a t i n g  those p a r t i a l  pressures  i.e.  v V "  P  h+=(  ^ '- i )Ph  and s i m p l i f y i n g , a q u a d r a t i c  2-a„  2  Iy+  ?  PH  =(  , P H  e q u a t i o n i n p^ r e s u l t s .  p  Therefore, •P  ^ p  p  H-  X  l  H-  1 *  p  p  1  ^e  2C2-a )/K H  V  H "  H  f  2  P  "Pp  + 4a„(2-a„)( -^-) F  H  H  K  H  2  (3-8)  and al so  p„ 2  H  K  =  P*  yield  Pu  =  P  +  P  H  -  Z  P  E  improved e s t i m a t e s for these  ii)  quantities  C,N,0,  The mass balance equations f o r C i s  a  - C* . p* P  C  P  C  +  P  a  C * P  =  P  CV  +  P  H  KT„  CH  ,  +  P  C0 p* p  +  l  'XO  XH  C  P  . C +  0  K7,  +  +  P  e  and  a p* c  CH f o r 0: K  Similarly,  *0 a p* 0  0  J l p*  „  p  K  0  +  C0  p  p  QH  e  H 0 p*  +  p  2  +  H H C = p ll + IT— + T ) — + °\ OH H 0 C0 n  P  P  P  K  P  °V  , "H K  0H  , H  G'  P  P  H 0 ?  0+  —  e  a  „  where the above e x p r e s s i o n f o r p  P  +  +  ,  P  K  K  2  C0  +  r t  I,  equation.  p  P  c  H  C *  .  P  +  p  0  +  V,  V' p  e  has been s u b s t i t u t e d i n the  After manipulation, this yields  latter  the f o l l o w i n g equation f o r  "2" P  ( 1  H  .  IT  +  K  P  K  +  OH  K  H  w  .^O+w,  2  +  P  H K  H 0  P  2  +  OH  +  ,  » CO  P  I, 0  which i s r e a d i l y  K  0H  p  s o l v e d to y i e l d  (3-10)  e  the i n i t i a l  estimate  Equation  (3-9 ) above then y i e l d s  Finally,  the N abundance equation can be w r i t t e n :  Pjj  a.  A  W  =  P  _  P0 N  or  P  P  N  H  +  2  N  +  P  This is  +  N  N  +  0  +  P N  +  2  Other  P  Q  f o r p^.  N+  +  I C „  +  NH  p"  1  P  ^e  N  a p* = 0  (3-11)  N  N'  elements  The o n l y other C and 0 are S i  p .  the c o r r e s p o n d i n g e s t i m a t e  s o l v e d to o b t a i n the d e s i r e d e s t i m a t e o f p  iii)  for  important elements which couple i n the manner o f  and S ( t h r o u g h ' t h e  formation o f S i S ) .  As b e f o r e ,  the  mass-balance e q u a t i o n f o r S y i e l d s  «cP* ! 1  where  p  s i  +  PH_  K k+ HS  i s as yet  P  K  H  2  H S 2  P  K  5i Si.S  (3-12) .  A  S+  P  e  undetermined.  32 Similarly,  the Si equation  SiP  a  p  1+ K  yields  SiO  K  s  !iL_  +  SiS  K  ^Si  +  SiH  e  p  S u b s t i t u t i o n o f the S equation into the expression f o r p ^ yields  J  eventually  the quadratic,  1  Si  K  SiS  (1 +  /,  'Si  p  K  fl- !sii,  '0 K  SiO  K  0  p  SiO  K  +  SiH  H  • ^i +w^  SiH  p  P  H  P  +  K  H2  HS  £  I  S  +  "H S 0  P  u  e  r  S  , P* ,  Si'K . SiS c  c  (3-13)  which  is readily  p<. t h e n  immediately follows  Initial in  solved to give  estimates  an a n a l o g o u s  the estimate  from e q u a t i o n  f o r several  a  and t h e r e f o r e ,  p  P  CI  C1  P  p'  a  CI  C1  additional  Cl  1+ HC1 p  HC1  +  P  C1  p  V  Similarly,  + P  i r e  +  p  C  a  Ti  Ti  1 +  P  +  K  TiO  The e s t i m a t e f o r  (3-12).  manner.  CI :  Ti :  o f p^^.  P  e  elements  c a n be o b t a i n e d  33  V*  P  K  V0  p  e  a p* y  -  :  P  in  ~  Y  K  Y0  K  a  K  Finally,  any o t h e r elements  P  Z r  ZrO  (e.g.  T  2  Y0  P  2  e  P  K  Zr0  P  2  e  He, Ne, Na, Mg, A l , F e , e t c . )  initial  estimates are r e q u i r e d are assumed to be present o n l y i n  neutral  atomic or s i n g l y i o n i z e d s p e c i e s ,  p  n  p  n  +  p  for  which  the  i.e.  n+  From the abundance e q u a t i o n  n p*-  p  a.  n  p  n  +  n+ p* p  a p* n  I  2.  obtain  N  U  V  p  e  Economized S o l u t i o n o f Chemical  Equilibrium  The d e t e r m i n a t i o n o f the chemical  e q u i l i b r i u m o f a hot gas was  s o l v e d i n a s a t i s f a c t o r y manner by the approach d e s c r i b e d i n S e c t i o n Nevertheless,  this  technique does have the drawback o f r e q u i r i n g  solution of a linear  system o f equations o f o r d e r n+1  per  the  iteration  i n o r d e r to determine the chemical e q u i l i b r i u m o f a gas c o n s i s t i n g o f  2-2.  34 n elements.  This i s computationally rather  expensive f o r l a r g e n  s i n c e the number o f o p e r a t i o n s r e q u i r e d to s o l v e a l i n e a r dimension n i s o f o r d e r n , and f o r reasonable s t e l l a r it  system o f  compositions  i s n e c e s s a r y to i n c l u d e at l e a s t 15 elements merely to o b t a i n  correct electron  pressure.  But most o f the c o u p l i n g i n the  the  non-linear  system o f equations d e s c r i b i n g the e q u i l i b r i u m occurs between the latively tion  few abundant s p e c i e s which r e a d i l y  (i.e.  H, C , N, 0 ) , w h i l e the remaining elements are  essentially is  undergo m o l e c u l a r  totally  i n e i t h e r the n e u t r a l  t h e r e f o r e r e a s o n a b l e to ask  re-  associa-  presently  or s i n g l y i o n i z e d s t a t e .  It  whether or not the e q u i l i b r i u m o f such  a system can be d e s c r i b e d by means o f a c o n s i d e r a b l y s m a l l e r s e t o f e q u a t i o n s , without demonstrate, i s  loss of accuracy.  i)  I  have separated the elements i n t o  the  classes:  Major e l e m e n t s : partial  now  yes.  To accomplish t h i s , f o l l o w i n g three  The answer, as I s h a l l  These elements i n c l u d e those f o r which  p r e s s u r e s o f the atoms bound i n t o molecules  present a s i g n i f i c a n t f r a c t i o n o f the t o t a l  the  re-  fictitious  * pressure p .  T h i s requirement b a s i c a l l y s t a t e s  that  those  elements be both abundant and c o u p l e through m o l e c u l a r association.  Note t h a t He, e . g . which i s abundant but does  not couple w i t h any o t h e r element, i s category.  Metals:  this  I have i n c l u d e d the s i x elements H, C, N, 0 ,  and S i n t h i s ii)  not i n c l u d e d i n  Si  class,  This class  i n c l u d e s those elements t h a t are r e a s o n -  a b l y abundant, but do not undergo m o l e c u l a r a s s o c i a t i o n to a s i g n i f i c a n t degree so t h a t e s s e n t i a l l y a l l given element are present e i t h e r  o f the atoms o f a  i n the n e u t r a l  or  singly  35 ionized state.  I have i n c l u d e d He, Ne, Na, Mg, A l , K, C a ,  Fe and Ni i n t h i s  iii)  Minor e l e m e n t s :  class.  This c l a s s  i n c l u d e s those elements o f low  abundance, which t h e r e f o r e c o n t r i b u t e n e g l i g i b l y toward the t o t a l  p r e s s u r e or the e l e c t r o n  p r e s s u r e , but  may r e p r e s e n t s i g n i f i c a n t o p a c i t y s o u r c e s . are s i m p l y not abundant enough to a f f e c t the major  either  nevertheless  These elements  the e q u i l i b r i u m among  s p e c i e s and the m e t a l s , and so the chemical  e q u i l i b r i u m c a l c u l a t i o n s can ignore elements i n the c l a s s . additional  molecular equilibrium c a l c u l a t i o n s  required  o p a c i t y sources can then be c a l c u l a t e d d i r e c t l y partial  p r e s s u r e s o f the major  Sr,  Y, Z r ,  known  (or m e t a l ) s p e c i e s u s i n g the  A somewhat a r b i t r a r y  important minor elements  for  from the  a p p r o p r i a t e mass balance equation o f the p a r t i c u l a r element i n v o l v e d .  Any  includes CI,  list  minor  o f the more  S c , T i , V, C r , Mn, Co,  etc.  The economization o f the system o f equations d e s c r i b i n g the chemical equilibrium is  performed by the f o l l o w i n g a p p r o a c h .  First,  major elements c o u p l e s s t r o n g l y to the o t h e r s and so each partial  p r e s s u r e must be t r e a t e d as a separate  independent  each o f  elemental variable.  And, as mentioned b e f o r e , the minor elements are o f n e g l i g i b l e on the e q u i l i b r i u m and can be i g n o r e d . w h i l e abundant, a l l  behave s i m i l a r l y  p r e s e n t o n l y i n the n e u t r a l all  the metals  element which  Finally,  the metal  the  influence  elements,  i n t h a t they do not c o u p l e and are  or s i n g l y i o n i z e d s t a t e .  T h e r e f o r e , I group  t o g e t h e r and t r e a t them as a s i n g l e f i c t i t i o u s I have d e s i g n a t e d Z , w i t h an abundance equal  metal  to the sum  36  o f the abundances o f the i n d i v i d u a l m e t a l s , i . e .  ou Z  = E a  where  m  m  the ct 's r e p r e s e n t the v a r i o u s metal abundances. Then  p* = p * a L  7  L  = p*Ea  = E(ap*)  m  M  m  m  = Ep* , where m m  p* = p L  L  7  + p L  7 +  .  Now I can r e w r i t e the abundance equations f o r the major e l e m e n t s , lumping all  metals t o g e t h e r i n t o Z, and e x p l i c i t l y summing over the major  only ( i . e .  H, C, N, 0 , S i ,  S),  rather  than over a l l  elements  elements  involved  as b e f o r e . I nk n n E N p + p* n n n Z  p\ k  N  p*  v  p  '  V  K  where s u b s c r i p t s k and n now r e f e r to major elements and s p e c i e s formed s o l e l y from major elements  0  r  I K n  "  N  with p  n  K  „ ^  Sf°k n  p  p  n  V z  +  nk  n^  =  ,  «-14)  0  as  previously  nPe  This l i n e a r i z e s  with  nk>  I  =  N  N  respectively.  "  N  to  nk  }  6 p  n  +  , nk 6p„ = p (E - — 6 p  a  k  Z  =  n  ( N  nk"  e - q„ - — ) ,  N  n^  6 p  6 p  n  k  a  k n > Pn" N  V *  again unchanged from b e f o r e .  *e  S u b s t i t u t i o n y i e l d s the m o d i f i e d l i n e a r i z e d abundance equation f o r element  k,  37  Ka.N n  k  T -Hi  - N . ) p  n  n  k  6  k Pn,  n  n  yt (a.k Nn -  _ _L  P  k  P  k  y  (N  L  n LINEARIZED  ABUNDANCE  EQUATIONS  For the f i c t i t i o u s metal  a  Z  " E n  Z  I Vn  N  N . ).p q nk' n^n  6p  K  e  + a 6p* k  e  FOR  .  nk  - a,kNn')  MAJOR  p p  K  a.r  p Z  v  (3-15)  ELEMENTS  Z, the abundance equation becomes  p + p* n n Z K  K  or a  "  (  1  "  a  Z  J  P  Z  =  (3-16)  0  T h i s l i n e a r i z e s to the f o l l o w i n g  a y 7  1  n  n  N  y  p  N  6  k Pn  n  result.  -  P n  k  k  Pe  ^ (I N P q ) 6 p n n  n  n  e  - (1  n LINEARIZED  ABUNDANCE  The t o t a l  IP +  P  I  PZ  p  n  +  Z  + P  +  + P  Z +  Pe  which l i n e a r i z e s  y  v FOR  FICTITIOUS  p r e s s u r e e q u a t i o n now m o d i f i e s  n  or  EQUATION  =  - a )  Sp*  z  *  n METAL  to  = P  e  (3-18)  P  to  N  P  n LINEARIZED  n  y— k n p  TOTAL  (3-17)  Z.  6  + (l- -  P n  k  PRESSURE  p  y  ~ e- n  q ) 6  P n  n  EQUATION  e  P  •+ 6p  P"  I  Pn"  p  Z"  P  c  (3-19)  38  The e q u a t i o n r e p r e s e n t i n g c h a r g e  I n n n P  q  +  P  Z+  =  with  chosen as t h e a p p r o p r i a t e  Z  +  =  ^ m m p  '  +  ^  m '  K  ^  K  r  e  t  independent  n  m  m  * V  elements  M  M  nr Pz  a n d so p* =  =  z  the metal  +  p;  a  variable.  summation i s o v e r  e  have  p+  m But  e x p r e s s e d i n t e r m s o f p*., w h i c h I  z +  m e ' m+  U,  m 'm  e  p  * p  N  w  now becomes  e  p  T h i s m u s t be r e w r i t t e n  P  neutrality  —  pi a l m ' ZL r \ III m III +' a7 I I + p Z m m e r  t  h  u  s  p  Z+  =  +  +  The m o d i f i e d c h a r g e p*  y ^ n  p q p  n n M  neutrality  a I +  + L i y mjLL_ a  e q u a t i o n i s now  =  p  £ I + p Z m m e m+  7  +  r  H  Q  (3-20)  e  and t h e c o r r e s p o n d i n g l i n e a r i z e d r e s u l t  i s , after  some m a n i p u l a t i o n ,  g i v e n by  *  IP q I n  n  n  p - P n. " K n k n, L  r  6  5  p  n  k  1  +  , p p  . £ ^ nn nn v  e  t  p P  q q  +  z  a_I_+ +  m  P f i  )  P  1  e  . . A A I ( n +  +  P f i  '  P  Z  k  p  v =  LINEARIZED  Py «  CHARGE  NEUTRALITY  " I n  p  n n q  Z  - aZ  EQUATION  v  °>nV I m I  m+  +  +  n +  P  p  e  p  (3-21 )  39  Again,  in this  notation, k refers  s p e c i e s , and m t o a m e t a l  to a major element, n to a major  element.  S i n c e T a, + a = 1, the s e t o f abundance e q u a t i o n s d e s c r i b e d here k r e d u n d a n t and s o I have c h o o s e n t o e l i m i n a t e t h e h y d r o g e n a b u n d a n c e 7  K  is  equation.  L  This leaves a set of  k abundance e q u a t i o n s , the t o t a l  e q u a t i o n and t h e c h a r g e n e u t r a l i t y e q u a t i o n , e a c h c o n t a i n i n g t h e  pressure k+2  * unknowns p^,  p^ and p .  Thus I  g  have a c l o s e d e c o n o m i z e d s e t o f  o f c o n s i d e r a b l y s m a l l e r order than b e f o r e .  S i n c e my c h o i c e o f t h e  o f major e l e m e n t s c o n t a i n s o n l y t h e s i x e l e m e n t s H, C , N, 0 , S i this  e c o n o m i z e d method i n v o l v e s  repeatedly  equations  and  s o l v i n g a l i n e a r system  class S, of  o r d e r 8 f o r t h e c o r r e c t i o n s S p ^ , S p ^ , 6 p ^ , S p g , S p ^ , Sp<,, 6 p * and <5p  g  and i t e r a t i n g u n t i l One d e t a i l pressures p  o f the  m  and  =  a  m  is  i n d i v i d u a l metal  a =  attained. to recover  a  m  (  a ^  =  ^  m\ P  *  Z  so, 1+ I  Pm ,/p m e + /  K  where a g a i n m r e f e r s  I  PePm + + p„ m ^e +  to a metal  the actual  elements a f t e r  T h i s i s done s i m p l y as  * p  is  r e m a i n s , and t h a t  has been f o u n d . * Pm  convergence  element.  follows:  partial  the above s o l u t i o n  40  CHAPTER FOUR  DISCUSSION  1.  Details  of  OF  RESULTS  Results  The a l g o r i t h m f o r t h e s o l u t i o n o f t h e e q u a t i o n o f described  i n the  the general package  previous chapter  has been i m p l e m e n t e d ( i n  and e c o n o m i z e d v e r s i o n s )  (GAS).  a s a FORTRAN  The e c o n o m i z e d v e r s i o n r u n s t h r e e  does the g e n e r a l  s o l u t i o n (71 m s e c . v s .  o f a gas o f s o l a r c o m p o s i t i o n w i t h Amdahl 4 7 0 V/8 c o m p u t e r ) .  state both  subroutine times f a s t e r  223 m s e c , f o r t h e  solution  T = 1 2 0 0 K , l o g P=5 on t h e  The p a r t i a l  These w o r s t c a s e s  c o n t a i n i n g Mg o r A l , w h i c h v i o l a t e economized s o l u t i o n t h a t not deplete  the major  essentially all slightly  general  species.  of these  are  molecules  the  do n o t a s s o c i a t e , and t h e r e b y  are  bound up i n t h e  species 0.  Nevertheless,  i n many o f t h e e q u i l i b r i u m c o n s t a n t s  insignificant.  the  occur for  t h e a s s u m p t i o n made i n  the  do  Under t h e above c o n d i t i o n s ,  elements  d e p l e t i n g the major  the u n c e r t a i n t i e s errors  "metals"  U.B.C.  p r e s s u r e s o b t a i n e d by  two m e t h o d s t y p i c a l l y a g r e e t o 3 s i g n i f i c a n t f i g u r e s , w i t h w o r s t d i s c r e p a n c i e s a r o u n d 2%.  than  Also the d i s c r e p a n c i e s  and e c o n o m i z e d s o l u t i o n w e r e much l e s s  hydroxide  hence  considering i n v o l v e d , these  between  the  f o r c o n d i t i o n s more  typical  of late-type  giants, i.e.  The GAS r o u t i n e s r e a d a l l a species for  specification f i l e ,  T i n the range 2500-4000K,  o f the necessary  which contains a s i n g l e l i n e  e a c h s p e c i e s t o be i n c l u d e d i n t h e c h e m i c a l  gas.  Each l i n e c o n t a i n s  chemical  symbol o f  number o f atoms c o m p r i s i n g t h e  If  the entry  then  it  is  is  i n c l u d e d , and i f  potential  is  included, i f  (form the n e u t r a l  state)  number  is a neutral  an i o n t h e n i t s  m o l e c u l e y . then the c o e f f i c i e n t s o f T s u j i ' s  (1973)  of  Each s p e c i e s  i n d i c a t i n g whether minor species  it  also contains  ionization  it  is  a  polynomial  is  a p r i o r i t y code  fit  u s e r t o r e a d i l y add ( o r  remove) a d d i t i o n a l  This  species  the  to the l i s t  or  FORTRAN  code.  An i n i t i a l  estimate  free atomic species linearization  is  of the  partial  never  solution.  o f cases with  neutral  to s t a r t  S i n c e my e s t i m a t e s ^ w e r e d e r i v e d  the  assuming  C/0 < 1 was  r e q u i r i n g more t h a n 2 o r 3 i t e r a t i o n s .  true of those cases with as t o p r e v e n t  pressures o f the  r e q u i r e d by t h e GAS r o u t i n e s  s o l a r c o m p o s i t i o n , the convergence rapid,  3)  or  scheme a l l o w s  t o m o d i f y abundances o f e l e m e n t s w i t h o u t any change t o t h e source  be  ( 1 , 2 , and  t o be c o n s i d e r e d as a m a j o r , m e t a l  i n the economized s o l u t i o n .  of  atom,  t o t h e e q u i l i b r i u m c o n s t a n t as a . f u n c t i o n o f t e m p e r a t u r e m u s t included.  the  species  each o f t h e c o n s t i t u e n t e l e m e n t s . abundance  the  the  (ionization state)  f o l l o w e d b y p a i r s o f t h e number o f atoms and t h e a t o m i c  its  entry  equilibrium of  the c h a r a c t e r s t r i n g r e p r e s e n t i n g  the s p e c i e s , the charge  s p e c i e s , and t h e t o t a l  i n f o r m a t i o n from  T h i s was  very also  C/0 > 1 w h i c h w e r e s u f f i c i e n t l y warm s o  formation of  polyatomic carbon molecules.  For  42  temperatures  T < 3 0 0 0 K w i t h C/0 > 1.  become i n c r e a s i n g l y i n a c c u r a t e polyatomic species  however,  rather  this  (particularly  HCN and S i C g ) .  not a f a i l i n g of the  equation  The r e a s o n f o r t h i s of state  w h i c h C/0 < 1 .  routines  this will  (for  over a range o f temperatures  it  remains a f a i r l y f o r the  A total  from A l l e n  and l o g P = 5 ( r e p r e s e n t a t i v e ; o f  (1976).  the  These  are d e t a i l e d  o f graphs  (Figs.  b u t i o n o f each o f the i m p o r t a n t elements and CI  among t h e v a r i o u s  temperature.  in Table  pressures  scale  for  1.  lategrid  The r e s u l t s  i n d i c a t i n g the  H, C, N, 0 , S i ,  these f i g u r e s  the  log P = 2  S,  e q u i l i b r i u m s p e c i e s as a f u n c t i o n  The v e r t i c a l  from  abundances,  over a temperature  1-9)  outer  formed  line-forming regions of  respectively)  the  (C/0 = 0.58)  equilibrium, with  r a n g i n g f r o m T = 1200K t o T = 6300K by 100K s t e p s . in a series  for  C/0 > 1 c a s e ;  o f 109 s p e c i e s  The s o l u t i o n s have been o b t a i n e d f o r b o t h t h e  presented  these  straightforward  and p r e s s u r e s a p p r o p r i a t e t o t h e  along w i t h the complete l i s t o f species  stars  with  has been s o l v e d u s i n g  25 e l e m e n t s w e r e c o n s i d e r e d i n t h e c h e m i c a l  t y p e g i a n t and d w a r f  s h o u l d be  future.  stars.  taken  It  the case  e c o n o m i z e d s o l u t i o n m e t h o d f o r a s o l a r c o m p o s i t i o n gas  abundances  lineariza-  from the i n t e n t i o n to use  For t h i s w o r k , t h e e q u a t i o n o f s t a t e  elemental  important  GAS r o u t i n e o r a l g o r i t h m , b u t  s t a r t i n g estimates  be d o n e i n t h e n e a r  atmospheres o f l a t e - t y p e  estimates  i n model a t m o s p h e r e s o f M - g i a n t s ,  Nevertheless,  to improve these  stems  and so t h e  to converge.  a c o n s e q u e n c e o f bad s t a r t i n g e s t i m a t e s  C/0 > 1 ) .  matter  is  initial  due t o t h e n e g l e c t o f s e v e r a l  t i o n s o l u t i o n r e q u i r e s many more i t e r a t i o n s noted t h a t  these  are  distriA l , Mg of  indicatesthe  43  l o g a r i t h m o f the r a t i o o f each s p e c i e s  P /P,  i.e.the  n  to the t o t a l  gas  A s i m i l a r s e r i e s o f graphs  r a t i o o f the  partial  pressure  pressure. (Figs.  10-13) a g a i n  indicate  d i s t r i b u t i o n o f t h e e l e m e n t s H , C , N a n d 0 among t h e v a r i o u s as a f u n c t i o n o f t e m p e r a t u r e ,  i n these cases  abundance o n l y , w h i l e  Finally, effect  the  of varying  (Figs.  over  all  a star with  C/0 < 1 ( i . e .  which f r e e C.remains  after  These  figures  is  calcu-  be l e f t  f r e e C, a f t e r  the.carbon  into  strikingly  C/0 < 1 and This  component i n i t s  is a  direct  essentially  formation.  with a  So  reasonable  CO f o r m a t i o n  is  f o r f o r m a t i o n o f many s t a r case  CO f o r m a t i o n , and so i s  formation of a plethora of  the  and  broken  CO m o l e c u l e w h i c h  T h i s oxygen i s then a v a i l a b l e holds for  This  (arbitrarily  an M - s t a r ) w i l l  o x i d e s . ? The r e v e r s e  the  unity.  illustrate  the cooler temperatures.  of the 1 e s s a b u n d a n t  slightly  the range 2000-4000K w i t h  presentation).  amount o f f r e e 0 , b u t v e r y l i t t l e considered.  14-18)  the d i f f e r e n t chemistry o f the regimes  p a r t i c u l a r l y at  C-  abundances  e q u i l i b r i u m of the gas.  r e s u l t o f the extreme s t a b i l i t y o f the depletes  variation  C/0 r a t i o , e a c h f o r a f i x e d t e m p e r a t u r e  two g r o u p s f o r c l a r i t y o f  C/0 > 1 ,  other elemental  f o r the most abundant s p e c i e s  demonstrate  The  sum o f t h e a b u n d a n c e s a s  f o r a s e r i e s o f temperatures  log P = 3,  f o r l o g P = 3.  p e r f o r m e d by i n c r e a s i n g t h e  a t h i r d set o f graphs  p r e s s u r e , upon t h e c h e m i c a l lated  is  all  scaling all  downward so a s t o p r e s e r v e  species  b u t now f o r t h e c a r b o n e n r i c h e d c o m -  p o s i t i o n s C/0 = 1 and C/0 = 1 . 5 , i n C/0 r a t i o  the  ( C / 0 > V) available  p o l y a t o m i c carbon compounds.  for' for  '  44  2.  Astrophysical  Implications  The s o l u t i o n o f t h e m o l e c u l a r e q u i l i b r i u m p r o b l e m  enters  i n t o t h e l a r g e r model a t m o s p h e r e p r o b l e m i n two w a y s . o b t a i n the o v e r a l l stant  pressure  d e p e n d e n c e o f d e n s i t y upon t e m p e r a t u r e  in order that  may be i n t e g r a t e d  in fact  the equation of h y d r o s t a t i c  inward i n t o the atmosphere.  s t r a i g h t f o r w a r d as species,  One i s  it  requires  This  is  is  considered.  problem.  It  is  the s o l u t i o n o f the  important to r e a l i z e that  ( p e r h a p s a c c o u n t i n g f o r o n l y 10  pressure)  in certain  and y e t  spectral  certain  region.  t h e v i s i b l e a t 3000K f o r  partial -8  o r even 1 0 "  it  the  from the c h e m i c a l  that  total  all  gas  infrared  3000K f o r M - s t a r s ,  and Cg i n  C-stars.  equilibrium will  solution obtained for a l l  is essential  o f the  E x a m p l e s a r e H~ i n t h e n e a r  S i n c e t h e o m i s s i o n o f a s p e c i e s more a b u n d a n t t h a n  validate  species  dominate the a b s o r p t i o n c o e f f i c i e n t o f the  at 5000K, TiO i n the v i s i b l e r e g i o n at  o p a c i t y sources  important  radiative  i n terms o f 6  gas  atmospheres  to determine the e q u i l i b r i u m abundances o f the  may be p r e s e n t as v e r y m i n o r c o n s t i t u e n t s  pressure  good  CO, N g ,  +  o p a c i t y sources c o n s i s t e n t with transfer  con-  the s p e c i e s H, C, N, 0 , H g , H, H ,  The s e c o n d , more c h a l l e n g i n g a s p e c t o f t h e s t e l l a r problem,  abundant  A very  +  and HgO a r e a l l  reasonably  a t o l e r a b l y good s o l u t i o n i s o b t a i n e d by  is obtained i f  con-  equilibrium  knowledge o f o n l y the most  s i d e r i n g o n l y h y d r o g e n i n t h e f o r m s o f H g , H and H . solution  at  to  species with  these  in general  o f the l e s s abundant s u s p e c t e d abundance  in-  species,  greater  45  than or comparable to the v a r i o u s v  the e q u i l i b r i u m c a l c u l a t i o n s . species  necessary  o p a c i t y s o u r c e s be i n c l u d e d  I n c o n t r a s t t o t h e mere h a n d f u l  t o o b t a i n an a c c u r a t e  d e n s i t y , we now m u s t  on t h e o r d e r o f one o r two h u n d r e d s p e c i e s o b t a i n an a c c u r a t e c o n s i d e r the cool its  light cycle,  i n the o p t i c a l  SC M i r a v a r i a b l e bands o f CaCl  P  CaCl ' /  P  o f VX A q l this  accurate  At c e r t a i n  In  fact,  And y e t  this  it  at  1977).  c a l c u l a t i o n s , but i t  evident  case,  is  never  lower  limit  find  from t h e o p t i c a l  demonstrates  and c o m p r e h e n s i v e e q u a t i o n o f s t a t e  p a r t o f a n y r e a l i s t i c model a t m o s p h e r e  of  features  t h e need f o r c o d e a s an  of  log  spectra  s p e c i e s m u s t be a s i g n i f i c a n t o p a c i t y  T h i s example c l e a r l y  to  C a C l j_s  T = 2500K, l o g P = 3 I  is  include  phases  become t h e s t r o n g e s t band  n  that  star.  As an e x t r e m e  abundant t o appear above t h e l o g P /P  -8.  =  VX A q l .  s p e c t r u m ( C l e g g and W y c k o f f ,  -9 on my f i g u r e s .  of  i n the c a l c u l a t i o n s  absorption c o e f f i c i e n t .  i n c l u d e d i n my e q u a t i o n o f s t a t e sufficiently  in  source  in  an  integral  program f o r l a t e - t y p e  stars.  TABLE  I  SPECIES CONSIDERED IN EQUATION OF STATE  ELEMENT  Z  ABUNDANCE  H  1  9 . 323E-01  He C  2 6  6 . 526E-02 4 .941E-04  N 0  7 8  8 .950E-04 8 .484E-04  Ne  10 1 1 12 13 14 16 17 19 20 21 22 23 24 25 26 27 28 38 39 40  Na Mg A1 Si  s  CI K Ca Sc T1 V Cr Mn Fe Co Ni Sr Y Zr A TOTAL OF  7 1 2 2 3 1 3 8 1 1  738E-05 678E-06 424E-05 238E-06 077E-05 492E-05 729E-07 298E-08 865E-06 492E-09 1 212E-07 2 331E-08 6 619E-07 2 331E-07 3. 729E-05 1 .119E-07 1 .865E-06 6. 6 1 9 E - 1 0 5. 874E-1 1 2 .9 8 3 E - 1 0  109 S P E C I E S  SPECIES CONTAINING THIS ELEMENT H OH HCO He C  sic  N 0 SIO ZrO Ne Na Mg Al SI  s  CI K Ca Sc T1 V Cr Mn Fe Co N1 Sr Y Zr  H+ H20 MgOH He+  c+  S1C2 N+ 0+ SO Zr02 Ne+ Na+ Mg+ A1 + S1 + S+ C1 + K+ Ca+ Sc+ T1 + V+ Cr+ Mn+ Fe+ Co+ N1 + Sr+ Y+ Zr+  FORMED FROM 25 ELEMENTS  HMgH A 1 OH  H2 A1H CaOH  H2 + S1H CH2  CH HS CHS  C2H2 H2S C2H  NH HC 1  NH2  C2 HCN N2 02 CaO HCO  C3 HCO NH OH ScO MgOH  CH CH2 NH2 H20 Sc02 A) OH  C2H2 CH3 NH3 NO TiO CaOH  CN C2H CN' CO VO  CO  C02  CS  NO C02 V02  HCN MgO YO  A10 Y02  NaCl Mg+ + A1H S1H HS CI -  MgH A10 SIO H2S HC1  MgO A1C1 SIS SO  MgCl A 10H S 1C CS MgCl  MgOH S1C2 SIS A1C1  Ca++ ScO T10 VO  CaH Sc02 T1S V02  CaCl  CaOH  Y0 ZrO  Y02 Zr02  NaCl CaO  T1S CaCl  CaH  NH3 HCN  log P = 2  CHEMICAL EQUILIBRIUM OF CARBON COMPOUNDS H  LOG(P)=5  I I I I I I I [ 1 I I I I 1 I I I [ I I I | || I 1 I | I I I I I I I I I |I I I 1 1 1 1 I I | I 1 H  Figure 2(b):  C/O = 0 . 5 8 log P = 5  (solar),  CHEMICAL EQUILIBRIUM OF OXYGEN i i| i i i n i i i i |  ^  I ' i i i i i i i | i i i i i i i  n  LOG(P)=2 i \ fi i  i i i i i r i j i i M  | i i  H  2CO  H,0  SiO  ^SiO  AlOH CC Ti(  XO  _liO  W*  -8-  VO'  1000  2000  Figure 4(a):  H  1  1  3000  1  I I  1  1  I |  1  I  1 1  1  1  I I I |  1  I I I I  5000  6000  T(K)  C/0 = 0 . 5 8 ( s o l a r ) , log P = 2  CHEMICAL EQUILIBRIUM OF OXYGEN I  T  40&0  1  I I I | I  1  1  I I I I  1  L0G(P^5  I | I I I I I I I I I |  1  I  Ij  0 ^  2-q  SiO  4000 Figure 4 ( b ) :  C/0 = 0 . 5 8 ( s o l a r ) , 1og P = 5  5000 T(K)  i i Ii i i 6000  Figure 5(b):  C/0 = 0 . 5 8 l o g P = 5.  (solar)  T(K)  =2  CHEMICAL EQUILIBRIUM OF SULFUR i i n  rTTTTTTTTT i i i I I | II T i i i i | i i i i i i i i i | i i i i i i i i i | i i i i i 'i f i i [  11  11 H  2-q  I IIII IIIII III I  1000  2000  Figure 6(a):  I '1 I'T'l I "| I i i I i I I I i j i i r i i ' i i  3000 C/0 = 0 . 5 8 log P = 2  4000 (solar),  5000  II|II I  6000  T(K)  CHEMICAL EQUILIBRIUM OF SULFUR  'n  2000 Figure 6(b):  3000 C/0 = 0 . 5 8 ( s o l a r ) , log P = 5  4000  5000 T(K)  i r 6000  i i | i  54  55  CHEMICAL EQUILIBRIUM OF CHLORINE L0G(P)=2 i | i 11 i i i i i i | i i i i i ' i 1 i i | I i  i i i ii i i i i [ i i i i i i i i i | i i i i i i n  H  0 ^  0, \  -2-  o o -4-  CI  -8-q  NaCl M i l  1000  2000 Figure 9 ( a ) :  n 3000  i "i n i i1 I i i i i M i i r i i i i i i I i i 'i  C/0 = 0 . 5 8 ( s o l a r ) , log P = 2  4000  11 i i 6000  5000 T  (  K  )  CHEMICAL EQUILIBRIUM OF CHLORINE lI I I I I I I I I 1 I II I I I1 1 1 1 I I I 1 1 I 1 I 1T  1 1 I 1 I I I  L0G(P)=5  if IIIII 1 I I I I I I'M  H  0 ^  \ k *—'  o o  -2-g -4-3  j^i i i i m Figure 9(b):  C/0 = 0 . 5 8 ( s o l a r ) , log P = 5  T(K)  i i  ii  I T  '56  Figure 10(b):  C/0 = 1 . 5 ,  Log P = 3  T(K)  Figure 11(b):  C/0 = 1 . 5 ,  log P = 3  T(K)  CHEMICAL EQUILIBRIUM OF NITROGEN i I i i i i i i i i i I i i i i i i i i i I i i  C/0=1 Iffiffi?, t. • ! . • q  T i i i M I  i i i i i i i |  o-3 -2-3  o -4-  -6-3  -8-3q NH, i i i i i  2000  1000  Figure 12(a):  3000 C/0 = 1 , l o g P  4000 3  5000  T(K)  o-q \  o o  -3  -6-3  h i  1000  i I'M  q 2000  i i  Figure 12(b):  3000  4000  C/0 = 1 . 5 , l o g P = 3  5000 ?(K)  6000  EQUILIBRIUM iOF i i nCHEMICAL M ii ii ii i1 ii |i ii ii ii ii ii ii ii ii ii |III ii ii ilOXYGEN ii ii ii ii ii |l ii ii ii iiC/ ii 11 m  C/0=1 |  0^  L O GI'I ^ S| i i  ii iTi i t  «j  _C0  4-^  -8AIOH\  i i i Vi i i i i I iTT 1000 2000 Figure 13(a):  3000  1000  n f i n | i i Vi n n i ] 2000 3000  Figure 13(b):  5000  6000  T(K)  C/0 = 1 , l o g P  CHEMICAL EQUILIBRIUM OF OXYGEN  H I  p i i ri i i i i | i i r  4000  C/0=1.5  11 i i i i i i i i | i i i  T 4000  C/0 = 1 . 5 , l o g P = 3  L0G(P)=3  5000 T(K)  6000  ABUNDANCE OF IMPORTANT SPECIES -( n t i n i i i i I I I I i i i i i i i i i i i i i i"i i i i  m  0.4  i n i ii  1  M  rT=2000K i 1 1 i i i i i i i iL0G(P)=3 i i ri 1 i i i i i  i i i i ii ii i j i l i ii  0.6 Figure 14(a):  0.8  1.0  T = 200QK, log P = 3  ABUNDANCE OF IMPORTANT SPECIES  Figure 14(b):  T = 2000K, log P = 3  1.2 C/0 RATIO T=2000K  C/0 RATIO  H  61  0.4  0.6 Figure 15(a):  0.4  0.6 Figure 15(b):  0.8  1.0  T = 2 5 0 0 K , l o g P=3  1.4  Q/Q RATIO  T = 2500K, log P = 3  0.8  1.2 '  1.0  1.2  C/0 RATIO  1.4  ABUNDANCE OF IMPORTANT SPECIES  Figure 16(a):  T = 3000K, • log P = 3  Figure 16(b):  T = 30QOK, log P = 3  T=3000K  C/O RATIO  C/O RATIO  L0G(P)=3  63  ABUNDANCE OF IMPORTANT SPECIES  Figure 17(a):  0.4  0.6 Figure 17(b):  T = 3500K, log P = 3  0.8  T=3500K  L0G(P)=3  C / O RATIO  1.0  T = 3 5 0 0 K , l o g P=3  1.2  c / 0 RATIO  1.4  ABUNDANCE OF IMPORTANT SPECIES  i i i i i i i i i i i i r i i i i i i I i i i i i i i i i I 11  11  T=4000K  LOG(P)=3 11  i i i i i I i i i i i i i i i  i i I I  64 I N  H OH  \  -2CO  o  17  -8^ 0.4  i  i | i i i i  0.6  n  i i i | i i i i i i i i i | i i i i i i i i i]  0.8  1.0  1.2  Figure 18(a):  T = 4000K, log P = 3  C/0 RATIO  Figure 18(b):  T = 4000K, log P = 3  C/0 RATIO  i I I I i i  iTT  | i i i "i H  1.4  r  65  REFERENCES  A l l e n , C.W. Press,  1976, A s t r o p h y s i c a l London.  Auman, J . R .  1 969, A p . J .  Clegg,  R.  and W y c k o f f ,  Hall,  D.N.B.  Irwin,  A.W.  and N o y e s ,  1977, M.N.R.A.S. 1970, A p . J .  R.W.  1972, A p . J .  thesis,  1981, A p . J .  S u p p l . 45_, 6 2 1 .  H.R.,  Johnson,  H.R.  Bernat, A.P.  179, 417.  Lett.  University  of  and K r u p p , B . M .  and S a u v a l , A . J .  1982, A s t r .  Tatum,  J.B.  Tsuji,  T.  Vardya,  1934, Ap. J .  1_75,  L95.  Toronto.  1980, A p . J .  Suppl. 42,501.  and A p . S u p p l . 49^, 7 7 .  K e e n a n , P . C . and M c N e i l , R . C . 1 9 7 6 , An A t l a s S t a r s , Ohio S t a t e U n i v e r s i t y Press. R u s s e l l , H.N.  Athlone  1_61_, 5 3 3 .  1978, Ph.D.  Johnson,  3rd e d . ,  157_, 7 9 9 . S.  G o o n , G. and A u m a n , J . R .  Quantities,  of Spectra  of the  80, 317.  1 9 6 6 , P u b l . Dom. A p . O b s . V i c t o r i a ,  ]3_, 1 .  1 9 7 3 , A s t r o n . and A p . 23_, 4 1 1 .  M.S.  1966, M.N.R.A.S. 1967 , Mem. R o y .  White, W.B., Johnson, 2 8 , 751 .  S.M.,  134, 347. A s t r o n . Soc.  D a n t z i g , G.B.  71_, 2 4 9 . 1958, J .  Chem.  Phys.  Cooler  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085777/manifest

Comment

Related Items