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

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A FAST , ROBUST ALGORITHM FOR THE SOLUTION OF THE EQUATION OF STATE FOR LATE-TYPE STELLAR ATMOSPHERES by PHILIP DESMOND BENNETT B . S c , Simon Fraser Un i ve r s i t y , 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE •in THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1983 © P h i l i p Desmond Bennett, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department Of Geophysics and Astronomy The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 12. 1983 i i ABSTRACT A f a s t , but accurate procedure to solve the equation of state for late-type s t e l l a r atmospheres i s an essent ia l component of a r e a l i s t i c model atmosphere code. This requires the determination of the chemical equ i l ibr ium of a gas conta in ing s i g n i f i c a n t amounts of perhaps one to two hundred spec i es , over a wide range of temperature, pressure and composi t ion. A general method of s o l u t i o n , based on a l i n e a r i z a t i o n approach, is der ived f i r s t . This i s accurate but has the disadvantage of requ i r ing the so lu t ion of a l i n ea r system of equations of order N, where N is the number of elements considered in the equ i l i b r i um, for each i t e r a t i o n . I then show that the order of th i s system can be reduced to 8 by the in t roduct ion of a " f i c t i t i o u s " metal element, thereby t r i p l i n g the so lu t ion t iming without s i g n i f i c a n t loss of accuracy. Both the general and economized algorithms make no assumptions as to the pa r t i cu l a r species to be considered in the equ i l i b r i um ; a l l such information i s read from a s p e c i f i c a t i o n f i l e at execution t ime. F i n a l l y , the equi l ibr ium abundances of s i g n i f i c a n t species are d isplayed g raph i ca l l y over a range o f temperature, pressure and com-pos i t ion (C/0 r a t i o ) , with these resu l t s p lot ted both as a funct ion of temperature and C/0 r a t i o . The importance of obta in ing accurate equ i l ibr ium abundances of the various opac i ty sources , and the imp l i c a -t ions for model atmosphere const ruct ion are d i scussed . i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS ix CHAPTER ONE THE EQUATION OF STATE FOR LATE-TYPE STELLAR ATMOSPHERES 1 1 . Introduct ion ^ 2. H i s t o r i c a l Development 3 CHAPTER TWO A GENERAL SOLUTION OF THE EQUATION OF STATE 9 1. Chemical Equ i l ib r ium of a Gas o f A rb i t r a r y Composition 9 2. Evaluat ion of Equi l ibr ium Constants 16 CHAPTER THREE AN OPTIMIZED SOLUTION OF THE EQUATION OF STATE PROBLEM 20 1. I n i t i a l Estimates' of Par t ia l Pressures 20 A; S impl i f y ing Assumptions 20 B. Estimates of E lectron Pressure 22 C. L i nea r i za t i on o f Charge Neu t ra l i t y Equation 26 D. Estimates of Atomic Par t i a l Pressures 29 2. Economized Solut ion of Chemical Equi l ibr ium 33 iv Page CHAPTER FOUR DISCUSSION OF RESULTS . 40 1 . Deta i l s o f Results 40 2. Astrophys ica l Impl icat ions 44 REFERENCES 65 L I ST OF TABLES TABLE I. Species considered in Equation of State vi L I ST OF FIGURES Page FIGURE 1. Chemical Equi l ibr ium of Hydrogen: a) C/0 = 0.58 ( s o l a r ) , log P=2 47 b) C/0 = 0.58 ( s o l a r ) , log P=5 47 2. Chemical Equi l ibr ium of Carbon: a) C/0 = 0.58 ( s o l a r ) , log P=2 48 b) C/0 = 0.58 ( s o l a r ) , log P=5 48 3. Chemical Equi l ibr ium of Ni t rogen: a) C/0 = 0.58 ( s o l a r ) , log P=2 49 b) C/0 = 0.58 ( s o l a r ) , log P=5 49 4. Chemical Equi l ibr ium of Oxygen: a) C/0 = 0.58 ( s o l a r ) , log P=2 50 b) C/0 = 0.58 ( s o l a r ) , log P=5 50 5. Chemical Equi l ibr ium of S i l i c o n : a) C/0 = 0.58 ( s o l a r ) , log P=2 51 b) C/0 = 0.58 ( s o l a r ) , log P=5 51 6. Chemical Equ i l ibr ium of Su l f u r : a) C/0 = 0.58 ( s o l a r ) , log P=2 52 b) C/0 = 0.58 ( s o l a r ) , log P=5 52 7. Chemical Equi l ibr ium of Aluminium: a) C/0 = 0.58 ( s o l a r ) , log P=2 53 b) C/0 = 0.58 ( s o l a r ) , log P=5 53 v i i Page FIGURE 8. Chemical Equi l ibr ium of Magnesium: a) C/0 = 0.58 ( s o l a r ) , log P=2 54 b) C/0 = 0.58 ( s o l a r ) , log P=5 54 9. Chemical Equi l ibr ium of Ch lo r ine : a) C/0 = 0.58 ( s o l a r ) , log P=2 55 b) C/0 = 0.58 ( s o l a r ) , log P=5 55 10. Chemical Equ i l ibr ium of Hydrogen: a) C/0 = 1, log P=3 . 56 b) C/0 = 1.5, log P=3 56 11. Chemical Equi l ibr ium of Carbon: a) C/0 = 1 , log P=3 ' • 57 b) C/) = 1.5, log P=3 57 12. Chemical Equi l ibr ium of N i t rogen: a) C/0 = 1, log P=3 58 b) C/0 = 1.5, log R=3 58 13. Chemical Equi l ibr ium of Oxygen a) C/0 = 1 , log P=3 59 b) C/0 = 1.5, log P=3 59 14. Abundance o f Important Spec ies : a) T = 2000K, log P=3 60 b) T = 2000K, log P=3 60 15. Abundance of Important Spec ies : a) T = 2500K, log p=3 61 b) T = 2500K, log P=3 61 vi i i Paje FIGURE 16. Abundance of Important Spec ies : a) T = 3000K, log P=3 62 b) T = 3000K, log P=3 62 17. Abundance of Important Spec ies : a) T = 3500K, log R=3 63 b) T = 3500K, log P=3 63 18. Abundance of Important Spec ies : a) T = 4000K, log P=3 64 b) T = 4000K, log P=3 64 ix ACKNOWLEDGEMENTS I would l i k e to express my thanks to my superv i sor , Dr. Jason Auman for introducing me to the realm of late-type s t a r s , and in p a r t i c u l a r , for his advice and encouragement throughout th i s p ro jec t . Many thanks a lso go to Dr. Harvey Richer and Dr. Hoi l i s Johnson for the i r s t imulat ing d iscuss ions concerning the atmospheres of late-type s t a r s . I am indebted to Dr. Corr ie Kost for his cont inuing support of my dec is ion to return to graduate school under rather t r y ing c ircumstances. I a lso must express my grat i tude to Carmen de S i l v a both for her beaut i fu l typing of th is t h e s i s , and her perserverance with the pages of equations contained w i th in . F i n a l l y , I wish to dedicate th i s work to my c h i l d r e n , Max and Athena, in the hope that they too may someday feel the sense of wonder about the universe that has insp i red me over the years . 1 CHAPTER ONE THE EQUATION OF STATE FOR LATE-TYPE STELLAR ATMOSPHERES 1. Introduct ion The basic s t e l l a r atmospheres problem is to determine the run of temperature, pressure and rad ia t ion f i e l d s through the atmosphere, which i s considered to be the outer region o f the s ta r from which photons may escape d i r e c t l y to space without intervening absorpt ion or s c a t t e r i n g . I f the assumption that the s t e l l a r mater ia l is in loca l thermodynamic equ i l ibr ium (LTE) i s made, then a wel l-def ined temperature of the gas ex is ts ( e . g . the k ine t i c temperature = e x c i t a -t ion temperature = i on i za t i on temperature) and the composition of th is gas w i l l be well descr ibed by equi l ibr ium processes. It must be noted, however, that the temperature of the rad ia t ion f i e l d in the outer atmosphere genera l ly d i f f e r s from th i s gas temperature, and in fact is of ten s u f f i c i e n t l y non-Planckian that an unique rad i a t i on temperature cannot be de f ined . The essent ia l compl icat ion of the s t e l l a r atmosphere problem ar i ses from the coupl ing between the rad ia t ion f i e l d and the s t e l l a r mater ia l v i a the opac i t y . This s i t ua t i on i s fur ther aggravated for the case of late-type stars s ince the i r atmospheric temperatures are 2 low enough to permit molecular assoc ia t ion to occur . Many of these molecules exh ib i t strong e l ec t ron i c t r ans i t i ons in the v i s i b l e spectra l r eg ion , or strong ro ta t i ona l-v ib ra t i ona l t rans i t ions in the in f ra red and so are very s i g n i f i c a n t opac i ty sources. For example, the c l a s s i c a l MK c l a s s i f i c a t i o n scheme (Keenan and McNei l , 1976) for M stars i s based pr imar i l y on the strength o f TiO bands, which completely dominate the op t i ca l spectra of l a te M s t a r s . The presence of T iO , for example, a f f ec t s the atmosphere in two basic ways. One i s the l i n e blanket ing a f f e c t which serves to channel the emergent f lux into the "windows" between the absorpt ion features . The other is the backwarming of the photosphere that r esu l t s from the absorpt ion and subsequent thermal izat ion of photons in the TiO bands. Since the t yp ica l energy of the photons absorbed by these bands (p r imar i l y in the blue and v isua l spectra l regions) exceeds the mean thermal energy of the gas p a r t i c l e s , the net a f f e c t i s to t rans fe r energy from the r ad i a t i on f i e l d to the gas r e su l t i ng in a warming of the photospheric l a ye r s . Furthermore, the abundance of TiO in the atmosphere (which determines the magnitude of the above e f f ec t s ) i s i t s e l f a strong funct ion o f temperature (and to a l esse r extent , pressure ) . This example i l l u s t r a t e s well the basic d i f f i c u l t i e s of con-s t ruc t ing model atmospheres for late-type s t a r s . As a b a s i s , i t i s necessary to have an accurate , yet reasonably fas t procedure to determine the chemical equ i l ibr ium ( i . e . the equation of state) of a gas for temperatures ranging from 1500-5000K. It i s th i s problem that I address in th i s work, and as such i t i s intended to be the f i r s t step in the development of a model atmosphere code fo r l a te-type giant stars with extended atmospheres. 3 2. H i s t o r i c a l Development The c a l cu l a t i on of the chemical equ i l ibr ium in cool s t e l l a r atmospheres remained a basic obstac le for many years to the formula-, t i on o f model atmospheres for these s t a r s . The pioneering e f f o r t o f Russel l (1934) y ie lded q u a l i t a t i v e l y cor rec t r e s u l t s , but remained l im i t ed by the lack of basic thermochemical and spectroscopic da ta , and by const ra in ts of manual c a l c u l a t i o n . Russel l determined the chemical equ i l ib r ium of a gas at a spec i f i ed temperature and pressure through the use o f equ i l ib r ium constants to express mass balance among the species in terms of the par t i a l pressures of the free atomic spec ies . While the problem can a lso be approached through the minimizat ion o f the tota l free energy of the system (White e t . a l . , 1958), most inves t iga t ions of chemical equ i l ibr ium in the s t e l l a r atmospheres context have used the c l a s s i c a l Russell equat ions. The numerical complexity of the f u l l so lu t ion of the problem, however, discouraged fur ther work un t i l the advent o f high speed e l ec t ron i c computers rendered the ca l cu l a t i ons t r a c t a b l e . The f i r s t inves t iga tor to perform a de ta i l ed ana lys is of the molecular equ i l ibr ium problem coupled with a model s t e l l a r a t -mosphere was Vardya (1966), who computed equ i l ib r ium concentrat ions of one hundred chemical species ( inc lud ing neutral atoms, molecules , pos i t i ve and negative ions) formed from f i f t e e n elements, for temperatures and pressures representat ive of l a te K and M s t a r s . 4 Va rdya used t h e i n v e r s e t e m p e r a t u r e G = 5 0 4 0 . 3 9 / T and the * 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 hyd rogen p^, as h i s i n d e p e n d e n t s t a t e v a r i a b l e s . The 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 an e l e m e n t r e f e r s t o t h e 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 t h e a t o m i c s p e c i e s a s suming t h e c o m p l e t e d i s s o c i a t i o n o f a l l m o l e c u l e s and t h e c o m p l e t e r e c o m b i n a -t i o n o f a l l i o n s i n t o f r e e n e u t r a l a t o m s . For e x a m p l e , i f hyd rogen i s c o n s i d e r e d t o be p r e s e n t i n t h e gas o n l y i n t h e forms o f H , H + , H~ and W^, t h e n t h e 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 p r e s s u r e o f * hyd rogen p^ wou ld b e : PH = P H + P H + + PH- + 2 p H 2 Assuming f o r an i n i t i a l a p p r o x i m a t i o n t h a t the e l e c t r o n s i n t h e a tmosphe res o f t h e s e l a t e - t y p e s t a r s a l l a r i s e f rom t h e f i r s t i o n i z a t i o n o f t h e e l emen t s H, N a , K, S i and Mg , Va rdya o b t a i n e d an e s t i m a t e o f t h e e l e c t r o n p r e s s u r e p g g i v e n by p e I I ^ / d i + P e ) , where i = H, N a , K, S i , Mg and I., i s t h e 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 t h e f i r s t i o n i z a t i o n o f e l emen t i , i . e . , A -> A + + e " , I. = P A + ( P e /PA ) 5 Vardya s i m p l i f i e s the molecular equ i l ibr ium equations by re ta in ing only the more abundant species ( for approximately so lar composi t ion) , y i e l d i n g simple equations which can be solved d i r e c t l y to give i n i t i a l estimates of , p c , p Q , , . . . e t c . The complete set of f i f t e e n simultaneous equations (one equation for each element considered) were then solved exact ly by applying the Newton-Raphson method to th i s system, and i t e r a t i n g to convergence. These ca l cu l a t i ons ind icated that W^, CO, H 2 O , OH, Ng. Si0 and HC1 are the most abundant molecules in M giant atmospheres. Model atmospheres were ca l cu la ted using th is equation of s tate under the assumption of a grey atmosphere, and the resu l t s published in a se r ies o f very useful graphs i l l u s t r a t i n g the march of mole-cu lar abundances with op t i ca l depth for l a te K and M dwarf, g iant and supergiant s t a r s . This work was a s i g n i f i c a n t advance over the old s ing le-layer models, s ince temperature and pressure va r i a t i on in the atmosphere are now accounted f o r . S t i l l , the assumption of a grey atmosphere r e -presents a considerable source of er ror s ince the opac i ty in late-type stars is h igh ly non-grey. A re la ted study by Vardya (1967) concerned the importance of negative ions in cool s t a r s . Another ser ies o f graphs were presented showing the va r i a t i on of pa r t i a l pressure of these ions with op t i ca l depth for l a te K and M s t a r s , again under the assumption o f a grey atmosphere. The most prevalent ion ic species were found to be H~, C l ~ , 0~, and S~, although the large abundance o f CI" is an a r t i f a c t o f the high ch lo r ine content in these models. A CI log number dens i ty of 7.2 on a scale with log n^ = 12 was assumed; a more reasonable current value i s log n ^ = 5.4 (Hal l and Noyes, 1972). Even so , the par t i a l pressure of C l~ w i l l s t i l l be comparable to that o f H~ in the outer layers of an M6 III s t a r . 6 Goon and Auman (1970) ca l cu la ted column number dens i t i es of molecules above an opt i ca l depth T^= 3 for wavelengths at which the c h a r a c t e r i s t i c spectra l features o f that molecule occur . The non-grey atmospheres o f Auman (1969) for late-type stars with e f f e c t i v e temperatures between 2000K and 40Q0K, that included the atomic, mole-cu la r and ion ic species considered by Vardya (1966), were used in th i s study. The conc lus ion reached by the authors was that HC1, HS, S i S , NH^, HCO, H^S and COr, should be po t en t i a l l y detectable in s t e l l a r spec t ra . Tsu j i (1973) published a more extensive ana lys is o f the s t e l l a r atmosphere chemical equ i l ibr ium problem, cons ider ing 36 elements and a tota l of 232 species over a range of temperatures and pressures appropr iate to late-type s t a r s . This paper contains a tabu la t ion o f c o e f f i c i e n t s o f quart ic polynomial approximationsto log as a func-t ion of the inverse temperature 0, where is the equi l ibr ium constant descr ib ing the d i s s o c i a t i o n of the pa r t i cu l a r chemical species i into i t s const i tuent atoms. I have used these values for my molecular equ i l ib r ium ca l cu l a t i on in th i s research. While Tsu j i invest igated the chemistry of d i f f e r e n t temperature and pressure regimes in con-s iderab le d e t a i l , no attempt was made to incorporate these resu l t s into a model atmosphere. Irwin (1978, 1981) presented an even more complete tabulat ion of polynomial approximations to the p a r t i t i o n funct ions for 344 atomic and molecular spec ies . The interna l pa r t i t i on funct ion of a gaseous chemical species i s def ined by Q = I g. expC-E./kT) 1=o • where the summation 1 1s over the bound e l e c t ron i c s t a t e s , each with 7 energy E. above the ground state and s t a t i s t i c a l weight g^. The summation i s truncated at state n with a cu t-of f energy A E , where AE i s the depression of the continuum due to the in te rac t ion with ne igh-bouring pa r t i c l e s in the gas. This t runcat ion i s necessary to prevent the p a r t i t i o n funct ion from diverg ing to i n f i n i t y . As shown by Irwin (1978), given a polynomial approximation to the logarithm of the p a r t i t i o n f unc t i on , i . e . 5 . In Q = I a . ( l n T ) 1 , i=o then the logarithm of the equ i l ib r ium constant for th i s species can be wr i t ten as 5 In K = I. b . ( l n T ) 1 + b g /T , i =o where the c o e f f i c i e n t s b. can be ca l cu la ted from the values a^ and the d i s soc i a t i on energy D. This procedure ensures that the p a r t i t i o n funct ion (required for the opac i ty ca l cu l a t i ons ) and the equi l ibr ium constant ( required for the equation of state) are i n t e r n a l l y cons i s ten t . Un-fo r tuna te l y , s ince Tsu j i d id not publ ish his p a r t i t i o n func t ions , i n -cons is tenc ies between h is molecular equ i l ibr ium constants (which I have adopted in th i s work) and independently derived o p a c i t i e s , are i n e v i t a b l e . Recent ly , an exhaustive study of the chemical equ i l ib r ium of nearly 1600 species formed from a l l of the 92 elements H through U was performed by Johnson and Sauval (1982). The authorsca lcu lated weighted column dens i t i es for these species based on a gr id of red giant model atmospheres o f so lar composition ca l cu la ted by Johnson, Bernat, 8 and Krupp (1980). These column d e n s i t i e s , which included a f lux weighting func t i on , were evaluated at s p e c i f i c wavelengths corresponding to the pos i t ion of p r inc ipa l molecular absorpt ion f ea tu res , as well as for the standard wavelength of 1 ym. The r esu l t s reaf f i rmed e a r l i e r studies regarding the dominance of molecules such as H^, CO, OH, N 2 , H 2 0 , S iO , HC1, HF, HS, SiH in the atmospheres o f late-type g i an t s . It a lso demonstrates that most of the remaining po t en t i a l l y de tec tab le , but cu r ren t l y undetected molecules in Tate M giants are e i ther ionized metal monohydrides + + or ion ized monoxides, e . g . CaH and TiO . 9 CHAPTER TWO A GENERAL SOLUTION OF THE EQUATION OF STATE PROBLEM 1 . Chemical Equi l ibr ium of a Gas o f A rb i t r a r y Composition In th i s sect ion I w i l l descr ibe the development of an algorithm to solve the chemical equ i l ibr ium of a gas of a rb i t r a r y composi t ion, for which an a rb i t r a r y l i s t of chemical species are to be considered in the equ i l i b r i um. Assume the gas cons is ts of N chemical species (atoms, molecules , and ions) formed from K elements. In the fo l l ow ing , i t is to be assumed that the index n, n = l . . . , N may re fe r to any a rb i t r a r y species present in the gas, while the index k, k = 1...K re fers to the neutral free atomic species of element H-Let P N = par t i a l pressure o f species ny PL. = pa r t i a l pressure o f neutral free atomic species k. * p k 5 f i c t i t i o u s par t i a l pressure of element k, i . e . the r e su l t i ng pa r t i a l pressure of the free atomic species k i f a l l molecules in the gas were d i s s o c i a t e d . P E = e lec t ron pressure . p = tota l gas pressure , inc lud ing e lect ron pressure. * _ P = f i c t i t i o u s to ta l pressure , i . e . r e su l t i ng par t i a l pressure of the free atoms (not inc lud ing the e lect rons) i f a l l molecules were d i s s o c i a t e d . 10 q n = charge ( i on iza t i on state) of species n. N ^ = number o f atoms of element k in species n. N n = E N ^ = to ta l number of atoms in species n. = f rac t iona l abundance of element k, by number, with £ a^= 1. I = i on i za t i on equ i l ib r ium constant r e l a t i ng the par t i a l pressure of ion ized species n with charge q n to that of i t s neutral parent spec ies . K n = molecular equ i l ibr ium constant r e l a t i ng the par t i a l pressure of the neutral molecular species n to the par t i a l pressures of i t s const i tuent neutral f ree atoms. A l s o , T w i l l be used throughout th i s chapter to denote the system temperature (assumed to be in thermodynamic equ i l i b r i um) , and h and k re fe r to the Planck and Boltzmann constants r e spec t i v e l y . Some re la t ions between these quant i t i es fo l low immediately. P = P + £ P K K e n K n * Pk = n N nk Pn p = E N D a n n n * y N . p Pk I nk^n k p* E N p n n Hn E E N . p E p E N , E p N tr „• n k n n n k nk n *n n * • 2 — — — -j * n n*n n n Kn n n r n If n' is the neutral parent species of species n with charge q n , the i on i za t i on equation is n 1 n + q e" M n and the corresponding i on iza t i on equi l ibr ium constant I is 11 I = P P /P n H n H e / p q n n or P = I P'/P n V H e For a neutral spec ies , q n = 0 and thus 1 = 1 . S i m i l a r l y , i f n is a neutral molecule, then the d i s soc i a t i on equation can be wr i t ten where n k indexes the k-th element present in species n. The corresponding d i s soc i a t i on equ i l ib r ium constant K is then given by For a f ree neutral atomic species n th i s reduces to K = 1. I have now obtained an expression r e l a t i ng the pa r t i a l pressure of an ion ized (poss ib ly molecular) species to the pa r t i a l pressure of i t s neutral parent and the e lec t ron pressure, and s i m i l a r l y , I have an expression r e l a t i n g the pa r t i a l pressure o f a neutral molecule to the par t i a l pressures of i t s free atomic cons t i tuents . 'These expressions may be combined to y i e l d a r e l a t i on between the equ i l ib r ium par t i a l pressure of an a rb i t r a r y species and the pa r t i a l pressures of i t s const i tuent f ree atoms, along with the e lec t ron pressure. Therefore , for any species n, or n „ _ nk (2-1) — - — n p K n P e q " k »k 12 The equations of mass-balance among the species in chemical equ i l ibr ium can how be imposed in terms of the spec i f i ed elemental abundances. Therefore , for each of the k elements, Pt I N n k p n Jk n a. = —+ = -k p* . Y N p - Y N , p = 0 k t n Kn L nk pn n or I(a kN n - N n k ) P n = 0 , k=2, . . .K (2-2) n Note that the complete set o f K equations are not l i n e a r l y independent, s ince Y a. = 1 and so any one equation may be derived from the remaining k K K-l. I have therefore chosen the mass balance equations for elements 2 , . . . , K - 1 , omitt ing the equation for K=l (normally hydrogen). Charge neu t r a l i t y of the gas y i e l d s another r e l a t i o n , which I sha l l consider as equation 1 o f my se t . This is simply expressed by I P A - P e = 0 (2-3) n It should be r eca l l ed that each of the pa r t i a l pressures p n is to be wr i t ten in terms of the pa r t i a l pressures p^ o f the f ree const i tuent atoms, and so therefore there ex i s t a tota l of K+l unknowns ( p k , k = l , . . . K and p e ) to be determined, given the composition a k , k = l , . . . , K and the tota l gas pressure p. Hence one add i t iona l equation (K+l of my set) is required to c lose the system, and th i s is jus t the tota l pressure r e l a t i o n . 13 P e + I P n = P n I P A + IPn = p n n I P n ( q n + 1) = P n (2-4) This system of K+l equations must be solved for the K+l pa r t i a l pressures to determine the chemical equ i l ib r ium of the gas. One model o f approach is to assume that an approximate so lu t ion is a v a i l a b l e . The h ighly nonl inear system of equations descr ib ing the chemical equ i l ib r ium of the gas is then l i n e a r i z e d , and the so lu t ion of th is r e su l t i ng l i nea r system used to improve the o r i g i na l est imate. This procedure can then be i t e ra ted to convergence provided the s t a r t i ng so lu t ion was s u f f i c i e n t l y good. I w i l l show l a t e r that i t i s indeed r e l a t i v e l y s t ra ightforward to obtain su i t ab le s t a r t i ng so lu t i ons . Let us proceed with the l i n e a r i z a t i o n , beginning with the mass balance equat ions, I assume that the p n a va i l ab le is only an estimate of the exact so lu t ion p^  with P n = P n + SPn- T n e ob ject ive is to solve for the co r rec t ion term 6p . T a. N - N , ) p' = 0 t k n nk' *n n n N n k ) ( P n + 6 p n ) = 0 Nnk> 6 p n = K N nk " < W P n n (2-5) 14 This expresses the l i n e a r i z e d mass balance equation in terms of the changes in the pa r t i a l pressures 6p p of a l l the spec ies . Our independent va r iab les , however, are the par t i a l pressures of the neutral free atoms p^  and the e lec t ron pressure p g on l y , and so the co r rec t ion 6p n must be expressed in terms of these quan t i t i e s . This is accomplished by l i n e a r i z i n g equation (2-1) I N . „ - n T T n k n K p n k n k n e and p + <5p = - g- 1 1 (Pn + 6 p n ' " " K ( p + ' S o }q" k n k n k n ' e I Nnk & \ N 'n . Q E p (1 + -)' n ' K n P e N i + ^ ) q n k v V KP> 0 P e k ^ ^ Pn k K"fe n r e I N . 6p N , n nk /, K e , , nk \ K p % k n k ^ P e k P n r \ n Ke K 6 P P N nk p (1 - q — }^(1 + ? — Sp ) Nnk 6 p e p (1 + E 6p - q — - ) n K P^ \ q n p e N . 6p p + p (E — 6p - q — ) Pn Pn ^ k p Pn k % p ' k N nk X n „ ^Pe, 5>n ^ / i f V U ' ' (2"6) k e 15 where i t has been assumed that the quant i t ies 6p , <Sp are sma l l , n k e i . e . Sp /p.. « 1 , 6p /p « 1 in order that the var ious f i r s t order n k n k e e expansions performed above be v a l i d . Then, subs t i tu t ing back into the l i n ea r i z ed equation of mass balance, (2-5), N nk 6 p e E (a. N - N , )p (Z — 6p • - q — ) = E N . - a. N ) p n v k n nk y t n v i< p ' n . ^n p ' k nk k n K n n k k e N nk 1 Z (a. N - N , )p E — Sp - -n k n n k / H n k p n H n k p g S n ( a k N n " N n k ) p n q n = 5 ( Nnk a, N )p LINEARIZED MASS BALANCE EQUATIONS (2-7) This i s the l i n ea r i z ed equation of mass balance. Next consider the charge neu t r a l i t y equat ion. E p'q - p 1 = 0 n K n M n 'e where aga in , the primed quant i t ies re fe r to the exact so lu t ion p n = p n + 6 p n p; = P e + 5 p e •"• E n ( p n + 6 P n > q n " ( p e + 6 p e> = 0 I q n 6 p n " 6 p e = P e " \ p n q n and subs t i tu t ing in the expression for 6p n (Equation 2-6), 16 N nk 6 P P Z p q (Z Sp - q —-) - Sp = p - Z p q N. Z p q Z -nk Sp - 1 •+ — Z p q 2 P e n n v n 5p„ = p - Z p q , *e *e n *n^n LINEARIZED CHARGE NEUTRALITY EQUATION (2-8) y i e l d s the l i n e a r i z e d equation of charge n e u t r a l i t y . F i n a l l y , the to ta l pressure equation is l i n e a r i z e d , Z p'(q + 1) = p s(p n + « p n ) (q n + D = P 2 ( q + - D 6p n = P - Z p ( q n + 1) n n - n n n " and therefore I P n K + 1 > N , Sp v nk « K e f p p n . q n p k K n . k K e N = p - Z p (q + 1) K n n X M n ' Z p (q + 1) J — Sp - - Z p q (q + 1) Sp n K n V H n £ P n ^ p f i[n H n V H n ;J 0 f e Z p (q + 1) n *n ^n ' LINEARIZED TOTAL PRESSURE EQUATION (2-9) 2. Evaluat ion of Equi l ibr ium Constants Consider the reac t ion AB -»- A + B r e f l e c t i n g chemical equ i l ib r ium between the reactants and the products. This could re fe r to the d i s soc i a t i on of a diatomic molecule AB into the atoms A and B, or equal ly w e l l , to the i on i za t i on of a neutral atom AB into a s i ng l y charged ion A and a free e lec t ron B. As given by Tatum (1966), i f N^, Ng, N^g are the to ta l number of the respect ive spec ies , 17 and Z . , Z n , Z n D are the i r respect ive tota l p a r t i t i on func t ions , then A D A D for a system in equ i l i b r i um , ¥ B . V B P V K T N Z AB -AB where E Q i s the energy d i f fe rence between the reactants in the i r ground states and the products in the i r ground s ta tes . This would correspond to the d i s s o c i a t i o n or i on i za t i on energy for the two examples mentioned prev ious ly . The to ta l p a r t i t i on funct ion Z of a species contained in a volume V i s the product of i t s interna l ( r o t a t i o n a l , v ib ra t iona l and e l ec t ron i c ) p a r t i t i on funct ion Q, and i t s t r ans l a t i ona l p a r t i t i o n funct ion given by 2 T r m k T \ 3 / 2 v where m is the species mass. Then /W\ 3 / 2 V " E o / k T NAB "I h 2 ^AB where m i s now the "reduced mass" M^ Mg/M^ g. Using pV = NkT, the desired expression for the equ i l ib r ium constant (the genera l ized Sana equation) i s obta ined. K„CT) - VB =(l™hlY/2KT W E-V * i { AB \ h Z / Q A B 18 For the s p e c i f i c case o f i on i za t i on of a species X : X X + + e~, the i d e n t i f i c a t i o n s AB=X, A=X+ and B=e~ can be made.* A l s o , E Q = x (the i on i za t i on energy) , K^g = 1^, m = m g and Q g = 2. With these reduc t ions , the standard Saha equation is obta ined, where m e i s the e lec t ron mass, or numer ica l l y , (2-11) log Ix = 2.5 log T - 0.48 + log I —^— I - P 0 3 ^- 9 Jx (2-12) For atomic spec ies , at low temperatures, normally Q x - g x ^ , the ground state s t a t i s t i c a l weight. An analogous equation can be wr i t ten for molecular d i s s o c i a t i o n , but now the evaluat ion o f the molecular pa r t i t i on funct ion i s much more d i f f i c u l t . This c a l c u l a t i o n involves summing over the var ious r o t a t i o n a l , v ib ra t iona l and e l e c t ron i c l e ve l s o f the molecule ; the interested reader is re fer red to the exce l l en t a r t i c l e by Tatum (1966) on th i s subject . The most cons is tent approach to so lv ing the chemical equ i l ib r ium problem would be to express the various equ i l ib r ium constants d i r e c t l y in terms of the polynomial f i t s to the p a r t i t i o n funct ions required in equation (2-10), as suggested by Irwin (1981). This has the prev ious ly mentioned advantage that the same pa r t i t i on funct ions may be used in the opac i ty c a l c u l a t i o n s , ensuring consistency between the opac i ty and the equation of s t a t e . It a lso decouples the pa r t i t i on funct ion from the d i s s o c i a t i o n energy in the expression for the equi l ibr ium constant . This r e ad i l y al lows for r e v i s i on of the often poorly determined d i s soc i a t i on energy (usua l ly the l a rges t source of er ror in the equ i l ib r ium constant) without the necess i ty of f i t t i n g new paratnetrizations to these equ i l ib r ium constants . Unfortunate ly , I rwin's (1981) compi lat ion o f p a r t i t i o n funct ion approximations was not ava i l ab le un t i l th i s work was already underway and so th i s approach was not fo l lowed. Instead, the i on i za t i on equi l ibr ium constants were evaluated d i r e c t l y from the Saha equation (2-12) using tabulated p a r t i t i o n funct ions for 9 = 1.0 ( A l l e n , 1976). The molecular equ i l ib r ium constants , fo r now, were ca l cu la ted d i r e c t l y from the polynomial approximation of Tsu j i (1973). These w i l l , however, be expressed in terms of Irwin's polynomial pa r t i t i on funct ion c o e f f i c i e n t s for future work. 20 CHAPTER THREE AN OPTIMIZED SOLUTION OF THE EQUATION OF STATE PROBLEM 1 . I n i t i a l Estimates of Par t ia l Pressures A. S impl i f y ing assumptions The 1 i near iza t ion method of so lu t ion of the chemical equ i l ibr ium descr ibed e a r l i e r in the previous chapter requires reasonably good i n i t i a l estimates o f the so lu t ion to converge r e l i a b l y . Furthermore, s ince th is technique is equivalent to general ized Newton-Raphson method and thus exh ib i t s quadrat ic convergence near a s o l u t i o n , i t is des i rab le to achieve as good an i n i t i a l estimate as poss ib le to reduce the number o f i t e r a t i ons in the l i n e a r i z a t i o n loop . As I w i l l now show such i n i t i a l estimates are r ead i l y obta ined, but at the pr ice of i n t r o -ducing assumptions about the composition of the gas. The basic assumptions needed to render the problem computat ional ly t r ac tab le a re : i ) p « p H : where n represents any species except H, i i } P C N ' ^ PC H + , Hr, or He. This i s c e r t a i n l y v a l i d for a l l "normal" s t e l l a r composit ions, th i s favours M-stars (with n^ < n^) over C-stars , although the s t a r t i ng estimates a c tua l l y obtained remain useable even for the C-star case . 21 i i i ) p<-.Q « PQ : t h i s i s c e r t a i n l y v a l i d for C-stars , but somewhat less so in the M-star case. S t i l l , good res-timates are obtained for M-stars. iv) molecule formation among the elements other than H , H G , C 1c 1c 1c 1c 1c N , 0, S, Si obeys p n « p c , p Q , p N , p $ , p$. , where p n is the par t i a l pressure of any molecular species invo lv ing such an element with any of C, N , 0, S or S i . This states that molecular assoc ia t ion of H , C, N , 0, S, Si atoms with others does not s i g n i f i c a n t l y deplete these elements. With these assumptions, the f i c t i t i o u s tota l pressure can be expressed s imply . P* * PH + 2 P H 2 + V + PHe and since P K PH + PH 2 + P H + + PHe + Pe P* 88 P+ p H - P e (3 - D The var ious f i c t i t i o u s par t i a l pressures can now be approximated as fo l l ows : P H " P H + 2 p H 2 + P H + + PH-p He" pHe P C K P C + PCH + P C 0 + P C + P * " p0 + P0H + PH20 + PCQ + P 0 + 22 ~- P N + P N H + 2 P N 2 + P N + * P S i s P S i + P s i o + P S i S + P S iH + p S i+ * Ps * p s + P H S + P H 2 S + P S i S + P S + Pci a p c i + P H C I + Pci -* PTi * P T i + P T iO + p T i + * Pv * p v + p v o + P v+ * P Y HY HYO PY02 + PY+ * K P Z r + p Z r 0 + p Z r O ? + pZr+ For a l l other elements assume PK = PK + P K + B. The Estimates of E lectron Pressure The most d i f f i c u l t quant i ty to estimate accurate ly is the e lec t ron pressure p g , s ince for the outer regions o f cool s t a r s , most of the e lectrons are suppl ied by a large number of metals each of r e l a t i v e l y low abundance. Deeper in the atmosphere, however, with r i s i n g temperature hydrogen begins to contr ibute to the e lec t ron pressure, and at s u f f i c i e n t l y warm temperatures dominates the metals as an e lec t ron donor because of i t s large abundance. The equation of s tate code for a cool model atmosphere must be able to handle the f u l l range of dependence of p e upon temperature. The procedure I have used to obtain an i n i t i a l estimate o f p g i s ou t l ined below. B a s i c a l l y , estimates o f p g are derived for the more r ead i l y treated extremes of high temperature, 23 where e s s e n t i a l l y a l l of the e lectrons a r i se from ion iza t i on o f H and so p g - p H +, and o f low temperature, ./Where most of the e lectrons are due i on i za t i on of the r e l a t i v e l y abundant metals Na, Mg, A l , S i , K, Ca, Fe along with C. A s imple , though -crude, estimate of p g over a wide range of temperature is then obtained by merely taking the maximum of these two l i m i t i n g values o f the e lec t ron pressure. This crude estimate of p e is subsequently improved by a l i n e a r i z a t i o n method to y i e l d a very good estimate (usua l ly wi th in ^ 2% of the cor rec t va lue ) , i ) The high temperature l i m i t : F i r s t , the cont r ibu t ion from H» denoted by p e ^, to the tota l e lec t ron pressure is estimated in the high temperature l i m i t where H suppl ies most of the e l ec t rons . The abundance of H, by number i s : \ _ PH + 2 P H 2 + P H + + PH- a H P* P + P H 2 V P e or a H (p + p^- p e ) = p R + 2 p H 2 + p H + + p H-In th i s l i m i t , p u « p and can be neglected, and (always) p u _ « p, so a H (P - P e) = PH + PH+ " PH + ( Pe / r H+ + 1 } where the equi l ibr ium r e l a t i on p^  = p^+(pe/I^+) has been subs t i tu ted . A l s o , in th i s l i m i t , e s s e n t i a l l y a l l of the e lectrons come from H and therefore. p e '=- p H + . This y i e lds 24 a H ( p - p e ) - P e ( P e / I H + + 1-) Pe + IH+(1 + V pe " AHVP = 0 0 r P e * 1 " V ( 1 + T H + ( 1 + a H ] 2 + H L i (3-2) i i ) The low temperature l i m i t : This case in which the metals supply e s s e n t i a l l y a l l of the e lectrons cannot be treated so d i r e c t l y , s ince now more than one e lec t ron donor is invo lved . I approached th i s problem by t rea t ing a l l the re levant ea s i l y ionized metals (C, Na, Mg, A l , S i , K, Ca, Fe) as one f i c t i t i o u s species Z with an e f f e c t i v e i on i za t i on potent ia l and abundance = | a^, where the summation runs over the above meta ls . The choice of an " e f f e c t i v e " i on i za t i on potent ia l for Z i s rather a r b i t r a r y s ince the best value w i l l vary with temperature and composi t ion, but I found x^  = 7.3 eV to y i e l d a s a t i s f a c to r y estimate of p (the e lec t ron pressure cont r ibut ion from e M the above meta l s ) , and th i s value was used in a l l my work. It should be noted that th is value of p g i s only an i n i t i a l estimate to supply the subsequent l i n e a r i z a t i o n s tep . The pa r t i cu l a r value of x^  adopted has no bearing on the f i n a l p g value obtained a f t e r the l i n e a r i z a t i o n s tep , provided that th i s s t a r t i ng estimate obtained was s u f f i c i e n t l y c lose for the l i n e a r i z a t i o n procedure to converge. Aga in , the abundance equation for H is A H - P * P + P H 2 - P E P H P H + 2 P H 2 + P H + + P H -a, l ( p + P H " Pe } = P H + 2 p H „ + PH+ + PH" 25 In the cool temperature l i m i t , hydrogen i on i za t i on is n e g l i g i b l e , and the e lec t ron pressure p g a r i ses from ion i za t i on of meta ls , a l l much less abundant than hydrogen. Therefore , p H + , p^_, p g « p and can be ignored, but formation may be very s i g n i f i c a n t . A H ( P + P H 2 } = P H + 2 P H 2 Since = P^/p^ , PH = ^ / K h p H and t h i s equation may be expressed in terms o f , and solved for p^  . a H (p + p ) = p + 2p 2 "2 This y i e lds the quadratic equation ,2_2 (2 - a H ) ' p ^ - [2a H(2-a H) p + K^] p ^ + a 2 p 2 = 0 with 2a H(2-a H) p + 2(2-a H)' '2a H(2-a H)p 2a H(2-a H) p + Kj. .2 2J A l s o , for the f i c t i t i o u s metal Z, (3-3) Z p* Pz P Z + Pz+ Here the assumption is made that Z does not take part in molecule formation and i s present in the gas only in the neutral atomic or s ing l y ion ized s t a t e s . Then, invoking a f i c t i t i o u s i on i za t i on equi l ibr ium constant 1^ +, which can be expressed in terms o f the prev ious ly mentioned i on i za t i on potent ia l xz by means o f the Saha r e l a t i on (2-11), t h i s abundance equation 26 for Z can be wr i t ten as fol lows P e % P z+P e P z+P e azp*= pz+^  + r"+ ' since rz+= — p ~ a n d pz= i — lZ+ In the cool temperature l i m i t , p* - p + p g and p 7+ - p P e 2 L 6 So, ^ ( p + p ^ ) = p e ( l + I - ) or p 2 + I z + p e - a z I z + ( p + p^) = 0 This y i e l d s I Z + + ^ I Z + 4 a z I z + ( p + p H ) p e 7 = - = i- (3-4) with p u given by the estimate of equation (3-3). H 2 A useable , though crude, estimate p g o of the e lec t ron pressure v a l i d over a wide range of temperatures can now be obtained simply by using the e lec t ron pressure a r i s i n g from the dominant source (hydrogen or meta ls ) , i . e . p = max (p , p ) e o e Z e H C. L i nea r i za t i on of charge neu t r a l i t y equation A very good estimate of the e l ec t ron i c pressure can now be obta ined, using the above rough estimate as a s t a r t i ng guess, by l i n e a r i z i n g the equation of charge neu t r a l i t y and i t e r a t i n g un t i l convergence. Aga in , only the major e lec t ron donors , (H, He, C, Na, Mg, A l , S i , K, Ca, Fe) are included in the l i n e a r i z a t i o n . S tar t with the equation of charge neu t r a l i t y p e = I pnV 27 Assume, as before , that the e lectrons a r i s e from s ing l y ionized donors that are not s i g n i f i c a n t l y depleted by molecular a s soc i a t i on . This is a very good approximation since hydrogen assoc iates only at temperatures at which i t i s unimportant as an e lectron donor. Therefore , the charge neu t r a l i t y equation can be w r i t t en , summing over only the re levant spec i es , p e = I pk+ since qk = 1 for s i n 9 l y ionized species 0 for neu t ra l s . By the above assumption, pk + V * ) p k+ and 1 + P e /I k + The equation of charge neu t r a l i t y then becomes P e = E k - P k + = p * I r a k V k V + pe (3-5) which I now proceed to l i n e a r i z e in p g . F i r s t , however, note that P* * P + PH - P e and since the tota l pressure p i s f ixed at a spec i f i ed va lue , p* w i l l vary with p g . Therefore , I def ine P1 = P + P^  > which w i l l remain constant as p g v a r i e s . So I now have p* = p' - p£ y i e l d i n g the cor rec t dependence of p* upon p g . The charge neu t r a l i t y equation (3-5) now becomes P P = (P " P J I T + ~ n ~ e e k Ik+ + P e (3-6) To l i n e a r i z e , I assume p g is an approximate so lu t ion of th i s equation and so seek a cor rec t ion 6pg such that p g + 6pg i s an exact s o l u t i o n . The assumption i s made here that 6pg « p g . Then P + 6p = (p* - p - 6p) y ^ . . H e H e K W ^e ve' fe I k + + p g + 6pe v 6pp V k + a k V V k + Cp ' -P e)I T +~n~ " 5 P e I T ~ ( P ' " Pe } 6 p e I 11 + - n - ) 2 6 k V J pe e k V pe 6 6 k u k + p e ; , , y a k V , n > v V k + 0 a k T k + 5 p e ^ ' - P e i ^ - P^  or *P< * 7 ^ k V _ P ? T +~TT " Pp k l.k+ + P e e 1 + & h + Pe with p* = p' - p. (3-7) An exce l l en t estimate of the e lec t ron pressure i s obtained in several i t e r a t i ons by th i s method s t a r t i ng with the prev ious ly derived rough 29 est imate, for a range of temperatures from 1500K to 30000K, and pressures from log P = 0 to 10. D. Estimates of Atomic Par t ia l Pressures i ) Hydrogen Knowing the e lect ron pressure p g , an accurate estimate of p^, the par t i a l pressure of atomic hydrogen (and a better value for p,, ) M 2 can be obta ined. Aga in , the mass balance equation for H i s : P* _ PH + 2 p H 2 + p H_ a H p* P + P H 2 " Pe a H ( P + P H 2 " Pe } = PH + 2 P H 2 + PH+ + p H-Using the various equi l ibr ium expressions r e l a t i ng those par t i a l pressures to that of p^, i . e . v V " Ph+ =(^)Ph' PH- = ( i , P H and s i m p l i f y i n g , a quadratic equation in p^  r e s u l t s . 2-a„ ? I y + p There fore , p p X H - 1 p p 1 H " • P 2 P " P p ^ f + 4 a „ ( 2 - a „ ) ( F - ^ - ) l H- * ^e V H H K H 2 2C2-a H)/K H (3-8) Pu and al so p„ = 2 K H P * = P + P H Z - P E y i e l d improved estimates for these quant i t ies i i ) C,N,0, The mass balance equations for C is a - P C * . P C + PCH + p C0 + P C + C p* p * P H , P 0 . lC a C P * = P C V + KT„ + K7, + XH 'XO P e and a c p * KCH K C0 p e S i m i l a r l y , for 0: J l „ p 0 + pQH + p H 2 0 + p C0 + P0+ *0 p* p* n P H P H P C + a 0 p * = p r t l l + I T — + T ) — + + — 0 ° \ OH K H 2 0 K C 0 P e I, , "H , P H , a C P * . V' P°V K0H K H ? 0 „ P H + p 0 + V, p e where the above expression for p c has been subst i tuted in the l a t t e r equat ion. Af te r manipulat ion, th i s y i e lds the fo l lowing equation for P G ' " 2 " w P H . P H 2 . ^ O + w , + P H + + , » ( 1 + IT + K KOH K H 2 0 P KOH P CO I, 0 K0H p e (3-10) which is r ead i l y solved to y i e l d the i n i t i a l estimate for p Q . Equation (3-9 ) above then y i e lds the corresponding estimate for p^. F i n a l l y , the N abundance equation can be wr i t t en : a. Pjj _ P N + P N H + 2 P N 2 + P N + AW P = PN0 + N + or P N + 0 + I C „ + p " 1 P N NH ^e a N p* = 0 (3-11) This is solved to obtain the des i red estimate of p N' i i i ) Other elements The only other important elements which couple in the manner of C and 0 are S i and S (through'the formation of S i S ) . As before , the mass-balance equation for S y i e l ds «cP* ! + P H _ P H 2 P 5 i . A S + 1 k- + (3-12) K HS K H 2 S KSi.S P e where p s i is as yet undetermined. 32 S i m i l a r l y , t h e S i e q u a t i o n y i e l d s a S i P 1 + p s + ! i L _ + ^Si KSiO KSiS KSiH p e S u b s t i t u t i o n o f t h e S e q u a t i o n i n t o t h e e x p r e s s i o n f o r p ^ e v e n t u a l l y y i e l d s t h e q u a d r a t i c , J S i 1 K S i S (1 + '0 KSiO KSiH fl- + !sii, ' S i /, p 0 pH • ^ i + w ^ PH + PH2 I S + u r , P* , K S i O K S i H p £ K HS "H0S P S S i ' K c . c e S i S (3-13) w h i c h i s r e a d i l y s o l v e d t o g i v e t h e e s t i m a t e o f p^^. The e s t i m a t e f o r p<. t hen i m m e d i a t e l y f o l l o w s f rom e q u a t i o n ( 3 - 1 2 ) . I n i t i a l e s t i m a t e s f o r s e v e r a l a d d i t i o n a l e l e m e n t s can be o b t a i n e d i n an a n a l o g o u s manner . CI : a PC1 P C 1 + PHC1 + PC1 CI p' and t h e r e f o r e , p CI a C l p 1 + VHC1 + i Cr p e T i : S i m i l a r l y , p a T i P T i 1 + + KTiO P e 33 P V* K V0 p e a y p * - : P Y ~ in P 2 T K Y0 K Y 0 2 P e a Z r P K ZrO K Z r 0 2 P e F i n a l l y , any other elements ( e .g . He, Ne, Na, Mg, A l , Fe, e tc . ) for which i n i t i a l estimates are required are assumed to be present only in the neutral atomic or s i ng l y ionized spec i es , i . e . p n p n + pn+ From the abundance equation p n p n + p n+ a. n p*- p* a n p * I obtain N U V p e 2. Economized So lut ion of Chemical Equi l ibr ium The determination of the chemical equi l ibr ium of a hot gas was solved in a s a t i s f a c t o r y manner by the approach descr ibed in Sect ion 2-2. Nevertheless , th i s technique does have the drawback of requ i r ing the so lu t ion of a l i n ea r system of equations of order n+1 per i t e r a t i on in order to determine the chemical equ i l ib r ium of a gas cons i s t ing of 34 n elements. This i s computat ional ly rather expensive for large n s ince the number of operat ions required to solve a l i nea r system of dimension n i s of order n , and for reasonable s t e l l a r compositions i t i s necessary to include at l e as t 15 elements merely to obtain the cor rec t e lec t ron pressure. But most of the coupl ing in the non-linear system of equations descr ib ing the equ i l ib r ium occurs between the r e -l a t i v e l y few abundant species which r ead i l y undergo molecular a s soc i a -t ion ( i . e . H, C, N, 0 ) , while the remaining elements are present ly e s s e n t i a l l y t o t a l l y in e i the r the neutral or s i ng l y ionized s t a te . It i s therefore reasonable to ask whether or not the equi l ibr ium of such a system can be described by means of a cons iderably smaller set of equat ions, without loss o f accuracy. The answer, as I sha l l now demonstrate, i s yes . To accomplish t h i s , I have separated the elements into the fo l lowing three c l a s s e s : i ) Major elements: These elements include those for which the pa r t i a l pressures of the atoms bound into molecules r e -present a s i g n i f i c a n t f r a c t i on o f the to ta l f i c t i t i o u s * pressure p . This requirement b a s i c a l l y states that those elements be both abundant and couple through molecular a s s o c i a t i o n . Note that He, e .g . which i s abundant but does not couple with any other element, i s not included in th i s category. I have included the s ix elements H, C, N, 0, Si and S in th i s c l a s s , i i ) Meta ls : This c l ass includes those elements that are reason-ably abundant, but do not undergo molecular assoc ia t ion to a s i g n i f i c a n t degree so that e s s e n t i a l l y a l l o f the atoms of a given element are present e i ther in the neutral or s i ng l y 3 5 ion ized s ta te . I have included He, Ne, Na, Mg, A l , K, Ca, Fe and Ni in th i s c l a s s . i i i ) Minor elements: This c l ass includes those elements of low abundance, which therefore contr ibute n e g l i g i b l y toward e i ther the tota l pressure or the e lect ron pressure, but nevertheless may represent s i g n i f i c a n t opac i ty sources. These elements are simply not abundant enough to a f f e c t the equ i l ib r ium among the major species and the meta ls , and so the chemical equ i l ib r ium ca l cu l a t i ons can ignore elements in the c l a s s . Any add i t iona l molecular equ i l ib r ium ca l cu l a t i ons required for opac i ty sources can then be ca l cu la ted d i r e c t l y from the known par t i a l pressures of the major (or metal) species using the appropriate mass balance equation of the pa r t i cu l a r minor element invo lved . A somewhat a r b i t r a r y l i s t of the more important minor elements includes CI , Sc , T i , V, C r , Mn, Co, S r , Y, Z r , e t c . The economization of the system of equations descr ib ing the chemical equ i l ib r ium is performed by the fo l lowing approach. F i r s t , each of the major elements couples s t rongly to the others and so each elemental pa r t i a l pressure must be treated as a separate independent v a r i ab l e . And, as mentioned before , the minor elements are of neg l i g i b l e inf luence on the equ i l ib r ium and can be ignored. F i n a l l y , the metal elements, while abundant, a l l behave s i m i l a r l y in that they do not couple and are present only in the neutral or s i ng l y ion ized s t a te . Therefore , I group a l l the metals together and t reat them as a s ing le f i c t i t i o u s metal element which I have designated Z, with an abundance equal to the sum 36 of the abundances of the ind iv idua l meta ls , i . e . ou = E a where Z m m the ct 's represent the various metal abundances. Then p* = p*a 7 = p*Ea = E ( a p * ) = Ep* , where p* = p 7 + p 7 + . L L M m m m m m L L L Now I can rewrite the abundance equations for the major elements, lumping a l l metals together into Z, and e x p l i c i t l y summing over the major elements only ( i . e . H, C, N, 0, S i , S ) , rather than over a l l elements involved as before . p\ I N n k p n n k p* E N p + p* ' v n n Kn VZ where subscr ipts k and n now re fer to major elements and species formed so l e l y from major elements r e spec t i v e l y . 0 r I K N n " Nnk> p n + V z = 0 « - 1 4 ) I „ nk with p n = ^ p n ^ , as prev ious ly KnPe This l i n e a r i z e s to S f ° k N n " N n k } 6 p n + a k 6 p Z = n ( N n k " a k N n> Pn" V * , N nk 6 p e with 6p„ = p (E - — 6p n - q„ -—) , again unchanged from before . n^ k *e Subs t i tu t ion y i e l d s the modif ied l i n e a r i z e d abundance equation for element k, 37 Ka.N - N . ) p T - H i 6 P _ _ L n k n n k n k Pn, n k P e k y (a. N - N . ).p q t k n nk ' K n^n 6p e + a k6p* y ( N . - a, N ) p - a. p v  L nk k n' K p r Z n LINEARIZED ABUNDANCE EQUATIONS FOR MAJOR ELEMENTS (3-15) For the f i c t i t i o u s metal Z, the abundance equation becomes a Z " E N p + p* n n Kn K Z or a Z I V n " ( 1 " a Z J P Z = 0 (3-16) This l i n e a r i z e s to the fo l low ing r e s u l t . N a 7 y N p y 6 P - ^ 1 n n n k Pn k n k Pe n (I N n P n q n ) 6p e - (1 - a z ) Sp* n * v n LINEARIZED ABUNDANCE EQUATION FOR FICTITIOUS METAL Z. (3-17) The to ta l pressure equation now modif ies to I Pn+ P Z + P Z + + P e = P or I p n + PZ + Pe = P (3-18) which l i n e a r i z e s to N y P y — 6 P + ( l - - y P q ) 6 P •+ 6p n n k p n n k p ~ - n n e e n P" I Pn" p Z " Pc LINEARIZED TOTAL PRESSURE EQUATION (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 n e u t r a l i t y now becomes I P n q n + P Z + = p e n T h i s must be r e w r i t t e n w i t h p z + e x p r e s s e d i n te rms o f p*., w h i c h I have chosen as t h e a p p r o p r i a t e i n d e p e n d e n t v a r i a b l e . P Z + = ^ p m + ' w ^ e r e t n e summat ion i s o v e r t he me ta l e l e m e n t s m * pm m ' K e ' m+ ^ Km m + m 'm U + p, N * V M M m nr p; Pz But a z = and so p* = — p i a l rZ r m m + ' L \ III III ' t h u s p Z + = a7 I I + + p Z m m + e The m o d i f i e d c h a r g e n e u t r a l i t y e q u a t i o n i s now p* a I + y p q + L i y mjLL_ = p (3-20) ^ p n M n a 7 £ Im+ + p Q H e n Z m m + r 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 , a f t e r some m a n i p u l a t i o n , g i v e n by * , v. t Py a_I_+ 1 . . A A L 6 P n . " 1 + p ^ P n q n + k r n , I P n q n I p - 5 p n k p e £ p n q n « z m + P f i ) P e I ( n + + P f i ' P Z n K n k v p Z v °>nV + n = " I p n q n - aZ I + p e n m Im+ + P p LINEARIZED CHARGE NEUTRALITY EQUATION (3-21 ) 39 A g a i n , i n t h i s n o t a t i o n , k r e f e r s t o a ma jo r e l e m e n t , n t o a ma jo r s p e c i e s , and m t o a m e t a l e l e m e n t . S i n c e T a, + a 7 = 1 , t h e s e t o f abundance e q u a t i o n s d e s c r i b e d he re k K L i s r e d u n d a n t and so I have choosen t o e l i m i n a t e t h e hydrogen abundance e q u a t i o n . T h i s l e a v e s a s e t o f k abundance e q u a t i o n s , t he t o t a l p r e s s u r e 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 , each c o n t a i n i n g t h e k+2 * unknowns p^, p^ and p g . Thus I have a c l o s e d economized s e t o f e q u a t i o n s o f c o n s i d e r a b l y s m a l l e r o r d e r t han b e f o r e . S i n c e my c h o i c e o f t h e c l a s s o f ma jo r 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 and S , t h i s e conomized method i n v o l v e s r e p e a t e d l y s o l v i n g a l i n e a r s y s t em o f o r d e r 8 f o r t h e c o r r e c t i o n s Sp^ , Sp^ , 6 p ^ , S p g , S p ^ , Sp<,, 6 p * and <5pg and i t e r a t i n g u n t i l c o n v e r g e n c e i s a t t a i n e d . One d e t a i l r e m a i n s , and t h a t i s t o r e c o v e r t h e a c t u a l p a r t i a l p r e s s u r e s p m o f t h e i n d i v i d u a l me ta l e l emen t s a f t e r t h e above s o l u t i o n has been f o u n d . T h i s i s done s i m p l y as f o l l o w s : * * a m \ * Pm = a m p = a m ( a ^ = ^ P Z and s o , Pm PePm 1+ I , /p I + + p„ m + / K e m + ^e where a g a i n m r e f e r s t o a me ta l e l e m e n t . 40 CHAPTER FOUR D I S C U S S I O N OF R E S U L T S 1. D e t a i l s o f R e s u l t s 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 s t a t e d e s c r i b e d i n t h e p r e v i o u s c h a p t e r has been imp lemented ( i n b o t h t h e g e n e r a l and economized v e r s i o n s ) as a FORTRAN s u b r o u t i n e package (GAS ) . The economized v e r s i o n r u n s t h r e e t i m e s f a s t e r t han does t h e g e n e r a l s o l u t i o n (71 m s e c . v s . 223 m s e c , f o r t h e s o l u t i o n 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 T = 1200K , l o g P=5 on t h e U . B . C . Amdahl 470 V/8 c o m p u t e r ) . The p a r t i a l p r e s s u r e s o b t a i n e d by t h e two methods t y p i c a l l y ag r ee 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 t h e w o r s t d i s c r e p a n c i e s a round 2%. These w o r s t c a s e s o c c u r f o r m o l e c u l e s c o n t a i n i n g Mg o r A l , wh i ch v i o l a t e t h e a s s u m p t i o n made i n t h e e conomized s o l u t i o n t h a t " m e t a l s " do no t a s s o c i a t e , and t h e r e b y do no t d e p l e t e t h e ma jo r s p e c i e s . Under t h e above c o n d i t i o n s , e s s e n t i a l l y a l l o f t h e s e e l emen t s a r e bound up i n t h e h y d r o x i d e hence s l i g h t l y d e p l e t i n g t h e ma jo r s p e c i e s 0 . N e v e r t h e l e s s , c o n s i d e r i n g t h e u n c e r t a i n t i e s 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 i n v o l v e d , t h e s e e r r o r s a r e i n s i g n i f i c a n t . A l s o t h e d i s c r e p a n c i e s between the g e n e r a l and economized s o l u t i o n were much l e s s f o r c o n d i t i o n s more t y p i c a l o f l a t e - t y p e g i a n t s , i . e . T i n t h e r ange 2500-4000K , The GAS r o u t i n e s r e a d a l l o f t he n e c e s s a r y i n f o r m a t i o n f rom a s p e c i e s s p e c i f i c a t i o n f i l e , w h i c h c o n t a i n s a s i n g l e l i n e e n t r y f o r each 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 e q u i l i b r i u m o f t h e g a s . Each l i n e c o n t a i n s t h e 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 t h e c h e m i c a l symbol o f t he s p e c i e s , t h e c h a r g e ( i o n i z a t i o n s t a t e ) o f t h e s p e c i e s , and t h e t o t a l number o f atoms c o m p r i s i n g t h e s p e c i e s f o l l o w e d by p a i r s o f t h e number o f atoms and t h e a t o m i c number o f 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 . I f t h e e n t r y i s a n e u t r a l a t o m , t h e n i t s abundance i s i n c l u d e d , i f i t i s an i o n t hen i t s i o n i z a t i o n p o t e n t i a l ( f o rm t h e n e u t r a l s t a t e ) i s i n c l u d e d , and i f i t i s a m o l e c u l e y . t h e n t h e c o e f f i c i e n t s o f T s u j i ' s (1973) p o l y n o m i a l f i t 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 must be i n c l u d e d . Each s p e c i e s a l s o c o n t a i n s a p r i o r i t y code ( 1 , 2 , and 3) i n d i c a t i n g w h e t h e r i t i s t o be c o n s i d e r e d as a m a j o r , me ta l o r m i n o r s p e c i e s i n t h e e conomized s o l u t i o n . T h i s scheme a l l o w s t h e 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 s p e c i e s t o t h e l i s t o r 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 FORTRAN s o u r c e c o d e . An i n i t i a l e s t i m a t e o f t h e p a r t i a l p r e s s u r e s o f t h e n e u t r a l f r e e a t o m i c s p e c i e s i s r e q u i r e d by t h e GAS r o u t i n e s t o s t a r t t h e l i n e a r i z a t i o n s o l u t i o n . S i n c e my e s t i m a t e s ^ were d e r i v e d a s suming s o l a r c o m p o s i t i o n , t h e c o n v e r g e n c e o f c a s e s w i t h C/0 < 1 was v e r y r a p i d , n e v e r r e q u i r i n g more t han 2 o r 3 i t e r a t i o n s . T h i s was a l s o t r u e o f t h o s e c a s e s w i t h C/0 > 1 wh i ch were s u f f i c i e n t l y warm so as t o p r e v e n t f o r m a t i o n o f p o l y a t o m i c c a r b o n m o l e c u l e s . Fo r 42 t e m p e r a t u r e s T < 3000K w i t h C/0 > 1. howeve r , t h e s e i n i t i a l e s t i m a t e s become i n c r e a s i n g l y i n a c c u r a t e due t o t h e n e g l e c t o f s e v e r a l i m p o r t a n t p o l y a t o m i c s p e c i e s ( p a r t i c u l a r l y HCN and S i C g ) . and so t h e l i n e a r i z a -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 t o c o n v e r g e . I t s h o u l d be n o t e d t h a t t h i s i s n o t a f a i l i n g o f t h e GAS r o u t i n e o r a l g o r i t h m , bu t r a t h e r a consequence o f bad s t a r t i n g e s t i m a t e s ( f o r t h e c a s e w i t h C/0 > 1 ) . The r e a s o n f o r t h i s stems f rom t h e i n t e n t i o n t o use t h e s e e q u a t i o n o f s t a t e r o u t i n e s i n model a tmosphe res o f M - g i a n t s , f o r w h i c h C/0 < 1. N e v e r t h e l e s s , i t r e m a i n s a f a i r l y s t r a i g h t f o r w a r d m a t t e r t o improve t h e s e s t a r t i n g e s t i m a t e s f o r t h e C/0 > 1 c a s e ; t h i s w i l l be done i n t h e near f u t u r e . 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 has been s o l v e d u s i n g t h e e conomized s o l u t i o n method f o r a s o l a r c o m p o s i t i o n gas (C/0 = 0 .58 ) o v e r a range o f t e m p e r a t u r e s 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 o u t e r a tmosphe res o f l a t e - t y p e s t a r s . A t o t a l o f 109 s p e c i e s fo rmed f rom 25 e l e m e n t s were c o n s i d e r e d i n t h e c h e m i c a l e q u i l i b r i u m , w i t h t h e e l e m e n t a l abundances t a k e n f rom A l l e n ( 1 9 7 6 ) . These a b u n d a n c e s , a l o n g w i t h t h e c o m p l e t e l i s t o f s p e c i e s a r e d e t a i l e d i n T a b l e 1 . The s o l u t i o n s have been o b t a i n e d f o r bo th t h e p r e s s u r e s l o g P = 2 and l o g P = 5 ( r e p r e s e n t a t i v e ; o f t he l i n e - f o r m i n g r e g i o n s o f l a t e -t y p e g i a n t and dwa r f s t a r s r e s p e c t i v e l y ) o v e r a t e m p e r a t u r e g r i d r a n g i n g f rom T = 1200K to T = 6300K by 100K s t e p s . The r e s u l t s a r e p r e s e n t e d i n a s e r i e s o f g raphs ( F i g s . 1-9) i n d i c a t i n g t h e d i s t r i -b u t i o n o f each o f t h e i m p o r t a n t e l emen t s H , C , N, 0 , S i , S , A l , Mg and CI among the v a r i o u 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 o f t e m p e r a t u r e . The v e r t i c a l s c a l e f o r t h e s e f i g u r e s i n d i c a t e s t h e 43 l o g a r i t h m o f the r a t i o P n / P , i . e . t h e r a t i o o f t h e p a r t i a l p r e s s u r e o f each s p e c i e s t o t h e t o t a l gas p r e s s u r e . A s i m i l a r s e r i e s o f g raphs ( F i g s . 10-13) a g a i n i n d i c a t e t h e 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 and 0 among t h e v a r i o u s s p e c i e s as a f u n c t i o n o f t e m p e r a t u r e , bu t now f o r t h e c a r b o n e n r i c h e d com-p o s i t i o n s C/0 = 1 and C/0 = 1 . 5 , a l l f o r l o g P = 3 . The v a r i a t i o n i n C/0 r a t i o i n t h e s e c a s e s i s pe r f o rmed by i n c r e a s i n g t h e C-abundance o n l y , w h i l e s c a l i n g a l l o t h e r e l e m e n t a l abundances s l i g h t l y downward so as t o p r e s e r v e t h e sum o f t h e abundances as u n i t y . F i n a l l y , a t h i r d s e t o f g r a p h s ( F i g s . 14-18) i l l u s t r a t e t h e e f f e c t o f v a r y i n g C/0 r a t i o , each f o r a f i x e d t e m p e r a t u r e and p r e s s u r e , upon t h e c h e m i c a l e q u i l i b r i u m o f t h e g a s . T h i s i s c a l c u -l a t e d f o r a s e r i e s o f t e m p e r a t u r e s o v e r t h e r ange 2000-4000K w i t h l o g P = 3 , f o r t h e most abundant s p e c i e s ( a r b i t r a r i l y b roken i n t o two g roups f o r c l a r i t y o f p r e s e n t a t i o n ) . These f i g u r e s s t r i k i n g l y d e m o n s t r a t e t h e d i f f e r e n t c h e m i s t r y o f t h e r e g i m e s C/0 < 1 and C/0 > 1 , p a r t i c u l a r l y a t t h e c o o l e r t e m p e r a t u r e s . T h i s i s a d i r e c t r e s u l t o f t h e ex t r eme s t a b i l i t y o f t h e CO m o l e c u l e w h i c h e s s e n t i a l l y d e p l e t e s a l l o f t h e 1 e s s a b u n d a n t component i n i t s f o r m a t i o n . So a s t a r w i t h C/0 < 1 ( i . e . an M-s t a r ) w i l l be l e f t w i t h a r e a s o n a b l e amount o f f r e e 0 , bu t v e r y l i t t l e f r e e C , a f t e r CO f o r m a t i o n i s c o n s i d e r e d . T h i s oxygen i s t h e n a v a i l a b l e f o r f o r m a t i o n o f many o x i d e s . ? The r e v e r s e h o l d s f o r t h e . c a r b o n s t a r c a s e (C/0 > V) for' ' w h i c h f r e e C . r e m a i n s a f t e r CO f o r m a t i o n , and so i s a v a i l a b l e f o r t h e f o r m a t i o n o f a p l e t h o r a o f p o l y a t o m i c c a r b o n compounds . 44 2 . A s t r o p h y s i c a l I m p l i c a t i o n s 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 rob l em e n t e r s i n t o t h e l a r g e r model a tmosphe re p rob l em i n two w a y s . One i s t o o b t a i n t h e o v e r a l l dependence o f d e n s i t y upon t e m p e r a t u r e a t c o n -s t a n t p r e s s u r e i n o r d e r t h a t t h e e q u a t i o n o f h y d r o s t a t i c e q u i l i b r i u m may be i n t e g r a t e d i n w a r d i n t o the a t m o s p h e r e . T h i s i s r e a s o n a b l y s t r a i g h t f o r w a r d as i t r e q u i r e s knowledge o f o n l y t h e most abundant s p e c i e s , i n f a c t 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 c o n -s i d e r i n g o n l y hydrogen i n t h e fo rms o f H g , H and H + . A v e r y good s o l u t i o n i s o b t a i n e d i f t h e s p e c i e s H , C , N , 0 , H g , H , H + , CO, N g , and HgO a r e a l l c o n s i d e r e d . 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 a tmosphe re s p r o b l e m , i s t o d e t e r m i n e t h e e q u i l i b r i u m abundances o f t h e i m p o r t a n t o p a c i t y s o u r c e s c o n s i s t e n t w i t h t h e s o l u t i o n o f t h e r a d i a t i v e t r a n s f e r p r o b l e m . I t i s i m p o r t a n t t o r e a l i z e t h a t c e r t a i n s p e c i e s 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 i n terms o f p a r t i a l 6 -8 p r e s s u r e ( pe rhaps a c c o u n t i n g f o r o n l y 10 o r even 1 0 " o f t he t o t a l gas p r e s s u r e ) and y e t domina t e t h e a b s o r p t i o n c o e f f i c i e n t o f t h e gas i n c e r t a i n s p e c t r a l r e g i o n . Examples a r e H~ i n t h e nea r i n f r a r e d a t 5000K , T iO i n t h e v i s i b l e r e g i o n a t 3000K f o r M - s t a r s , and Cg i n t h e v i s i b l e a t 3000K f o r C - s t a r s . 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 abundant t h a n t h e s e o p a c i t y s o u r c e s f rom t h e c h e m i c a l e q u i l i b r i u m w i l l i n g e n e r a l i n -v a l i d a t e t h e s o l u t i o n o b t a i n e d f o r a l l o f t h e l e s s abundant s p e c i e s , i t i s e s s e n t i a l t h a t a l l s p e c i e s w i t h s u s p e c t e d abundance g r e a t e r 45 t h a n o r c o m p a r a b l e t o t h e v a r i o u s 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 v t h e e q u i l i b r i u m c a l c u l a t i o n s . In c o n t r a s t t o t h e mere h a n d f u l o f s p e c i e s n e c e s s a r y 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 must i n c l u d e on t h e o r d e r o f one o r two hundred s p e c i e s i n t h e c a l c u l a t i o n s t o o b t a i n an a c c u r a t e a b s o r p t i o n c o e f f i c i e n t . As an ex t r eme c a s e , c o n s i d e r t h e c o o l SC M i r a v a r i a b l e VX A q l . A t c e r t a i n phases o f i t s l i g h t c y c l e , bands o f CaCl become t h e s t r o n g e s t band f e a t u r e s i n t h e o p t i c a l s p e c t r u m ( C l e g g and W y c k o f f , 1 9 7 7 ) . CaCl j_s 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 c a l c u l a t i o n s , bu t i t i s neve r s u f f i c i e n t l y abundant t o appea r above t h e l o g P n /P l o w e r l i m i t o f -9 on my f i g u r e s . In f a c t , a t T = 2500K , l o g P = 3 I f i n d l o g P C a C l / ' P = -8 . And y e t i t i s e v i d e n t f rom t h e o p t i c a l s p e c t r a o f VX Aq l t h a t t h i s s p e c i e s must be a s i g n i f i c a n t o p a c i t y s o u r c e i n t h i s s t a r . T h i s example c l e a r l y d e m o n s t r a t e s t h e need f o r an a c c u r a t e 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 code as an i n t e g r a l p a r t o f any r e a l i s t i c model a tmosphere program f o r l a t e - t y p e s t a r s . TABLE I SPECIES CONSIDERED IN EQUATION OF STATE ELEMENT Z ABUNDANCE SPECIES CONTAINING THIS ELEMENT H 1 9 . 323E-01 H H+ H- H2 OH H20 MgH A 1 H He HCO MgOH A 1 OH CaOH 2 6 . 526E-02 He He+ C 6 4 .941E-04 C c+ C2 C3 s i c S1C2 HCN HCO N 7 8 .950E-04 N N+ N2 NH 0 8 8 .484E-04 0 0+ 02 OH SIO SO CaO ScO Ne ZrO Zr02 HCO MgOH 10 7 738E-05 Ne Ne+ Na 1 1 1 678E-06 Na Na+ NaCl Mg 12 2 424E-05 Mg Mg+ Mg+ + MgH A 1 13 2 238E-06 Al A1 + A 1 H A10 S i 14 3 077E-05 SI S1 + S1H SIO s 16 1 492E-05 s S+ HS H2S CI 17 3 729E-07 CI C1 + CI - HC1 K 19 8 298E-08 K K+ Ca 20 1 865E-06 Ca Ca+ Ca++ CaH Sc 21 1 492E-09 Sc Sc+ ScO Sc02 T 1 22 1 212E-07 T1 T1 + T10 T1S V 23 2 331E-08 V V+ VO V02 Cr 24 6 619E-07 Cr Cr+ Mn 25 2 331E-07 Mn Mn+ Fe 26 3. 729E-05 Fe Fe+ Co 27 1 . 119E-07 Co Co+ Ni 28 1 . 865E-06 N1 N1 + Sr 38 6. 619E-10 Sr Sr+ Y 39 5. 874E-1 1 Y Y + Y0 Y02 Zr 40 2 . 983E-10 Zr Zr+ ZrO Zr02 H2 + S1H CH2 CH CH2 NH2 H20 Sc02 A) OH MgO A1C1 SIS SO NaCl CaO CH HS CHS C2H2 CH3 NH3 NO T iO CaOH MgCl A 10H S 1C CS MgCl CaCl C2H2 H2S C2H CN C2H CN' CO VO MgOH S1C2 SIS A1C1 CaOH NH HC 1 CO NO C02 V02 NH2 CaH C02 HCN MgO YO NH3 HCN CS A10 Y02 T1S CaCl A TOTAL OF 109 SPECIES FORMED FROM 25 ELEMENTS l o g P = 2 CHEMICAL EQUILIBRIUM OF CARBON COMPOUNDS LOG(P)=5 H 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 F i g u r e 2 ( b ) : C/O = 0 .58 ( s o l a r ) , l o g P = 5 CHEMICAL EQUILIBRIUM OF OXYGEN LOG(P)=2 ^ I ' i i i i i i i | i i i i i i i i i | i i i n i i i i | n i i i i i r i j i i M i \ f i i | i i H 2--8-CO H,0 SiO AlOH CC Ti( W* VO' _liO ^SiO XO 1000 2 0 0 0 3 0 0 0 4 0 & 0 T 5 0 0 0 6 0 0 0 F i g u r e 4 ( a ) : C/0 = 0 .58 ( s o l a r ) , l o g P = 2 T(K) 0 ^ 2-q CHEMICAL EQUILIBRIUM OF OXYGEN L0G(P^5 H 1 1 I 1 I I 1 1 I | 1 I 1 1 1 1 I I I | 1 I I I I 1 I I I | I 1 1 I I I I 1 I | I I I I I I I I I | 1 I Ij SiO F i g u r e 4 ( b ) : C/0 = 0 .58 ( s o l a r ) , 1og P = 5 4 0 0 0 5 0 0 0 T(K) i i I i i i 6 0 0 0 F i g u r e 5 ( b ) : C/0 = 0 . 5 8 ( s o l a r ) l o g P = 5. T(K) CHEMICAL EQUILIBRIUM rTTTTTTTTT OF SULFUR i i n i i i I I | I I T 11 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 H =2 2-q 1000 I I I I I I I I I I I I I I 2 0 0 0 I '1 I'T'l I "| I i i I i I I I i j i i r i i ' i i I I | I I I 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 F i g u r e 6 ( a ) : C/0 = 0 .58 ( s o l a r ) , l o g P = 2 T(K) CHEMICAL EQUILIBRIUM OF SULFUR 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 F i g u r e 6 ( b ) : C/0 = 0 .58 ( s o l a r ) , T(K) l o g P = 5 ' n i i | i i r 6 0 0 0 54 55 i i i i CHEMICAL EQUILIBRIUM OF CHLORINE L0G(P)=2 0 ^ i i i i i [ i i i i i i i i i | i i i i i i n i | i 11 i i i i i i | i i i i i ' i 1 i i | I i H 0, \ o o - 2 -- 4 -- 8 - q CI NaCl M i l n i "i n i i I i i i i M i i r i i i i i i I i i 'i 11 i i 1 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 1000 6 0 0 0 F i g u r e 9 ( a ) : C/0 = 0 .58 ( s o l a r ) , T ( K ) l o g P = 2 CHEMICAL EQUILIBRIUM OF CHLORINE L0G(P)=5 i f 1 1 I 1 I I I l I I I I I I I I I 1 I I I I I I 1 1 1 1 I I I 1 1 I 1 I 1 T 1 I I I I I I'M I I I I I H \ k *—' o o 0 ^ -2-g -4 -3 j ^ i i i i m i i i i I T F i g u r e 9 ( b ) : C/0 = 0 .58 ( s o l a r ) , T(K) l o g P = 5 '56 F i g u r e 1 0 ( b ) : C/0 = 1 . 5 , Log P = 3 T(K) F i g u r e 1 1 ( b ) : C/0 = 1 . 5 , l o g P = 3 T(K) o o-3 -2-3 - 4 --6-3 -8-3 CHEMICAL EQUILIBRIUM OF NITROGEN C/0=1 I f f i f f i? , T i i i M 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 i i i i | t . • ! . • q q NH, i i i i i 1000 2 0 0 0 3 0 0 0 F i g u r e 1 2 ( a ) : C/0 = 1 , l o g P 4 0 0 0 5 0 0 0 3 T(K) o-q \ o o -3 - 6 - 3 h i i I ' M i i q 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 F i g u r e 1 2 ( b ) : C/0 = 1 . 5 , l o g P = 3 ?(K) 6 0 0 0 CHEMICAL EQUILIBRIUM OF OXYGEN C/ n i i i 1 i i i i i i i i i i i III i i l i i i i i l i i i i i 1 C 0=1 LOG^S i i M 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 1 m | i i i T i i t I'I | i i «j 0 ^ 4-^ -8-_C0 AIOH\  i i i V i i i i i I iTT 1000 2 0 0 0 3 0 0 0 F i g u r e 1 3 ( a ) : C/0 = 1 , l o g P 4 0 0 0 p i i r i i i i i | i i r 5 0 0 0 6 0 0 0 T(K) CHEMICAL EQUILIBRIUM OF OXYGEN C/0=1.5 L0G(P)=3 H I n f i n | i i Vi n n i] 1000 2 0 0 0 3 0 0 0 T 4 0 0 0 11 i i i i i i i i | i i i 5 0 0 0 6 0 0 0 F i g u r e 1 3 ( b ) : C/0 = 1 . 5 , l o g P = 3 T(K) ABUNDANCE OF IMPORTANT SPECIES T=2000K L0G(P)=3 -( 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 r i 1 1 i i i i i i i i i i r i 1 i i i i i H m i n i i i 1 i i i i i i i i i j i l i i i 0.4 0.6 0.8 F i g u r e 1 4 ( a ) : T = 200QK, l o g P = 3 1.0 1.2 C/0 RATIO ABUNDANCE OF IMPORTANT SPECIES T=2000K F i g u r e 1 4 ( b ) : T = 2 0 0 0 K , l o g P = 3 C/0 RATIO 61 0.4 0.6 0.8 1.0 1.2 1.4 F i g u r e 1 5 ( a ) : T = 2 5 0 0 K , Q/Q RATIO l o g P = 3 ' 0.4 0.6 0.8 1.0 1.2 1.4 F i g u r e 1 5 ( b ) : T = 2500K , l o g P=3 C/0 RATIO ABUNDANCE OF IMPORTANT SPECIES T=3000K L0G(P)=3 F i g u r e 1 6 ( a ) : T = 3000K , C/O RATIO • l o g P = 3 F i g u r e 1 6 ( b ) : T = 30QOK, l o g P = 3 C/O RATIO 63 ABUNDANCE OF IMPORTANT SPECIES T=3500K L0G(P)=3 F i g u r e 1 7 ( a ) : T = 3 5 0 0 K , C/O RATIO l o g P = 3 0.4 0.6 0.8 1.0 1.2 1.4 F i g u r e 1 7 ( b ) : T = 3500K , l o g P=3 c / 0 RATIO 64 ABUNDANCE OF IMPORTANT SPECIES T=4000K LOG(P)=3 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 i i i i i I i i i i i i i i i 11 i i OH \ o -2-- 8 ^ 0.4 H CO 17 I I I N i 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 i i ] i I I I i i iTT | i i i " i H r 0.6 0.8 1.0 1.2 1.4 F i g u r e 1 8 ( a ) : T = 4 0 0 0 K , l o g P = 3 C/0 RATIO F i g u r e 1 8 ( b ) : T = 4 0 0 0 K , l o g P = 3 C/0 RATIO 65 REFERENCES A l l e n , C.W. 1 9 7 6 , A s t r o p h y s i c a l Q u a n t i t i e s , 3 r d e d . , A t h l o n e P r e s s , L o n d o n . Auman, J . R . 1 9 6 9 , A p . J . 157_, 7 9 9 . C l e g g , R. and W y c k o f f , S. 1 9 7 7 , M . N . R . A . S . 1 7 9 , 4 1 7 . Goon , G. and Auman, J . R . 1 9 7 0 , A p . J . 1_61_, 5 3 3 . H a l l , D .N .B . and N o y e s , R.W. 1 9 7 2 , A p . J . L e t t . 1_75, L 9 5 . I r w i n , A.W. 1 9 7 8 , P h . D . t h e s i s , U n i v e r s i t y o f T o r o n t o . 1 9 8 1 , A p . J . Supp l . 45_, 6 2 1 . J o h n s o n , H . R . , B e r n a t , A . P . and K r u p p , B .M. 1 9 8 0 , A p . J . S u p p l . 4 2 , 5 0 1 . J o h n s o n , H.R. and Sauva l , A . J . 1 9 8 2 , A s t r . and A p . S u p p l . 49^, 7 7 . Keenan , P .C . and M c N e i l , R .C . 1 9 7 6 , An A t l a s o f S p e c t r a o f t h e C o o l e r  S t a r s , Oh io S t a t e U n i v e r s i t y P r e s s . R u s s e l l , H .N . 1 9 3 4 , A p . J . 8 0 , 3 1 7 . Ta tum, J . B . 1 9 6 6 , Publ . Dom. A p . Obs . V i c t o r i a , ]3_, 1. T s u j i , T . 1 9 7 3 , A s t r o n . and A p . 23_, 4 1 1 . V a r d y a , M.S. 1 9 6 6 , M . N . R . A . S . 1 3 4 , 3 4 7 . 1967 , Mem. Roy . A s t r o n . S o c . 71_, 2 4 9 . W h i t e , W . B . , J o h n s o n , S . M . , D a n t z i g , G .B . 1 9 5 8 , J . Chem. P h y s . 2 8 , 751 . 

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