UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Radiation driven instabilities in stellar winds Carlberg, Raymond G. 1978

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

Item Metadata

Download

Media
831-UBC_1978_A1 C36.pdf [ 12.44MB ]
Metadata
JSON: 831-1.0085756.json
JSON-LD: 831-1.0085756-ld.json
RDF/XML (Pretty): 831-1.0085756-rdf.xml
RDF/JSON: 831-1.0085756-rdf.json
Turtle: 831-1.0085756-turtle.txt
N-Triples: 831-1.0085756-rdf-ntriples.txt
Original Record: 831-1.0085756-source.json
Full Text
831-1.0085756-fulltext.txt
Citation
831-1.0085756.ris

Full Text

RADIATION DRIVEN INSTABILITIES IN STELLAR WINDS by RAYMOND GARY CARLBERG M.Sc. , U n i v e r s i t y o f B r i t i s h Columbia, 1975 B.Sc., U n i v e r s i t y o f Saskatchewan, 1972  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOB THE DEGREE OF DOCTOE OF PHILOSOPHY  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 g u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1978  (c) Baymond Gary C a r l b e r g , 1978  In p r e s e n t i n g  this  thesis  in p a r t i a l  fulfilment of  an advanced degree at the U n i v e r s i t y of B r i t i s h the I  Library shall  f u r t h e r agree  for  scholarly  by h i s of  this  written  make i t  t h a t permission  It  is understood that  for financial  gain s h a l l  permission.  University of B r i t i s h  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  Odf  I agree  reference and  f o r e x t e n s i v e copying o f  Department of The  for  Columbia,  this  *?  jl%  Columbia  not  copying or  for  that  study. thesis  purposes may be granted by the Head of my Department  representatives. thesis  freely available  the requirements  or  publication  be al1 owed without my  ii  ABSTRACT T h i s t h e s i s i n v e s t i g a t e s the variability  which  high  observations  A program  of  optical  observations  found  t h e nature no  optical  of  the  fluctuations.  v a r i a b i l i t y over a time  p e r i o d of s i x hours and hence r e s t r i c t the v a r i a b i l i t y p e r i c d t o s i z e s c a l e s of l e s s than 5 x 1 0  comprised  of  a  neutron  star  variability  of s t e l l a r winds.  to  derive  estimates  of  A class  of  X-ray  of  i n f o r m a t i o n on  A theory of a c c r e t i o n onto a  neutron s t a r was developed which i s used data  this  o r b i t i n g a s t a r with a  strong s t e l l a r wind p r o v i d e s another source the  over  cm, but do confirm the  1 1  v a r i a t i o n s on time s c a l e s exceeding one day. sources  of the  time and s p e c t r a l r e s o l u t i o n was designed t o provide  q u a n t i t a t i v e i n f o r m a t i o n on These  nature  i s present i n the s t e l l a r winds of high l u -  minosity e a r l y type s t a r s . with  quantitative  with  X-ray  intensity  t h e d e n s i t y and v e l o c i t y o f the  s t e l l a r wind.  T h i s a n a l y s i s performed  the  i n t h e s t e l l a r wind i n c r e a s e s as the wind d e n s i t y  velocity  on Cen X-3 suggests  that  increases. A t h e o r e t i c a l a n a l y s i s o f the s t a b i l i t y of a is  made  to  determine  the wind i t s e l f .  are founds  amplify p r e - e x i s t i n g d i s t u r b a n c e s ,  and  which  w i t h i n the gas.  can grow from random motions  by  ones  that  instabilities I t i s found  (>10  11  strongly cm)  are  r a d i a t i o n i f the gas i s t h e r m a l l y s t a b l e , that i s i f  the net r a d i a t i v e energy termediate  absolute  those  d i s t u r b a n c e s <<10* cm) are always  damped by c o n d u c t i o n , and leng wavelength damped  wind  whether the v a r i a b i l i t y may o r i g i n a t e i n  Two types o f i n s t a b i l i t y  t h a t s h o r t wavelength  stellar  wavelengths  l o s s i n c r e a s e s with o f about  temperature.  In-  1 0 ~ cm are u s u a l l y s u b j e c t t o 8  9  iii  an a m p l i f i c a t i o n due radiation 10*  1  ca.  to the d e n s i t y g r a d i e n t of the  acceleration Absolute  wind.  a m p l i f i e s d i s t u r b a n c e s of s c a l e s 10  a temperature of s e v e r a l m i l l i o n the  basis  7  to  i n s t a b i l i t i e s are present i f the gas i s t h e r -  mally u n s t a b l e , i f the flow i s d e c c e l e r a t i n g , or i f the gas  On  The  of  has  degrees.  the i n f o r m a t i o n d e r i v e d on s t e l l a r  s t a b i l i t y i t i s proposed t h a t a complete theory should be on the assumption that the wind i s a n o n s t a t i o n a r y  flow.  wind based  CONTENTS  Chapter 1: I n t r o d u c t i o n  ....................................  Chapter 2: O p t i c a l Observations Chapter 3: Supersonic  Accretion  Chapter 4: P h y s i c a l D e s c r i p t i o n of The Gas Chapter 5: The S t a b i l i t y Chapter 6: C o n c l u s i o n s  analysis  .................  ..........................  .....................................  Bibliography Appendix 1: R a d i a t i v e E f f e c t s In Supersonic Appendix 2: Gas P h y s i c s  Accretion  ......  ....................................  Appendix 3: The D i s p e r s i o n R e l a t i o n  ........................  Appendix 4: The Major Computer Prcgrams  ....................  V  FIGURES  1.  Lambda C e p h e i : The E f f e c t Of R e s o l u t i o n  ................ 10  2.  lambda C e p h e i  3.  lambda C e p h e i Day T c Bay  4.  A l p h a C a m e l o p a r d a l i s Day To Day ........................ 16  5.  D e l t a O r i o n i s Day To Day  6.  S u p e r s o n i c A c c r e t i o n S c h e m a t i c .........................22  7.  Schematic  8.  The D e n s i t y  9.  I o n i z a t i o n 8 a l a nc € ..........»« .... . . . . . . . . . . « • . . . . . . . . .35  Time S e r i e s .............................. 14 ...15  ...17  X - r a y I n t e n s i t y V a r i a t i o n o f Cen X-3 .........27 V e l o c i t y V a r i a t i o n o f Can X-3 .............. 28  10.  S t a n d a r d H e a t i n g And C o o l i n g  11.  CNO A b u n d a n c e s 10 T i m e s S o l a r  12.  L o s s e s I n An O p t i c a l l y T h i c k Medium ................... 40  13.  Radiation  14.  Momentum B a l a n c e ...................................... 46  15.  Roots For A S t a t i c Pseudo I s c t h e r m a l  16.  H i t h H e a t i n g And C o o l i n g , V, dv/dz N o n z e r o ........... .60  17.  8 i t h The R a d i a t i o n F o r c e ..............................62  18 •  No  1.  CooXxncj  Rate  ..................... 36  H e a t i n g And C o d i n g  .....38  F o r c e As A f u n c t i o n Of T e m p e r a t u r e ..........42  A t m o s p h e r e ....... 55  • ••• • ••• •* • * • • • •* • • • • * • • • • * • « • • • •  Thermal I n s t a b i l i t y  * • • • • 63  T=5x10s K ......................... .65  19.  High Temperature I n s t a b i l i t y  20.  Decceleration  .......................... 66  I n s t a b i l i t y .............................67  vi  TABLES 1.  C a t a l o g u e Of O b s e r v a t i o n s  2.  Scales  3.  A t o m i c Abundances  4.  Maximum  5.  The D i s p e r s i o n  6.  Photoionization Cross  7.  Gaunt F a c t o r  In S u p e r s o n i c  9  Accretion  . . . . . . . . . . . . . . . . . . . . . . . . .23  ......................................32  V e l o c i t y For An A c c e l e r a t i n g S o l u t i o n ..........47 Relations  Plotted  Section  .......................68  P a r a m e t e r s . . . . . . . . . . . . . . . 90  F o r Hydrogen  92  8., The R e c o m b i n a t i o n Hate C o n s t a n t s .......................93 9.  I o n i z a t i o n P o t e n t i a l s And S u b s h e l l  10.  Gaunt  Factor  Constants  11.  The Resonance L i n e s  Populations  .........97  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103  vii  ACKNOWLEDGEMENTS The  completion  of  this  thesis  owes a g r e a t deal t o t h e  t e a c h i n g , a s s i s t a n c e , and encouragement t h a t My  s u p e r v i s o r . Dr.  support  when  have  Greg Fahlman, always provided  ear and a c a r e f u l c r i t i c i s m o f my i d e a s . of  I  needed.  received.  an i n t e r e s t e d  He was a l s o  a  scurce  Drs. Gordon Walker and Jason Auman  made a number o f suggestions  which  substantially  improved  the  p r e s e n t a t i o n o f the t h e s i s . My i n d u c t i o n i n t o the world of o b s e r v a t i o n a l astronomy came with  three  very  pleasant  summers  spent  in  V i c t o r i a a t the  Dominion A s t r o p h y s i c a l Observatory. . The f r u s t r a t i o n s spectrophotometry ated by Dr.  system Dr.  owe  The s u c c e s s f u l o b s e r v a t i o n s  use  Bruce  Campbell,  L e s t e r , and Mike C r e s w e l l .  obtained  Dr.  Chris  use of C h r i s P r i t c h e t t * s data  handling  A l a r g e part of t h i s t h e s i s r e l i e d completion.  with  The  OBC  Pritchett,  activity  this  a  go  Computing Centre has been a great  to  A.  C.  by t h e  on the computer f o r i t s  Hearn,  assis-  and the  l a r g e number o f w e l l documented u t i l i t y  P a r t i c u l a r thanks  Tim  package, RETICENT.  tance and deserves much p r a i s e f o r i t s f a c i l i t i e s , of  fieticon  The whcle data r e d u c t i o n process has  reduced t o a s t r a i g h t f o r w a r d and enjoyable  vision  his  a great d e a l to the advice and t e c h n i c a l support o f  Walker, Dr.  been  doing  with photographic p l a t e s were l a r g e l y e l i m i n -  Gordon Walker's encouragement to  d e t e c t o r system.  of  originally  pro-  programs. at t h e  U n i v e r s i t y of Utah, who wrote the program REDUCE. Financial  support  was  provided  by the N a t i o n a l Research  C o u n c i l of Canada and by a MacMillan Family  Fellowship.  1  CHAPT IB 1. The  INTE0DUC1I0N  o p t i c a l s p e c t r a of many hot s t a r s were  emission  components  on the Harvard  the Henry Draper Catalogue,  proposal  that  and  was  P i c k e r i n g 1918).  leading  to  A great d e a l of  {see  (1929) emission informa-  accumulated on the o p t i c a l s p e c t r a of emission  line  0  E s t a r s i n the f o l l o w i n g years, which i s summarized i n Eeals  (1951) and opened found  D n d e r h i l l (1960).  by  Morton  A  new  blue  observational  shifted  S , 6,  and £., had  to v e l o c i t i e s o f order 2000 Km  v e l o c i t i e s exceeded the t y p i c a l escape speed of 300 a  large  window^ was  (1967) using a rocket borne s p e c t r o g r a p h .  t h a t 4 o f the Orion s t a r s ,  lines  such  have  Many o f  Beal's  the l i n e p r o f i l e s could be explained by  from gas being e j e c t e d from the s t a r . tion  to  o b j e c t i v e prism p l a t e s  Cannon and  these s t a r s were s t u d i e d i n d e t a i l ,  noted  f a c t o r t h a t there was  He  absorption s  _ l  .  These  Km  s~  by  1  no question t h a t the s t a r s  were l o s i n g mass at a l a r g e r a t e . The  Snow and  Morton u l t r a v i o l e t survey  (1976)  showed  that  all  s t a r s with an e f f e c t i v e temperature g r e a t e r than about 3x10*  K,  and  l u m i n o s i t i e s g r e a t e r than a b o l o m e t r i c  magnitude o f  have a d e t e c t a b l e s u p e r s o n i c wind which c a r r i e s away a cant  amount  of  the s t a r ' s mass during i t s l i f e t i m e .  mass l o s s r a t e s have been (see  Hutchings 1976)  which now and  from  and OV o b s e r v a t i o n s  have been extended to i n f r a r e d  r a d i o wavelengths  provide  obtained  confirming  (Bright and and  Barlow  optical  -6,  signifiReliable  observations  (Snow and Morton  1976)  (Barlow and Cohen  1977)  1975),  all  of  which  complementary data on the magnitude of  the mass l o s s .  The d e r i v e d mass l o s s r a t e s f o r OB s t a r s l i e  the  10~  range  of  9  to  10~  s  in  M0/year, with t e r m i n a l v e l o c i t i e s  2  ranging  from 1000 t o 3000 km s-».  The is  s t e l l a r wind phenomenon poses s e v e r a l  questions:  the p h y s i c a l mechanism d r i v i n g the mass l o s s ?  what  how does the  mass l o s s e f f e c t the s t a r ' s e v o l u t i o n ?  and how do  luminous  medium and the e v o l u t i o n  s t a r s - % f f e e t the i n t e r s t e l l a r  of a galaxy?  The answers to a l l c f these  thorough understanding thesis  is  a  questions hinge  contribution  to  the  understanding  which occurs  galactic  1972 and Kippenhahn et a l .  with  line  profiles  time  and F r o s t  nuclei,  been  For  period makes i t d i f f i c u l t  more c l o s e l y  a  (Mushotzky,  et  known  to  show  ones some  1951, U n d e r b i l l  1974, Leep and C o n t i 1978, Brucato 1971,  t i o n of the v a r i a t i o n . to  field;  c f s t e l l a r wind s t a r s , e s p e c i a l l y  Snow 1S77, and Hosendahl 1S73). this  of the b a s i c  radiation  v a r i a b i l i t y over one day (see f o r i n s t a n c e Beals Conti  This  1975).  high mass l o s s r a t e s have long  1960,  a  i n a number of a s t r o p h y s i c a l s i t u a t i o n s ,  i n c l u d i n g quasars and a c t i v e  The  on  The p h y s i c a l problem i s to d e s c r i b e  the dynamics of a qas moving i n an i n t e n s e  al.  hot,  o f the p h y s i c s o f the s t e l l a r wind.  nature o f the s t e l l a r wind.  situation  these  ground  based  t o r e s o l v e the time e v o l u -  An o b s e r v a t i o n a l program  d e f i n e the nature  using a modern d e t e c t o r capable  observation  was  initiated  of these r e p o r t e d v a r i a t i o n s ,  of measuring very s m a l l changes.  T h i s w i l l be d i s c u s s e d i n Chapter 2. Besides o p t i c a l evidence  of v a r i a b i l i t y ,  of X-ray binary s t a r s where the X-ray source accreting sources which  mass  from  the  stellar  wind  there are a number isa  fCcnti  neutron  star  1978).  These  show a number of s c a l e s of v a r i a t i o n of t h e i r can  be  ascribed  to  intensity  v a r i a t i o n s of the s t e l l a r wind.  A  3  t h e o r e t i c a l a n a l y s i s of the a c c r e t i o n process the  ctserved  could  be  that  the  X-ray  wind a t the l o c a t i o n of the neutron s t a r .  performed on X-ray data  This  be d e s c r i b e d The  This  o f the theory  luminous s t a r was i n i t i a l l y  of a s t e l l a r  wind from  put f o r t h by Lucy and Solomon  (1970), who proposed t h a t the a c c e l e r a t i o n was produced of  photons  resonance l i n e s .  with  wavelengths t h a t f e l l  T h i s was a g e n e r a l i z a t i o n  of  by the  w i t h i n a few  Milne*s  (1926)  idea t h a t momentum t r a n s f e r from photons c o u l d s e l e c t i v e l y l e r a t e c e r t a i n ions. and  Klein  (1975,  T h i s was l a t e r extended by C a s t o r , referred  many l i n e s of many i o n s . agreement  will  i n Chapter 3.  basic formulation  scattering  an-  f o r the source Cen X-3 i n d i c a t e s  a c o r r e l a t i o n between the wind v e l o c i t y and d e n s i t y .  a hot,  data  used to d e r i v e the p r e v a i l i n g d e n s i t y and v e l o c i t y o f  the s t e l l a r alysis  i n t e n s i t y was made i n order  and how i t e f f e c t s  with  acce-  Abbott,  t o as CAK) t o i n c l u d e t h e f o r c e on  The  theory  provided  an  encouraqinq  the l i m i t e d data a v a i l a b l e on the v e l o c i t y as a  f u n c t i o n o f r a d i u s and mass l o s s r a t e s . Recently vealed  the u l t r a v i o l e t s a t e l l i t e  have r e -  t h a t some h i g h l y i o n i z e d s p e c i e s , i n p a r t i c u l a r 0 VI and  N V, are present to  observations  i n the wind.  These i o n s would not be  be i o n i z e d i n any observable  appropriate  to these s t a r s .  variations  in  as short as s i x  q u a n t i t y by the r a d i a t i o n f i e l d  York, j t a l .  the 0 VI l i n e i n three hours.  This  expected  (1977) have  observed  s t a r s over a time  observation  suggests  a  periods "slab"  moving outwards a t an i n c r e a s i n g v e l o c i t y . . The presence of 0 VI in  the  stellar  excitation.  wind presents  At the present  a puzzle as t o the source o f i t s  time  there  are three  proposals.  4  First,  Castor  (1978) has modified h i s r a d i a t i o n d r i v e n wind t o  an a r b i t r a r i l y s p e c i f i e d temperature  higher than r a d i a t i v e e q u i -  l i b r i u m , which p r o v i d e s a s u i t a b l e abundance of Lamers  and  Snow  (1978)  have  an  empirical  C VI.  Second,  "warm  radiation  p r e s s u r e " model, i n which they show t h a t the i o n s can be ded  if  the  s t e l l a r wind i s a t a temperature  Neither of these models s p e c i f y the heating.  Third,  Hearn  source  provi-  of aJsout 2x10  of  (1975) has proposed  the  million  Cassinelli  degrees.  Pursuing  that s t e l l a r  this  idea  winds  ionization ratios.  a corona, Hearn could The  and  produce  the  To provide a heating mechanism f o r  (1972) showed t h a t r a d i a t i o n d r i v e n sound  waves  be a m p l i f i e d while propogating outward i n the atmosphere.  waves grow t o a s a t u r a t e d amplitude  enough  shock  are two  d i f f i c u l t i e s with t h i s  (1975)  have  sufficient  h e a t i n g to maintain a corona  pointed  out  analysis.  (Hearn  to  same as the atmospheric  Berthomieu  t h a t Hearn*s s i m p l i f y i n g  amplification  scale  length.  Therefore  et  There al.  assumptions which i s the significant  only o c c u r s over l e n g t h s which i n v a l i d a t e the as-  sumption cf s m a l l v a r i a t i o n s of the zero order length  provide  1973).  r e s u l t i n a s c a l e l e n g t h f o r the wave a m p l i f i c a t i o n  the  Clson  cf  (1978) have shown t h a t a s m a l l corona, about 10% of a  s t e l l a r r a d i u s , generates enough thermal X-rays t o reguired  K*  additional  are i n i t i a l l y a c c e l e r a t e d i n a hot corona with a temperature several  s  for amplification.  quantities  over  In a d d i t i o n , the u n s t a b l e waves  that he f i n d s are a m p l i f y i n g i n s t a b i l i t i e s  (Castor  1977),  and  r e q u i r e some o s c i l l a t o r to i n i t i a t e the wave motion. Motivated  by t h e o r e t i c a l arguments and  f l u c t u a t i o n s I have performed  the o b s e r v a t i o n s of  a s t a b i l i t y a n a l y s i s on the  egua-  5  tions The  governing  the moving gas i n t h e s t e l l a r r a d i a t i o n  complete s e t of e g u a t i c n s governing the  ££ipri  simplifications  were  used.  motion  field.  with  no  An accurate d e s c r i p t i o n of  the gas physics was developed u s i n g an approximate treatment radiation  transfer  dependent  ccly  on l o c a l g u a n t i t i e s .  r e s u l t the s t a t e of the gas can be completely s p e c i f i e d local  radiation  the d e n s i t y stability  field,  by  and temperature, as described  this  i n chapter  4.  dispersion  The  v e r t i c a l d i s t u r b a n c e s was i n v e s t i -  relation.  solu-  Chapter 5 i s comprised of  discussion. The  tive the  the  the  the gas v e l o c i t y and i t s g r a d i e n t , and  of the gas a g a i n s t  to  of As a  gated with the a i d o f a computer t o provide the numerical tions  a  purpose o f t h i s t h e s i s i s to i n v e s t i g a t e  nature  This  s t e l l a r wind. areas  and t h e o r e t i c a l problems which have been  i s not an attempt to c r e a t e  Bather i t i s a d e t a i l e d i n v e s t i g a t i o n of  wind.  because not much i s known about the b a s i c which  dominate  certain  some of the p h y s i -  c a l mechanisms which are important i n a s t e l l a r  cesses  out-  a u n i f i e d t h e o r y of a  o f t h e q u e s t i o n i n order t o i l l u m i n a t e  required  quantita-  of i n s t a b i l i t i e s i n s t e l l a r winds and r e l a t e i t t o  observational  lined.  the  This  physical  is pro-  the observed v a r i a b i l i t y o f the s t e l l a r  wind. The which  i n v e s t i g a t i o n i s c o n f i n e d to the s t e l l a r  i s l o o s e l y defined  as t h e r e g i o n  and for  the  gas  is  moving  with  As has been emphasized by Cannon  Thomas (1978), i t i s p o s s i b l e the  itself,  where the o p t i c a l depth  i n tbe continuum i s l e s s than one and t h e g r e a t e r than s c n i c v e l o c i t i e s .  wind  that  some of the d r i v i n g  force  wind and hence seme of the wind i n s t a b i l i t i e s may  6  o r i g i n a t e w i t h i n deeper l a y e r s of t h e s t a r . I t i s assumed t h a t there i s no done  mostly  because  of  problem which r e s u l t s . for  the  magnetic  field.  This  is  tremendous s i m p l i f i c a t i o n of the  But t h e r e i s no  observational  evidence  a magnetic f i e l d , although i f the wind i s as c h a o t i c as t h i s  t h e s i s suggests, In stellar  a magnetic f i e l d would be d i f f i c u l t  summary  this  thesis  wind v a r i a b i l i t y ,  t c detect.  i s motivated by o b s e r v a t i o n s o f  and suggestions  by ether authors  that  i n s t a b i l i t i e s do e x i s t which may be r e s p o n s i b l e f o r the c r e a t i o n of  a high  temperature corona.  c a r r i e d out i n twe p a r t s . b i l i t y i s acquired fluctuations  The i n v e s t i g a t i o n s described a r e  o b s e r v a t i o n a l evidence o f the  which suggests length and times s c a l e s c f the  which  are  present,  and a c o r r e l a t i o n between the  The  analysis  wind v e l o c i t y and d e n s i t y .  theoretical  p h y s i c a l sources of s e v e r a l i n s t a b i l i t i e s stellar  wind.  From t h i s i n f o r m a t i o n  only  provide  I suggest t h a t the s t e l l a r  and  the  the  instabilities  the source f o r t h e observed v a r i a b i l i t y , but  a l s o can be used to provide ion  provides  which can e x i s t i n the  wind i s an extremely c h a o t i c medium i n which not  varia-  corona  as  an i o n i z a t i o n source postulated  by Hearn.  f o r the  ating guantities.  guantities  and  VI  The presence o f  these i n s t a b i l i t i e s means t h a t a model f o r a s t e l l a r wind be i n the form of mean flow  0  associated  should fluctu-  7  CHAPTER 2.  OPTICAL OBSERVATIONS  For many years s e v e r a l o f the l i n e s i n the o p t i c a l spectrum of  several  e a r l y type  s t a r s have been reported  r e f e r e n c e s c i t e d i n the I n t r o d u c t i o n ) . been paid t o the s t a r Lambda Cephei, s t a r i n the northern  sky.  as varying (see  P a r t i c u l a r a t t e n t i o n has  because i t i s a b r i g h t  06f  The H ©cline has been reported to vary  on time s c a l e s of one day (Leep and C o n t i 1978) and l o n g e r , no  apparent  systematic  tion i s t y p i c a l l y fairly  10%  variation.  of  the  The amplitude of the v a r i a -  intensity.  This  behaviour  t y p i c a l of the more luminous mass l o s s s t a r s .  t e s t period v a r i a t i o n s with a high confidence  made by the s a t e l l i t e Copernicus o f  and  (York e t a l .  about 150 Km s observations  - 1  at o p t i c a l wavelengths  p l a t e s , which have a photometric  to  i t s character.  scanner o b s e r v a t i o n s  analysis  of  the  error.  In  on t h e  was l e s s  However the e g u i v a l e n t  than  fact  basis  cf a  expected  width of a l i n e  averages  a l l m a t e r i a l e m i t t i n g at t h a t l i n e freguency,,  but at the c o s t of l o n g e r exposure classical  Lacy  the  t i o n s which r e s o l v e the l i n e can provide much more  The  accuracy  e r r o r s present i n the e g u i v a l e n t  width t h a t any v a r i a b i l i t y present  together  been  of some of the l i n e s i n s t a r s  that were r e p o r t e d as v a r y i n g and concluded  randcm  have  t o r e v e a l the presence of the v a r i a t i o n , l e t alone  made  statistical  S o r i A, T j Q r i ,  spaced about 6 hours a p a r t .  r e v e a l much i n f o r m a t i o n as (1977)  UV  was seen to "move" i n the 0 VI l i n e between two  made with photographic able  the  1977), where a small f e a t u r e o f width  Host of the o b s e r v a t i o n s  barely  is  The shor-  l e v e l are  observations £ Pup  with  description  Observa-  information,  times. o f l i n e formation  in a stellar  8  wind was given by Beals  (1951).  The observed  l i n e can be c o n s i -  dered to be made up o f three almost independent  p a r t s ; an under-  l y i n g abscrpti.cn l i n e formed i n the photosphere  of the  star,  superposed  emission l i n e with i t s c e n t r o i d at zero v e l o c i t y  duced  emission  by  of  photons i n the s t e l l a r  s h i f t e d a b s o r p t i o n l i n e which i s formed i n the stellar  analysis  by the Sobolev  that  the  emission  wavelength by  pro-  wind, and a blue portion  of the  wind which i s s i l h o u e t t e d a g a i n s t the s t a r .  The fied  a  o f l i n e formation w i t h i n the wind i s s i m p l i approximation  (Sobolev  1960),  which  says  and a b s o r p t i o n of photons i n a given narrow  i n t e r v a l , o u t s i d e o f t h e doppler c o r e , i s  determined  the amount o f gas moving a t a v e l o c i t y such that t h e l i n e of  s i g h t v e l o c i t y of t h e gas f a l l s w i t h i n the wavelength  interval.  T h i s approximation i s v a l i d i f the gas speed i s s u p e r s o n i c . assumption and  i s supported by t h e o b s e r v a t i o n s of Hutchings (1976  r e f e r e n c e s t h e r e i n ) who has shown that the wind has a  c i t y exceeding the sound speed the s t e l l a r The  The  velo-  f o r d i s t a n c e s g r e a t e r than 10$ o f  radius.  observations  were  undertaken  t o c o n f i r m the r e p o r t e d  v a r i a b i l i t y and were to be made with s u f f i c i e n t l y high s i g n a l t c n o i s e , s p e c t r a l r e s o l u t i o n and time r e s o l u t i o n solve  the  thought anism  variations,  as they developed.  to  clearly r e -  In p a r t i c u l a r i t was  that there might be evidence f o r the nature of the mech-  of the v a r i a t i o n ,  f o r i n s t a n c e , a spot r o t a t i n g  with  the  s t a r cr a " b l o b " moving cut through the wind. fill cope The  of  o b s e r v a t i o n s were c a r r i e d out with t h e 1.2 meter t e l e s the Dominion A s t r o p h y s i c a l Observatory, V i c t o r i a , E.C.  2.4 meter camera i n the ccude spectrograph was used  with  a  9  red  coated  crons. 25.4  image s l i c e r  giving a projected s l i t  The spectrum was detected micron  diodes  pixel  size  with a 10 24 element  array  1S76).  combination  The image s l i c e r and  in  the  .125A/diode. 1977,  first  order  Observations  de-  were chosen to g i v e a p r o p e r l y  oversampled spectrum. , A l l o b s e r v a t i o n s were c e n t r e d on line  of  (a B e t i c o n RL1024/C17) cooled to a tempera-  t u r e o f -80° C (Walker e t a l . tector  width c f 6C mi-  resulting  in  a  the  Ho<  dispersion  of  were made i n September and October o f  and are t a b u l a t e d below and shown i n t h e accompanying  fig-  ures. #  1 2 3 4 5 6 7 8 9  TABLE 1: CATALOGUE OF OESIBVATIONS Expos Time Date PSI secon 1977 2250 22: 13 Sept 11/12 lambda Cep u •i ii 22:55 tt it « 23:41 ii ti ti 00:23 i i •i n 01:05 ti n ti 01:48 ti ii II 03:10 ii tt n 03:54 i i II » 04: 36  Star  Oct  10 11 12 13 14 15  it  16 17 18  Alpha  Cam  Oct Oct Oct  19 20 21 22  Delta O r i  Oct  ti  Oct  ti  fied  Oct Oct  ti  H •t  II  ti  tt  ii  II  •i  it  The in  n  tt it  12/13 16/17  23:00 01:08 22:15 21:05 23:38 04:28  3000 30 00 3000 3000 30 00 3000  11/12 12/13 16/17  00:27 01:35 22:58  1500 2002 1800  12/13  02:31 03:01 23:00 03:41  600 600 2641 2830  11/12  16/17  l i n e s present i n t h e 100 A r e g i o n examined are F i g u r e 1., They i n c l u d e the s t e l l a r  the i n t e r s t e l l a r 6614 A f e a t u r e ,  and  a  identi-  and He I I 6527,  multitude  of  narrow.  10  Fig  1: Lambda C e p h e i : The E f f e c t of  Resolution  11  weak, t e l l u r i c their tra.  water vapour l i n e s .  relative  The  telluric  water l i n e s  e q u i v a l e n t widths are i n d i c a t e d above the  T h i s data was  taken  from Moore et a l .  (1966), and  and  spec-  may  not  give the exact r e l a t i v e i n t e n s i t i e s f o r these o b s e r v a t i o n s . cf the s p e c t r a have been f i l t e r e d que  to U0% of the Nyquist  resolution  of  by a F o u r i e r t r a n s f o r m  frequency,  which i s roughly  true  the s p e c t r a , , A l l s p e c t r a had c o n s i d e r a b l e  (10$)  mostly  the window of the d e t e c t o r .  T h i s was  removed by  spectrum of a lamp which was  taken  observation.  For  the  to a l i g h t f r o s t  immediately  lamp  calibration.  d i v i d i n g by  before  cr  time s e r i e s s p e c t r a the  shape of the spectrum d i d not vary w i t h i n the the  techni-  the  response changes along the a r r a y , due  the  All  on the  after  underlyinq  error  (0. }%)  of  A l l s p e c t r a were r e c t i f i e d t o a l i n e a r  continuum, F i q u r e 1 shows the absolute n e c e s s i t y to r e s o l v e luric  water  lines.  As  the  tel-  the F i q u r e 2 time s e r i e s of Lamda Cep  over 6 hours shows, the water l i n e s vary s i g n i f i c a n t l y over hour.  In  F i g u r e 1 the top spectrum shows the mean of the time  s e r i e s of high r e s o l u t i o n s p e c t r a . recorded G.  A.  bottom with  at KPNO i n June, 1978 H.  Below  it  ( c o c r t e s y of G.  are G,  two  spectra  Fahlman  Salker) u s i n g a lower r e s o l u t i o n spectrograph.  spectrum  a Gaussian  solution  one  i n Figure 1, i s the top spectrum but to give approximately  as the KfHO s p e c t r a .  i n an H*, p r o f i l e can be e n t i r e l y variations,  if  the  instrumental  due  The  convolved  the same i n s t r u m e n t a l  I t i s evident t h a t the to  telluric  resolution  and  re-  variation  water  line  i s inadequate  to  c l e a r l y separate these v a r i a t i o n s out. The  time s e r i e s of Lambda Cephei ( s p e c t r a l type 06f)  shown  12  in  Figure  2 covers  6.5 hours.  The average of t h i s time s e r i e s  of o b s e r v a t i o n s i s shown at the t o p below  of  Figure  1.  The  lines  are the i n d i v i d u a l s p e c t r a d i v i d e d by the mean, then  malized. (say  Although  about  t h e r e are  suggestions  10 A) changes, these  cf  nor-  underlying  broad  the noise  level.  are l e s s than  In the day t c day observations shown i n Figure 3, there i s c l e a r evidence at  of a v a r i a t i o n  velocities  velocities The  near  Km s  _ l  feature  , and on the a b s o r p t i o n s i d e a t  time s e r i e s d i f f e r e n c e s p e c t r a , number 1 to 9 of F i g u r e t o determine the s t a t i s t i c a l s i g n i f i c a n c e  variations.  The r e p o r t by York e t a l .  of FwHM 150 Km s ~ slightly  200  emission  near -300 Km s~*.  2, can be analyzed any  at t h e H o t l i n e of the  wider  1  changing over  than  seme  (1977) of a f e a t u r e  a p e r i o d of  of the t e l l u r i c  6  hours  is  The standard  d e v i a t i o n o f the s p e c t r a i s i n the range of 0.6 t o 0.8% other  than f o r spectrum 9.  an  l e s s than  amplitude  exceeding  of the  Assuming the noise t o have a  normal d i s t r i b u t i o n with t h i s v a r i a n c e , have  only  water f e a t u r e s , and  l e a d s to some d i f f i c u l t l y i n i n t e r p r e t i n g changes.  mean,  of  the  2.57 standard  fluctuations  must  d e v i a t i o n s to have a  1% p r o b a b i l i t y of chance occurence.  T h i s amplitude i s  i n d i c a t e d i n F i g u r e 2., The smoothed s e r i e s o f p l o t s i n F i g u r e 2 are the same s p e c t r a as those on the l e f t but averaged diodes.  This  root o f 11.  reduces  the  variance by a f a c t o r of the square  I t can be seen  f e a t u r e s which do vary s i g n i f i c a n t l y .  except  11  The l i n e s f o r a s t a t i s t i c a l s i g n i f i c a n c e c f 99% a r e  again drawn on the p l o t .  varying  over  a l l correspond  to  that  there  are  many  But the f e a t u r e s t h a t are  t h e wavelengths of t e l l u r i c  lines,  f o r the f e a t u r e a t a v e l o c i t y with r e s p e c t t o t h e Hot, l i n e  13  of +200 Km s ficance  (left  1  dotted l i n e ) .  T h i s exceeds the 99%  l e v e l i n records 1,2,3,6,7, and 8, going from  to a d e f i c i e n c y with r e s p e c t t o t h e mean. in the l i n e The  an excess  The v a r i a t i o n  occurs  (see Figure 1) near the top of the emission f e a t u r e .  s u b t r a c t i o n would be very s e n s i t i v e t o very s m a l l s h i f t s of  the l i n e i n t h i s r e g i o n .  There a r e two reasons  t h i s f e a t u r e may not be s t e l l a r i n o r i g i n . correlates  very  respect t o H ond,  signi-  well  with  of -700 Km s ~  to  think  that  F i r s t , i t s variation  the water l i n e s at a v e l o c i t y (dotted l i n e on r i g h t ) .  1  And  with sec-  the f e a t u r e shows no v e l o c i t y s h i f t over t h i s time p e r i o d ,  which might be expected tions  are  i n a wind.  I conclude t h a t r e a l  present i n t h e time s e r i e s , but they a r e most  varialikely  due t o t e l l u r i c f e a t u r e s . Alpha Cam ( s p e c t r a l type 09.51a) was chosen because c f spec t r a l type, and the presence  of the emission l i n e a t H$(. Of a l l  the s t a r s examined i t seems to have the most s i g n i f i c a n t  varia-  t i o n s , see Figure 4. Delta  Orionis  ( s p e c t r a l type 09.511) was observed  of the v a r i a t i o n r e p o r t e d by York, et a l . , (1977).  because  Observations  made within one n i g h t , Oct 16/17, have no r e a l i n d i c a t i o n change.  There  is  only  weak evidence f o r a p r o f i l e change i n  f i v e days, because of the c o n f u s i o n strength  of  the  telluric lines.  binary of p e r i o d of 5 days,  created  by  the  This star i s a spectroscopic  Sobclev approximation  but  T h i s i s shown i n F i g u r e 5. a l l o w s an estimate o f t h e s i z e o f  the r e g i o n producing the photons i n a given wavelength Although  different  which produces v e l o c i t y s h i f t s ,  probably not p r o f i l e changes. The  cf a  interval.  the t h i c k n e s s o f the s h e l l s o f equal l i n e o f s i g h t ve-  1000  JiJ_JJ.j_ 0  4? I  I  A  -1000 '  I I  D i f f e r e n c e S p e c t r a Smoothed over 11  points  15  1000  500  0  -500  -1000 Km s '  10 Oct 11/12 23:00  11 Oct 11/12 01:08  12  Oct 12/13  '\Jty&l  21:05  14 Oct 16/17 23:38  15 Oct 16/17 04:28  fiq.  ytfl 1 ni 3: Lambda Cephei day t c day  16  Fig.  4: Alpha Cam day to day  17  Fig.  5: D e l t a O r i day t o  day  18  locity  will  vary  with the  velocity  and  the c o n t r i b u t i o n to the p r o f i l e w i l l of  higher  density.  length i n t e r v a l , shell  will  To e s t i m a t e  be w e i g h t e d  , a l l integrals  of  The  area  constant  and  the  justified  emission  then  1.  total  by t h e  of  stellar  The  the radii,  These a s s u m p t i o n s  frcm  the  The  these  average s h e l l t h i c k n e s s , As,  line  2  $  i s the angle  by  approximations  will  c f s i g h t v e l o c i t y g r a d i e n t , dv/ds  + s i n ^ v / r , where  are  p r o f i l e s a s done  a s a f a c t o r o f f i v e , b u t d e p e n d s on t h e  considered.  the  constant  may  large  2z$.  of  (1978).  as  e r r o r of  true  Cassinelli, et a l . be  The  volume 2  thickness  a c t u a l c a l c u l a t i o n s of l i n e :  has  4trr^ Asf,  i s approximately  dominant e m i s s i o n .  the  I f the s h e l l the  t h e assumed d i s k o f two  f w i l l be assumed t o be  region  regions  over  d i f f e r e n c e between  constant l i n e of sight v e l o c i t y s h e l l s over  star,  towards  the wavelength i n t e r v a l i s approximately  where f i s a f a c t o r c o n t a i n i n g t h e emitting  the  be r e p l a c e d by a v e r a g e q u a n t i t i e s .  in  the  t h e t o t a l i n t e n s i t y i n a wave-  an a v e r a g e l i n e o f s i g h t t h i c k n e s s . As, emitting  d i s t a n c e from  be  line  estimated  (=cos # dv/dr 2  between t h e l i n e o f s i g h t  and  t h e s t a r ) as  (D  A  fluctuation  be o b s e r v e d integrating  i n t h e wind which changes the e m i s s i o n  a s some f r a c t i o n a l c h a n g e o f t h e over  tion  r a t e c h a n g e s by  s e c t i o n an e s t i m a t e w i l l  will by  range.  100%,  i n the  follow-  be made o f t h e s i z e  of t h e  fluctua-  causing a given f r a c t i o n a l  tuation  obtained  a l l r e g i o n s e m i t t i n g i n that wavelength  Assuming t h a t t h e e m i s s i o n ing  flux  rate  intensity  change.  i s a r e g i o n o f s i z e jf, c n l y t h a t p a r t of t h e  I f the  fluc-  fluctuation  19  which i s moving at an a p p r o p r i a t e v e l o c i t y sity  in  the  wavelength  to e f f e c t the  inten-  i n t e r v a l c o n t r i b u t e s t o the i n t e n s i t y  change. I f the f l u c t u a t i o n internal the  velocity  i s moving a t a uniform v e l o c i t y  dispersion  with  l e s s than the thermal speed,  f r a c t i o n a l change i n i n t e n s i t y i n  one  wavelength  an then  interval  would be  (2) Xn  where  velocity, units  the s i z e o f t h e f l u c t u a t i o n  i n units  of  which  is  10  1 2  of 1 0  cm,  cm, r ^ i s the s i z e o f  1 1  A> the s p e c t r a l  is  For  a fluctuation  same  wavelength cubed.  as  the  1 2  in  i n u n i t s of 0.3A,  A typical  which has a v e l o c i t y  gradient  velocity —  1  which  is  wind, the minimum volume e m i t t i n g i n a given  i n t e r v a l would be j u s t the v e l o c i t y  more  star  cm.  In t h i s case the i n t e n s i t y  A  used.  the  found by t a k i n g a t e r m i n a l v e l o c i t y o f 1000 Km s  reached over a d i s t a n c e o f 1 0  the  resolution  was the spectrograph r e s o l u t i o n  gradient  moving with a ccamcn  realistic  shell  thickness  fluctuation i s  situation  s i z e X has cnly  a t h i n slab  priate velocity  to be i n the d e s i r e d  might  be i f a f l u c t u a t i o n of  of t h i c k n e s s As moving at the approwavelength  interval.  In  t h i s case A I  >  -  **** Htr  AS  -  8*-°- A  2  r -*. n  (4)  20  X Cep time s e r i e s r e s t r i c t s the magnitude of an i n t e n -  The  sity fluctuation Equation  to  less  than  2%  over  the  s i x hour  span.  4 then l i m i t s the s i z e c f the l a r g e s t region to change  i n t h i s time to 5 x 1 0  11  cm.  These o b s e r v a t i o n s have confirmed the  variability  of  the  s t e l l a r Hoc l i n e p r o f i l e over times l o n g e r than one day, and conclusively  show  that  the v a r i a t i o n i s due t o the change i n the  profile,  net changing t e l l u r i c  tensity  change  in  lines.  any one p i x e l  The amplitude of the i n -  i s only s l i g h t l y g r e a t e r than  what might be due to noise, but c o n s i d e r i n g that groups o f than  10  pixels  show the same change gives c o n s i d e r a b l e c o n f i -  dence t o the p h y s i c a l r e a l i t y series  of  of  the  change.  one  The s i g n a l t o noise  the time s e r i e s s p e c t r a i s only about 50, which  system.  The  > Cep has no c o n v i n c i n g evidence f o r any short  v a r i a t i o n , or e v o l u t i o n o f t h e p r o f i l e .  imposed  more  time term in  was a c o n s t r a i n t  on the maximum i n t e g r a t i o n time by the d e t e c t o r c e d i n g  21  CHAPTER 3.  SUPERSONIC ACCRETION  Optical observations of v a r i a b i l i t y e n t i r e volume of emission fluctuations  in  the  are averages  a t that p a r t i c u l a r  wind  ever  wavelength.  the  I f the  c o n t a i n components cn a s m a l l s c a l e  compared t c the s c a l e o f the wind, the d e t e c t i o n of f l u c t u a t i o n s by way of technigues are  limited  X-ray  binaries  imbedded  in  Cen X-3 and 30170G-37 (=HD153919) allows  of using the X-ray source  stellar  of  a  The  winds,  the p o s s i b i l i t y  as a probe of the s t e l l a r wind.  the X-ray l u m i n o s i t y i s d i r e c t l y r e l a t e d t o the r a t e tion  observed,  by the s i g n a l t o n o i s e which can be a c q u i r e d .  discovery of two namely  i n which the i n t e g r a t e d l i g h t i s  Since  of  accre-  s m a l l f r a c t i o n o f the s t e l l a r wind onto the neutron  s t a r , the i n t e n s i t y of the X-ray source can be used with the a i d of a s u f f i c i e n t l y  d e t a i l e d understanding  cess  estimates  to  wind.  derive  of  of  the  paper  which  support  accretion  pro-  the d e n s i t y and v e l o c i t y i n t h e  T h i s was t h e s u b j e c t o f the  been attached as Appendix 1.  of the  published  paper  which  A summary o f the p r i n c i p l e  has  results  the c o n c l u s i o n s o f t h i s t h e s i s i s  given below. A schematic drawing c f the supersonic shown  in  Fiqure  the f i g u r e . with  a c c r e t i o n process  is  5 and the r e g i o n s r e f e r r e d to are numbered i n  The incoming gas, region  1, i s moving at a speed  r e s p e c t to the neutron s t a r o f mass M.  V  The s t r e a m l i n e s are  bent i n by the neutron star»s g r a v i t a t i o n a l f i e l d .  The mass and  the v e l o c i t y d e f i n e the a c c r e t i o n r a d i u s , R =2GM/V , which gives Z  A  (apart  from  an  efficiency  cross s e c t i o n f o r a c c r e t i o n strikes  a  shock  cone  factor of  trailing  which i s c l o s e to one) the  material. the  The  incoming  gas  neutron s t a r , c a l l e d t h e  Fiq.  6:  Supersonic  accretion  schematic.  23  sheath, r e g i o n  2.  3x10* ( r / 1 0 c m ) - * l4  gas c o o l s .  The gas i s shock heated to a K,  If  the density  temperature  of  i s s u f f i c i e n t l y high  the  In the sheath the gas l o s e s i t s component of  ty away from the neutron s t a r , j o i n s the starts  falling  the X-ray ture  down, r e g i o n  luminosity  may  3.  r e g i o n 4,  ters  the  which  the dynamics o f the flow are  magnetosphere  of  the  Eventually  a short  X-rays.  The  encoun-  by  the  magnetic  d i s t a n c e above the  6, where the k i n e t i c energy  of i n f a l l i s converted i n t o thermal energy as  tempera-  the flow  regulated  of the neutron s t a r , r e g i o n  away  the  neutron s t a r , region 5, below  The gas s t r i k e s another shock  diated  and  The column w i l l then expand cut t c an  almost s p h e r i c a l i n f l o w ,  surface  column  Wear t h e a c c r e t i n g neutron s t a r  be l a r g e enough t o r a i s e  by Compton heating,  field.  accretion  veloci-  which  t a b l e below gives  is  mostly  ra-  length and  time  s c a l e s c h a r a c t e r i s t i c o f the d i f f e r e n t r e g i o n s .  TABLE 2: SCALES IH SUPEBSONIC ACCBETION region  size scale  time s c a l e  star  10  1 day  a c c r e t i o n column  10*0-11 m  magnetosphere  10  neutron s t a r  10  cm  1 2  500  C  cm  8 — 9  6  seconds  1 second  cm  1 millisecond  An a n a l y s i s c f t h i s model y i e l d s s e v e r a l are  quantities  which  d i r e c t l y r e l a t e d to the major parameters o f i n t e r e s t i n the  stellar  wind, the s t e l l a r wind d e n s i t y , n , and  luminosity  of the unobscured  velocity.  source i s  L =4.7x103* n ,'v -3(fl/fl0)3(H /1O« ci)-»|S erg s~», 1  ft  The  l(  where p i s a f a c t o r u s u a l l y o f order one  giving  the  (5)  efficiency  24  of the a c c r e t i o n , n = n / ( 1 0 n  The  angle  that  1 1  6  c n r ) , and Vg=V/10 3  t h e shock  8  cm s ~ . l  cone makes with the a x i s of the  accretion cclunn i s 0=2.7°  <T |/106 K) V -2 CO  .  s  The temperature  i n the column, T  but  l i m i t can be o b t a i n e d by c o n s i d e r i n g the h e a t i n g  an  upper  C o  i,  (6)  i s not d i r e c t l y o b s e r v a b l e ,  and c o o l i n g p r o c e s s e s , (T |/106 K) < 1.9 n „ V * s to  This  V */»s. 0  can be used t o estimate the o p t i c a l depth  the a c c r e t i o n column, "Ccol , due t o e l e c t r o n TcoJ>2.2 n , , - * / The  ls  (  7)  up the c e n t r e of  scattering (8)  VQSZ/IS.  e l e c t r o n s c a t t e r i n g o p t i c a l depth up the sheath i s l e s s than  that  up  the column i f n V - < 3 . 5 , which 2  u  8  estimate of the column temperature.  i s independent  With these simple  relations  i n hand and some X-ray data o f an o b j e c t t h a t i s c l e a r l y by  a  stellar  wind  of the  fueled  i t i s p o s s i b l e t o confirm the model of the  a c c r e t i o n process o u t l i n e d above. , More i m p o r t a n t l y the observat i o n s can be used t o d e r i v e the  density  and  velocity  i n the  wind. Two f a i r l y good s e t s o f p u b l i s h e d data e x i s t f o r the source Cen a  X-3, which appears spherical  to be the c l e a r e s t case o f a c c r e t i o n  s u p e r s o n i c wind.  The source 3U1700-37  would appear t o be a very s t r o n g s t e l l a r wind optical though with  spectrum  <=HD153919) from i t s  (see Fahlman, C a r l b e r g , and Walker 1977)  t h e r e are s i g n i f i c a n t the  source  from  period  e f f e c t s i n t h e spectrum  of the neutron s t a r o r b i t i n g  al-  associated  the 06f primary.  be  These e f f e c t s may^due to a wake o f d i s t u r b e d neutrcn s t a r  (see Appendix  1).  gas  trailing  the  Or they may r e p r e s e n t a s i g n i f i -  25  cant d i s t o r t i o n of the s t e l l a r wind i t s e l f . ray  data  from  3U1700-37  has a lower count r a t e than Cen  hence g r e a t e r s t a t i s t i c a l e r r o r s . situation  is  In any case, the X-  In Cen X-3  the  X-3,  observational  almost the exact r e v e r s e ; the X-ray source i s one  of the b r i g h t e r sources i n the sky, but i t s o p t i c a l companion i s a 14th magnitude OB s t a r which has been  poorly  studied  (Conti  1S78) . The X-ray data f o r Cen X-3 shows s e v e r a l s c a l e s of v a r i a b i l i t y ; a 4.8 the  second p u l s a t i o n p e r i o d , a s c r i b e d to the r o t a t i o n of  neutron  period  star  and  sometimes  aperiodic  superposed  change  A  made by Pounds, et a l . every  state. due two  to  orbit  Jackson  magnetic with  f i e l d , a 2.1  "anomalous  day  dips",  orbital and  an  i n the mean i n t e n s i t y l e v e l with a time s c a l e  of order one month.  ring  its  particularly (1S75), who  during  a  exciting  observation  was  observed r e g u l a r d i p s o c c u r -  t r a n s i t i o n from X-ray low to high  (1975) proposed t h a t the two  d i s t i n c t dips  were  the r e d u c t i o n of the r e c e i v e d f l u x by s c a t t e r i n g i n the  s i d e s of the sheath of t h e a c c r e t i o n column.  He  deduced  a  v e l o c i t y of the wind with r e s p e c t to the neutron s t a r of between 375  and 620 km s  tion  (6) and the  column al.  - 1  , and a column semi-angle  velocities  temperature  guoted  i s i n the range  by  of 20°.  Jackson,  3,5-9. 6x10  (1S76) estimate the d e n s i t y i n the wind as  s  K.  1.5x10  6  K  from  Eguation  (7),  the  implied  S c h r e i e r j§t cm . -3  temperature  A c c e p t i n g Jackson's p r o p o s a l  that the double d i p s are due to s c a t t e r i n g i n using  the  1-5x10**  These two e s t i m a t e s are c o n s i s t e n t with the l i m i t i n g of  From egua-  the  sheath,  but  theory developed i n Appendix 1, more i n f o r m a t i c r can  be d e r i v e d from the o b s e r v a t i o n s .  Pounds, et a l .  (1S15)  note  26  that  the  relative  turns on.  depths  of  the d i p s decrease as the  Also, from i n s p e c t i o n of t h e i r published  see t h a t the d i p s appear to become s i n g l e as on.  A  Figure  schematic  tracing  variations  proposed  source  plotted,  which  turns  the X-ray i n t e n s i t y i s shewn i n  and  the  estimate cm- ,  s t a t e and  of S c h r e i e r gt  and  3  wind  B, but not  density  1  velocity  state.  car  cm~ ,  line.  the deep d i p s .  data  shows that  below the  the  ^,= 1 l i n e .  up  as  the  d e n s i t y and  be c l o s e to one  dips  have  a  in Tc=1  decreasing  v e l o c i t y must be dropping  v e l o c i t y are moving f u r t h e r  E v e n t u a l l y the d i p s become s i n g l e  sity  v e l o c i t y v a r i a t i o n i s such t h a t the a c c r e t i o n r a t e , and  going  from  low  to high s t a t e .  high s t a t e , point B, locity The  with  luminosity  respect  5  only  increase  combined den-  slightly  while  The source s e t t l e s down at  with a d e n s i t y of to  The  the  and  hence the i n t r i n s i c  TT > f c l i n e .  as  density and  v e l o c i t y c r o s s the  the  This puts point A near the  f r a c t i o n a l depth, which means that the that  1  a l l o w i n g the source to become v i -  As the wind d e n s i t y drops  enough  the  f i x e s the d e n s i t i e s  3  exceeds the column o p t i c a l depth, and  order t o provide  be  Adopting  s i b l e , the v e l o c i t y must be such that the o p t i c a l depth sheath  term  s t a t e d e n s i t y as 5x10*  the v e l o c i t y . T h e  decreases  long  At point A the source i s  f o r the low 10*  the  A rough t r a j e c t o r y of  a t B the high  al.  cf  wind d e n s i t y and  high s t a t e d e n s i t y of  at p o i n t A and  model  i s shown i n Figure 8.  i n the X-ray lew  fast  can  of  by S c h r e i e r , e t a l ,  v a r i a t i o n of the s t e l l a r  the  data one  7.  From these o b s e r v a t i o n s  the  the  source  10  11  cm  -3  and  a wind  the neutron s t a r of about 500  c p t i c a l depth up the column i s so s m a l l that  dips  the ve-  Km  s~ .  are  not  l  28  Fig.  8:  Density  Velocity  Variation  o f Cen  X-3  29  seen  regularly.  When  the  source s t a r t s to t u r n o f f the  suggests t h a t a d i f f e r e n t d e n s i t y v e l o c i t y lowed,  such  that  the  always remains s m a l l . it  optical  depth up  trajectory  is  the column and  folsheath  As the s o u r c e approachs the p o i n t A  i s obscured by the i n c r e a s i n g d e n s i t y of the s t e l l a r  data  again  wind.  I f the t r a j e c t o r y o f the v a r i a t i o n of the wind v e l o c i t y and d e n s i t y i s s c h e m a t i c a l l y c o r r e c t i t i s p o s s i b l e to draw c l u s i o n as t o the d r i v i n g f o r c e f o r the Figure  8  suggests  the wind d e n s i t y up  trajectory in  which would imply  the a c c e l e r a t i o n of  acceleration  of  Hearn  In the r a d i a t i v e l y d r i v e n the wind v a r i e s as n  l e s s than one.,  a c c e l e r a t i o n should (1975)  suggests  that  The  wind  the  i n Chapter 5, one  sent  the  is  thermal  which v a r i e s as n . - 2  the  other  wind i s i n i t i a l l y  acce-  corona would heated by shock waves  of the dominant  instability  enigua.  one  appears  e x p l a i n i t , so i t may (Cannon and The  may  provide  the  of  pre-  correla-  state v a r i a b i l i t y  i s an  be no n a t u r a l s c a l e i n the wind t o  be connected t o the subatmosphere  Thomas 1977  observations  dence that there  to  instabilities  be  density.  month s c a l e cf the high low  There  As w i l l  which grows cn a time s c a l e  This i n s t a b i l i t y  t i o n between wind v e l o c i t y and The  CAK  radiation  On  which grow from i n s t a b i l i t i e s w i t h i n the atmosphere. discussed  density  i s a con-  the  increases.  gas  of  , where  -  This implies that  drop as the density  l e r a t e d i n a hot corona.  star  the  to the l o c a t i o n o f the neutron s t a r i n c r e a s e s as the  stant s l i g h t l y  hand  The  con-  a c o r r e l a t i o n between the wind v e l o c i t y and  i n the wind i n c r e a s e s . the  wind.  a  and  the  (1976) c o n t a i n  evi-  Thomas 1973).  S c h r e i e r et a l .  are s m a l l s c a l e  of  (cf order  10  11  cm)  fluctuations  30  i n the wind. greater  The count r a t e c l e a r l y v a r i e s  that  the s t a t i s t i c a l  an  amplitude  e r r o r on time s c a l e s c f about one  hour.  T h i s time s c a l e , which has been  orbit  and  pointing  with  set  mode, i s much longer  by  the  s p a c e c r a f t e^*^  than the n a t u r a l  res-  ponse time of the a c c r e t i o n p r o c e s s , which i s about ten minutes, from Table 2. with  a  I t would be extremely i n t e r e s t i n g  to  have  data  time r e s o l u t i o n of a few minutes to see i f the f l u c t u a -  t i o n s i n the wind become time  resolved.,  In summary the t h e o r y of supersonic  a c c r e t i o n that was  de-  veloped and a p p l i e d t o a l i m i t e d amount o f data on Cen X-3 shows that and tc  there  i s a p o s i t i v e c o r r e l a t i o n between the wind  vind v e l o c i t y during high state..,•  a period  of t r a n s i t i o n from  density  X-ray  low  31  CHAPTER 4. The  stability  PHYSICAL DESCRIPTION OF THE of  the  Hind  linearized s t a b i l i t y analysis. prevailing  physical  will  The  conditions  be i n v e s t i g a t e d with a  analysis requires  be s p e c i f i e d .  devoted t c the d e r i v a t i o n of the r e g u i r e d ficult  p h y s i c a l q u a n t i t i e s are  of the gas of  and  GAS  that  T h i s chapter i s  quantities.  those d e s c r i b i n g  The  l e s s , and  the  t h i s i n t e r a c t i o n i s probably the  dif-  the i n t e r a c t i o n  the s t e l l a r r a d i a t i o n f i e l d , which are  energy gain and  the  rate  radiation acceleration. key  to  the  the  stellar  Since  wind,  an  accurate p h y s i c a l d e s c r i p t i o n must be used. In  order to d e r i v e  the c o o l i n g r a t e , heating  r a t e , and  d i a t i o n f o r c e i t i s necessary t o know the d i s t r i b u t i o n cf over  the  various  stages c f i o n i z a t i o n , and  t i o n and  emission of r a d i a t i o n by the i o n s .  reguire  atomic  constants  which then become f u n c t i o n s and  radiation The  t c describe of the  These  calculations  the r a d i a t i o n p r o c e s s e s ,  local  density,  temperature,  field.  radiation  knowing  f i e l d i s i n f l u e n c e d by the  the  d e t a i l s of the flow.  have taken the unattenuated, but radiation  flow of the field  As an  geometrically  would  re-  diluted  stellar  f i e l d . .. T h i s o f f e r s the advantage of r e t a i n i n g a com-  tive transfer.  The  overlapping  effect i s likely gas  the  radia-  approximation of the unattenuated f i e l d  have the e f f e c t of somewhat over e s t i m a t i n g  The  gas,  approximation I  p l e t e l y l o c a l a n a l y s i s at the cost of o v e r s i m p l i f y i n g  because  atcms  the r a t e of absorp-  so t h a t a good approximation to the r a d i a t i o n quire  ra-  l i n e s are ignored.  As d i s c u s s e d  to be at most a f a c t o r of  i s assumed t c be  the r a d i a t i o n  in ionization  will force  later  the  two. equilibrium,  which  32  is  valid  for  time  s c a l e , 30 (T/10* K) / 1  the  rate  scales nn  2  l o n g e r than the recombination seconds.  - 1  Equilibrium  implies  j i s determined  For element  i the r a t e out of i o n i z a t i o n  i o n i z a t i o n s t a t e i s determined i o n i z a t i o n s from below.  n c,j-i ( e n  where  j-1  c  +  The r a t e i n t o  by recombinations from above  Algebraically,  "5 hJ-i > *  1  n  n  e °S  j#  rate  rt£j{ng,T) i s the recombination rate from  level  These i o n i z a t i o n balance equations were solved cf  data was sumed  j  i s the c o l l i s i c n a l i o n i z a t i o n rate out of  in^,T)  ^ CJ i s the p h o t o i o n i z a t i o n  atoms  state  by the r a t e o f i o n i z a t i o n t o the next higher i o n  and the recombination r a t e t o the next lower i o n .  and  that  of t r a n s i t i o n s out of an i o n i z a t i o n s t a t e i s balanced  by the r a t e i n .  the  time  significant available.  stellar  j to j - 1 .  f o r as  abundance f o r which gccd  The elements  many atomic  used are shown with t h e i r  as-  abundances i n the accompanying t a b l e . , I t would have been  d e s i r a b l e t o have i n c l u d e d high cosmic  abundance and great  r e l i a b l e and c o n s i s t e n t could be  N i c k e l and  Iron  with  their  fairly  number of s p e c t r a l l i n e s , but  s e t of data f o r a wide temperature  found.  TABLE 3: ATOMIC ABUNDANCES ELEMENT  Z  ABUNDANCE  Hydrogen  1  1.0  Helium  2  8.5x10-2  Carbon  6  3.3x10-*  Nitrogen  7  9.1x10-5  no  range  33  Oxygen  8  6.6x10-*  Neon  10  8. 3x 10-s  Magnesium  12  2.6x10-s  Silicon  14  3.3x10-s  Sulfur  16  1.6x10-5  These abundances were taken from A l i e n Standard  r a t e s were used  s e c t i o n s , recombination But and  since  the  (1973).  f o r a l l the p h o t o i o n i z a t i o n c r o s s  r a t e s , and c o l l i s i o n a l i o n z a t i o n  gas has a f a i r l y  high density (order 10* c a 1  i s i n an i n t e n s e r a d i a t i o n f i e l d i t  some c o r r e c t i o n s . corrections  rates.  i s necessary  to  - 3  )  make  The d e n s i t y e f f e c t s are allowed f o r by adding  t o the recombination  t i o n s , and recombination  r a t e f o r three body recombinasmall  correction  f o r i o n i z a t i o n o u t o f upper l e v e l s i s a l s o i n c l u d e d .  The g r e a t -  est  difficulty  field  to upper l e v e l s .  A  i s a l l o w i n g f o r the e f f e c t of both the r a d i a t i o n  and the d e n s i t y e f f e c t s on the d i e l e c t r o n i c  rate.  This  process  depends  upcn c a p t u r e s to l e v e l s of l a r g e  guantum number, and i t i s p o s s i b l e reionized levels.  before  recombination  that  these  levels  may  be  they can s t a b i l i z e by cascading down t o lcwer  These e f f e c t s have been c r u d e l y allowed  f o r by c a l c u l a -  t i n g a m u l t i p l i c a t i v e c o r r e c t i o n f a c t o r , based on a f i t to the guantum  mechanical  calculations  c f Summers (1574),  A l l these  r a t e s and c o r r e c t i o n s are d i s c u s s e d i n Appendix 2.  The  Ionization The  simple  Balance  s o l u t i o n t o the i o n i z a t i o n balance since  the  lowest  l e v e l , and then the second  equations  is  very  l e v e l only i n t e r a c t s with the second level i s  linked  to  the  first  and  34  third,  and  ionization  sc on., T h i s g i v e s the r a t i o of the p o p u l a t i o n  state to the population  normalization nonlinear usually  or  three  iterations  ionization  X ij  fraction  over j i s u n i t y . product  Figure in  for ion  and  j of  in  terms  t  of  atom  of the s t a r ,  the density  10  1 1  In cm , -3  for a  I t i s found t h a t f o r the range of d e n s i -  and r a d i a t i o n f i e l d  recombination  effects i s significant  tends t o s h i f t the i o n i z a t i o n s l i g h t l y t o h i g h e r At very  1.  i s shown  of i n t e r e s t the r e d u c t i o n of the d i e l e c t r o n i c  ionization.  summed  t  9 the i o n i z a t i o n balance f o r a gas of d e n s i t y  by  o f the  x*y  atom i , where  X t j - A ^ n , where A ' i s t h e abundance  range of temperatures.  rate  given  To get the number of atoms of type i , j we take  the u n d i l u t e d r a d i a t i o n f i e l d  ties  but  s u f f i c e s f o r an accuracy o f  The r e s u l t s a r e  6  A  are weakly  dependence on t h e e l e c t r o n d e n s i t y ,  about 1 p a r t i n 1 0 .  the  level.  completes the s o l u t i o n . , The equations  through t h e i r two  i n t h e next lower  ina  stages  high d e n s i t i e s the d i s t r i b u t i o n  of  approaches  to the . d i s t r i b u t i o n expected f o r ITE. The  heating  and c o o l i n g r a t e f o r a gas of d e n s i t y  in a u n d i l u t e d r a d i a t i o n f i e l d  are  p l o t t e d q u a n t i t i e s a r e the c e d i n g respectively.  The  plotted  shown  in  Figure  10-* c m 1  - 3  10. ,  The  and heating r a t e s , - A and T ,  g u a n t i t i e s are t o be m u l t i p l i e d by  the d e n s i t y squared t o o b t a i n the r a t e s per c m . - 3  l i z e d c o o l i n g r a t e i s taken as Jt* =n { A - r ). 2  The  genera-  The q u a n t i t y  *€/n  2  is plotted. The  r a d i a t i v e e q u i l i b r i u m between the p h o t o i o n i z a t i o n cea-  t i n q and r a d i a t i v e l o s s e s holds for zero  d e n s i t i e s around  10* c m . 4  at temperatures of about 2x10* K  -3  ;  T h i s i s shown i n F i q u r e  v e l o c i t y o f the qas with r e s p e c t  to the s t a r .  10 f o r  Of i n t e r e s t  Sulfur  Silicon  Magnesium  He on  Oxygen  Nitrogen  Carbon  at Helium , — i — , — — ~ - j—,  6  5 Fiq.  9  .—• »  1  7  : Ionization  Log T Balance  36  o  Fig.  10: Heating and C e d i n g  Bates f o r Solar  Abundances  37  to the "warm r a d i a t i o n a c c e l e r a t i o n " the a  l o s s r a t e i n an o p t i c a l l y temperature  would  r e g u i r i n g an immense source.  model i s that near 2x10  t h i n medium i s at a maximum.  be very d i f f i c u l t input  of  Between 10* and 10  7  s  K  Such  t o maintain i n the gas,  energy  from  seme  other  heat  K the l o s s r a t e drops t o a minimum  where the r a d i a t i v e l o s s e s would be more e a s i l y  balanced.  The  r a d i a t i o n l o s s e s from such a hot gas would c o n s i s t l a r g e l y o f Xrays,  which would be s u i t a b l e f o r producing the 0 VI i o n , as has  been suggested by C a s s i n e l l i and Olson The  gas i s t h e r m a l l y unstable  (1978).  (see F i e l d  1965) t o both i s o -  c h o r i c and i s o b a r i c d i s t u r b a n c e s when the temperature of the g e n e r a l i z e d The  cooling  temperature gradient  d e r i v a t i v e of the c o o l i n g to  admit  isentropic  r a t e at constant density i s not quite  that  i f there  instability,  i s a slight  appear  Calculations  in  steep enough  (logarithmic  and  wherein o r d i n a r y lose  inaccuracy  this isentropic i n s t a b i l i t y  would  i s negative.  r a t e l e s s than about -3) at any  gain energy i n the r a r e f a c t i o n s Even  derivative  sound waves  compressions.  i n the c a l c u l a t i o n s  condition  a very narrowly defined  by Raymond et a l .  i t in  pcint  could  such  be  met, i t  temperature  interval.  (1S78) i n d i c a t e  that  with  the  i n c l u s i o n of the i r o n group elements the slope becomes even  less  steep, and the gas i s f u r t h e r away from i s e n t r o p i c The  loss  to the s t a b i l i t y plotted  r a t e and i t s d e r i v a t i v e t u r n s out t o be c r i t i c a l c f an a c c e l e r a t i n g  atmosphere, so i t  has  been  i t f o r the CNQ elements enhanced by a f a c t o r of 10 i n  Figure 11. coding  instability.  O b v i o u s l y the abundance has a strong e f f e c t  rate,  since  the  c o d i n g i n the range 10  s  CNO elements are r e s p o n s i b l e  to 10* K.  on  the  f o r the  39  The longer  s t e l l a r wind i s u s u a l l y o p t i c a l l y t h i n at  o p t i c a l and  wavelengths f o r continuum e m i s s i o n , but can become o p t i -  c a l l y t h i c k i n the resonance l i n e s , which provide the l i n e ling  as  w e l l as most of the r a d i a t i o n a c c e l e r a t i o n .  a l t e r a t i o n t o the l o s s rate of Hybicki reduced  cooling  r a t e i s shown i n Figure  d i e n t of dv/dz=10 .  density, s  - 1  a  - 3  s~ ) w i l l s t i l l l  s i n c e the l o s s e s  times the d e n s i t y very  Hummer  Using the (1978) the  12 f o r a v e l o c i t y  Note that the s p e c i f i c c o o l i n g r a t e  -3  of e r g c m  and  increase  coo-  gra-  (units  approximately l i n e a r l y  with  vary with the c o o l i n g r a t e i n erg cm+  squared, over t h e o p t i c a l depth., T h i s  rough c a l c u l a t i o n , s i n c e  3  is  no allowance has been made f o r  the change o f t h e l o c a l i n t e n s i t y due  to  the o p t i c a l l y  thick  lines.  Radiation The  Force r a d i a t i o n f o r c e i s defined as  ij where ffi = J A-iiii , ions. an  C and  mt  J  (9)  i s the atomic weight o f the v a r i o u s  I f the unattenuated r a d i a t i o n f i e l d  upper  l i m i t to t h e r a d i a t i o n f o r c e .  mate i s s u p p l i e d  i s used  i t provides  A more r e a l i s t i c  by the method used by Castor,  esti-  Abbott, and K l e i n  (1975) , which i s based on an a n a l y s i s o f the r a d i a t i v e  transfer  in  one  With t h e  aid  c f the Sobolev approximation the problem can be  it  s p e c t r a l l i n e o r i g i n a l l y done by Lucy (1971).  solved  and  i s found that the f o r c e due to l i n e s i s ^ rad  -  ^ rad  ——*  (10)  Fig.  12:  Cooling  with o p t i c a l l y  thick  lines.  41  where  T =  T t e / ( m c ) f ; J (1) h X ; J nc[ 1/2 2  L  (I*/* ) 2  {dv/dz-v/r) +v/r ]-» and  Tfe /(mc)=. 02654 2  9 "rad i  acceleration  s  f / j ( / ) i s the o s c i l l a t o r  i n an o p t i c a l l y strength  for  t h i n gas  line  1  of  atom i i o n i z a t i o n s t a t e j , c i s the speed of l i g h t JJL i s c o s i n e  of the angle subtended by the s t e l l a r  radius  from t h e p o i n t  In a d d i t i o n t o t h e f o r c e the  i n the gas.  on the l i n e s there i s the f o r c e on  electrons,  q J  f t *—*  -  —  -  e  rr* e  ^c  —  C  T l TYV  <12) where F i s the f l u x i n t e g r a t e d  over a l l f r e g u e n c i e s ,  the  The f o r c e on the e l e c t r o n s i n the  Thomson  undiluted cm  s  - 2  cross  section.  radiation f i e l d  .  There  also  i n a ecu t i e t e l y i o n i z e d gas  is  guite s m a l l ,  with the undiluted  density  o f 10** c m  i t i s 63.48 cm s  The  line  acceleration  l i n e s , and i n c r e a s e s  is  is  194.7  the f o r c e on the continuum, which i s  usually  -3  and  is  - 2  radiation  at  by  optically  with the v e l o c i t y  thick  gradient.  A schematic of the a c c e l e r a t i o n as a f u n c t i o n of dv/dz i s in  Figure  function  11  below.  In Figure  13, the a c c e l e r a t i o n  of temperature i n t h e range 10* <T< 3 x 1 0  temperatures decreases.  larger  than  2x10  s  The  The s l i g h t hump a t 2x10  rapid  fall  s  K,  shown  i s a weak but f o r  the f o r c e on the l i n e s r a p i d l y s  K i s due to  ments changing i o n i z a t i o n s t a t e and t h e entry lines.  a  .  dominated  almost l i n e a r l y  field  the  c u e ele-  c f some new  strong  o f f i s due to the removal of ions  that  42  vo  Temperature  Fig.  13: B a d i a t i o n Force as a Function of Temperature  43  have resonance l i n e s near the maximum o f the field.  The  only  force  left  beyond 10  7  stellar  radiation  K i s the f o r c e on the  electrons. The  acceleration  found by CAK.  found here can be compared with the r e s u l t  The a c c e l e r a t i o n  can be represented  in  the  same  fcrm as they have, 9>«d= 9 e where t - c r r , v e  e  i s the  = .067 t - Q - 9 *  f o r n=10io,  -3  are  leration. thin  thermal  whereas CAK f i n d  There  ft),  < > 13  (dv/dz)-*, ge i s the r a d i a t i o n f o r c e  t K  trons, and v  cm ,  8  two  velocity.  and  M(t)  .033 tr ,  I  find  that  K(t)  = .022 t~° •** f o r n = 1 0  which  0,7  on the e l e c -  reasons f o r the d e n s i t y  is  good  agreement.  dependence of the acce-  F i r s t , a few of the l i n e s go from o p t i c a l l y t h i c k  as the d e n s i t y  ance i s d e n s i t y  13  to  gees down, and secondly, the i o n i z a t i o n b a l -  dependent i n t h i s c a l c u l a t i o n , through the a l l o -  wance f o r c c l l i s i c n a l i o n i z a t i o n , and through the d e n s i t y  depen-  dence of the r a t e c o e f f i c i e n t s . The are  d e f i c i e n c i e s i n t h i s c a l c u l a t i o n o f the r a d i a t i o n  due t o a somewhat l i m i t e d l i n e l i s t ,  of any i r o n group elements, and mere treatment  of r a d i a t i o n  transfer.  these two d e f i c i e n c i e s c a n c e l tent. account  mostly due to the lack  seriously  a  very  simple  Within the approximation used  each other out t o  a  certain  ex-  The r a d i a t i o n f o r c e has been over estimated by not t a k i n g c f overlapping l i n e s ,  which would i n v o l v e  model c f the atmosphere i n t e r v e n i n g and  force  the s t a r .  The r a d i a t i o n f o r c e  formulating  between t h e point increases  a  i n the gas  with the number  of  l i n e s present, but t h e f l u x a v a i l a b l e decreases as the number of lines  goes  up.,  Klein  and C a s t o r  (1978) have r e p o r t e d on new  44  c a l c u l a t i o n s made by that the  o r i g i n a l CAK  s f e r schemes, and imation  to the  ment with the The  Abbott of the law  CAK  line  i s bracketted by  probably the  force.  The  CAK  law  force.  twc  He  finds  alternative  tran-  represents a good approx-  c a l c u l a t i o n s here are i n good agree-  law.  acceleration  ((dv/dz) /n,-j )*  radiation  where  varies  i s i n the  approximately  range 0.7  to 0.9.  as As  (n J/n) t  the  velo-  c i t y gradient increases  a l l l i n e s become o p t i c a l l y t h i n , and  the  force  the  the  levels  off  f o r c e depends on of  .1 to .3,  at  the  maximum value.  T h i s means that  abundances roughly t o a power i n the  which i s a very weak f u n c t i o n .  d i a t i o n f o r c e i s i n s e n s i t i v e to the in radiative equilibrium  range  T h e r e f o r e , the  ra-  assumed abundances f o r flows  because most of the  l i n e s are o p t i c a l l y  thick. One the  aspect of the  analysis  of  l i n e s , which can has  been  the  radiation  transfer  s t a b i l i t y of the  the  provide an immediate source of i n s t a b i l i t y ,  as  r e p o r t e d by Nelson and  Hearn  it  i s dependent on  the  flow i s the  to  shape of  they f i n d only a c t s i n subsonic f l o w . because  which i s important  (1978).  The  This has  d e t a i l s of the  instability  been  left  radiation  out  trans-  fer.  H2fSUium Balance The reguirinq  number of f r e e zero order q u a n t i t i e s can that  be s a t i s f i e d . ture  the In the  eguations of mass and  by  momentum c o n s e r v a t i o n  case o f r a d i a t i v e e q u i l i b r i u m  i s determined by the balance of heatinq and  wise the  be reduced  the  tempera-  coclinq,  temperature i s j u s t a r b i t r a r i l y s p e c i f i e d . ,  other-  45  l o i l l u s t r a t e the s o l u t i o n s c f the mass and momentum t i o n s the one dimensional tituted  into  the  equation  momentum  equa-  of mass c o n s e r v a t i o n i s subs-  balance  equation  (see CAK and i n  Chapter 5, below) with zero temperature d e r i v a t i v e s .  where t h e form of the mass c o n s e r v a t i o n equation cally  symmetric  system  has been used.  The  spheri-  S p h e r i c a l qecmetry has  been used p a r t l y because the d e n s i t y q r a d i e n t even  for a  remains  negative  i f the v e l o c i t y g r a d i e n t a c q u i r e s a small neqative p e r t u r b a t i o n s are i n the form of plane  value.  naves, s o a l l d e r i v a -  t i v e s w i l l be made with r e s p e c t t o the heiqht z, i n s t e a d o f r . The  independent v a r i a b l e i s chosen t o be dv/dz.  In  Figure  14 the two s i d e s of the momentum equation are shown as f u n c t i o n s of dv/dz.  For s u p e r s o n i c  celerating  As can be seen from Fiqure  The  11, i f the ve-  i s n * t too l a r q e there are two s o l u t i o n s i n which the qas  i s a c c e l e r a t e d outwards. the  2  s o l u t i o n , where the r a d i a t i o n f o r c e i s l e s s than t h e  qravitational field. locity  flow, v >2RT, t h e r e i s always one dec-  For t y p i c a l  stellar  two s o l u t i o n s have dv/dz approximately two s o l u t i o n s are acceptable  wind  conditions  equal t o 10-* and 1,  l o c a l l y , but boundary and  con-  tinuity  c o n d i t i o n s may r u l e out the hiqh q r a d i e n t s o l u t i o n .  imposinq  c o n t i n u i t y o f v e l o c i t y from subsonic t o s u p e r s o n i c  CAK r e s t r i c t themselves t c the low q r a d i e n t s o l u t i o n .  By flow  The s o l u -  t i o n with the l a r q e v e l o c i t y qradient i s a c c e l e r a t i n q so r a p i d l y that the wind becomes o p t i c a l l y t h i n This  means  i n the resonance  t h a t i f the a c c e l e r a t i o n c o u l d be maintained  d i s t a n c e o f 0. 151 o f t h e s t e l l a r  r a d i u s , t h e qas would be  lines. over a acvinq  acceleration  2RT  g + g  ,, low d e n s i t y  - — -  rad 2RT  g + 8  r a d  >  h  i  §  h  density  dv/dz < 0 dv/dz > 0 dv/dz > 0, h i g h g r a d i e n t s o l u t i o n  Fig.  14;  The  Momentum Equation  Solution  47  at  the t e r m i n a l  acceptable,  velocity,  observational  although t h i s s o l u t i o n i s p h y s i c a l l y evidence suggests that i t may  not  be  realized. As  can  l a r g e no  be seen from F i g u r e  14, i f the  a c c e l e r a t i n g s o l u t i o n can  be found.  v e l o c i t y becomes too The  maximum  velo-  c i t y f c r which a c c e l e r a t i n g s o l u t i o n s e x i s t v a r i e s with the vity  and  the  i n d i c a t e s the  density  of the gas,  A table i s given  maximum v e l o c i t y g i v i n g an  In Table 4 the Vmax column g i v e s the positive  dv/dz can be found, the  next column. the  Two  outward  gra-  below which  acceleration.  maximum v e l o c i t y at which a  value of which i s given  values f o r the g r a v i t y are used to  in  show  the that  maximum v e l o c i t y with dv/dz>0 i s mostly e f f e c t e d by the  density.  The  gas  was  chosen to  be  in  radiative  gas  equilibrium,  which gives a temperature of 2x10* K. ,  TABLE 4: LIMITING VELOCITY FOR  ACCELERATING SOLUTIONS  g=10*cm s - 2  g=4000cm s ~ 2  Vmax  dv/dz  Vmax  dv/dz  n=10»o  4.1x108  .16x10-3  4.45x10«  .84x10-*  n=10»>  4.7x107  .16x10-2  5.2x10*  .16x10-2  =10»2  5.7x10*  .15x10-2  6.0x10*  .64x10-2  n  Table  4  shows  t h a t the  i n v e r s e l y p r o p o r t i o n a l to the  maximum v e l o c i t y i s approximately density.  The  maximum v e l o c i t y f o r  a c c e l e r a t i o n decreases nearly t o the sound speed at a d e n s i t y IO  1 2  for  cm .  Below t h i s v e l o c i t y the Sobolev  -3  d e r i v i n g the If  shell,  a  large  approximation  of  used  radiation acceleration i s invalid. p o r t i o n o f the  flow, t h i c k e r than one  (the sound speed d i v i d e d by the  velocity  Scbclev  gradient)  ac-  48  quires tive  a velocity  velocity gradient  deccelerate. tic  t h a n t h e maximum f o r a p o s i -  i n momentum b a l a n c e , t h e n  the  gas  will  This s i t u a t i o n c o u l d a r i s e i f the flow i s a chao-  medium i n w h i c h e l e m e n t s o f t h e f l u i d a r e p r o p e l l e d t o v e l o -  cities sity The  which i s g r e a t e r  i n e x c e s s c f t h e maximum f o r a c c e l e r a t i o n , o r have a  increase  w h i c h makes t h e v e l o c i t y  g r e a t e r t h a n t h e maxiaum.  wind m i g h t c o n s i s t o f many, q u i t e l a r q e p a t c h e s ,  beinq  a c c e l e r a t e d and d e c c e l e r a t e d  Hhere t h e s e r e q i o n s ensure  shock heatinq  a t e t o 0 V I and l i k e comprise a small forminq  collide  would  their  which would ions.  This  with respect supersonic  which  heated  are  t o one a n o t h e r .  velocities  produce t e m p e r a t u r e s shock  den-  qas  vculd  appropri-  would  cnly  p o r t i o n c f t h e t o t a l q a s i n t h e f l o w , and a f t e r  be blown  away f r o m t h e s t a r . ,  49  CHAPTER  5.  THE STAEILITY ANALYSIS  The o b s e r v a t i o n s suggest ly  variable,  inhomogenous  that a s t e l l a r flow.  wind i s an extreme-  On s c a l e s o f a day to years  there are l a r g e g e n e r a l v a r i a t i o n s , which may the  star.  The X-ray o b s e r v a t i o n s suggest  t i o n s c f order 1 0 two  causes:  cm.  1 1  the  T h i s observed  originate  small scale fluctua-  variability  could  have  flow may s t a r t out i n the lower atmosphere as  smooth, and then enter a region of i n s t a b i l i t y where up;  within  or the e x i s t e n c e o f the flow  may  it  breaks  be depend upon some i n s t a -  bility. In t h i s s e c t i o n the l o c a l s t a b i l i t y investigated.  with wavelengths s h o r t compared stellar  wind.  This  i.e. a  is  flow  will  linearized  analysis, within  a n a l y s i s i s d i r e c t e d towards f i n d i n g i . e . the  growth  oscillations  cr  lead  f o r cne dimensional  wave propogation,  a v e l o c i t y component  possibility  the Rayleiqh T a y l o r i n s t a b i l i t y , ,  and Hearn 1978).  imations  bee* done  which i s o r i e n t e d along the d i r e c t i o n  Eor i n s t a n c e , t h i s immediately  Nelson  only  t h a t i s the waves can only  of propagation. cf  gen-  to "clumps" within the wind, . One  major l i m i t a t i o n o f t h i s a n a l y s i s i s t h a t i t has  have  time  s h o r t e r than the time t o move one s c a l e l e n g t h i n t h e  atmosphere; and a b s o l u t e i n s t a b i l i t i e s , which can a c t u a l l y erate  be  propogation  to the s c a l e of v a r i a t i o n  i n s t a b i l i t i e s t h a t are r a p i d a m p l i f i e r s , scale  the  T h i s w i l l be done by c o n s i d e r i n g the  of i n f i n i t e l y s m a l l d i s t u r b a n c e s ,  the  of  rules  out  (Krolik  the 1577,  S i m i l a r a n a l y s e s , but with more approx-  i n the l i n e a r i z a t i o n ,  have  been  performed  by  Hearn  (1972) and f o r guasars by H e s t e l e t a l . , (1976). The b a s i c equations  that apply are the c o n s e r v a t i o n c f mass  50  115) where -f i s the mass d e n s i t y servation  and v i s the gas v e l o c i t y .  of momentum n e g l e c t i n g  The con-  the v i s c o s i t y i s given by,  (16)  T  where  9(which  w i l l be sometimes abbreviated as g ) i s the r  a c c e l e r a t i o n due t o r a d i a t i o n , P i s the gas p r e s s u r e , and the  gravitational  acceleration.  The conservation  g  is  of energy i s  expressed,  0*  (17)  where e and h a r e are r e s p e c t i v e l y the s p e c i f i c i n t e r n a l and  enthalpy. ,  ~£ i s the g e n e r a l i z e d  of t h e gas, defined  c o o l i n g r a t e i n the frame  as / = L-(1-v/c) G, where v i s the v e l o c i t y o f  the gas r e l a t i v e to t h e s t a r and L and G are the l o c a l coding  are an eguation c f s t a t e  P = kT(n+.ne) « where n  e  =  (j-1)n *;.  (18)  t  sum i , j i s over the i o n i z a t i o n s t a t e s and the elements, r e -  spectively. n  The number density  o f atoms and e l e c t r o n s  are n and  The i n t e r n a l energy i s e = 3/2 kT(n+n ) + Zn ; f . £  where  :  t  ,  (19)  " X ; . i s the i o n i z a t i o n energy of i o n i , j with d e n s i t y J -  The enthalpy i s defined  the c o n d u c t i v i t y  n y. J t  as, h = e *P/^ .  For  specific  and heating r a t e s , r e s p e c t i v e l y . , The thermodynamic r e -  l a t i o n s reguired  The  energy  •% the standard value cf S p i t z e r  (20) (1962) has  51  been  used. The  above  equations  dispersion relation,  are  l i n e a r i z e d i n order to o b t a i n a  which i s a polynomial d e s c r i b i n g the  g a t i c n of waves of i n f i n i t e s m a l amplitude. . The t i o n s are o b t a i n e d by imposing  a perturbation  t u r e , d e n s i t y and v e l o c i t y c f the  prcpar  linearized on  the  egua-  tempera-  form  Q(z,t) = Q (z,t)+£gt <e4k)expfi{kz-cdt) ],  (21)  D  and s u b s t i t u t i n g i n t o the c o n s e r v a t i o n e q u a t i o n s .  It  assumed t h a t the s c a l e o f v a r i a t i o n of Q ( z , t ) and  the r a d i u s o f  0  the  star  tion. of  are  equations.  order i n  r e s u l t s i n the  and  powers  of  cD and k.  matrix g i v e s a polynomial c f t h i r d persion r e l a t i o n . through  be  zero  order  The determinant  of the  order i n Ld, which i s the  by hand and the roots c f the  dis-  extended  cubic  polynomial  derived  f a r e a s i e r and l e s s prone to e r r o r to do i t  with the a i d of a computer. to  their  although t h i s process could have been c a r r i e d  a n a l y t i c a l l y , i t was  ly  system  T h i s can be w r i t t e n as a c o e f f i c i e n t  matrix, c o n s i s t i n g of zero order g u a n t i t i e s , derivatives,  been  much l a r g e r than the wavelength of the perturba-  Equating terms of f i r s t  linearized  has  Besides, t h i s analysis i s eventual-  to more complex motions,  computer would have to be used, sc the t h i s simpler case w i l l be u s e f u l l y  i n which case  experience  a p p l i e d there.  obtained The  the in  method of  generating the a l g e b r a i c f e r a of the d i s p e r s i o n r e l a t i o n i s outl i n e d i n appendix To  4.  d e f i n e the c o e f f i c i e n t s of the polynomial, i t i s neces-  sary t o knew the d e n s i t y , v e l o c i t y and derivatives,  the  second  d i a t i o n f o r c e , c o o l i n g and  temperature,  their  d e r i v a t i v e of the temperature,  first the r a -  heating r a t e s , and the e l e c t r o n  den-  52  sity  with  their  temperature  and  g u a n t i t i e s were d e r i v e d i n Chapter The  These  4.  t o o t s of the d i s p e r s i o n equation are found usinq a com-  puter program which  f i n d s the r o o t s of complex p o l y n o m i a l s . / The  r o o t found i s improved the  density derivatives.  polynomial  and  i n accuracy by s u b s t i t u t i n q  i t back  doinq a Newton's method i t e r a t i o n  f r a c t i o n a l chanqe i s l e s s than  1 part i n 1 0 . l s  into  until  Since the  the  roots  are computed f o r a sequence of k, the r o o t f o r the next value o f k  is  then estimated from the r o o t j u s t found by  and the same i t e r a t i v e  improvement performed. . The  accuracy of the n u m e r i c a l s o l u t i o n s means that have  real  and  imaqinary parts d i f f e r e n t by  tude or the d i f f e r e n t r o o t s themselves smallest  quantities  s o l u t i o n chosen the  resulting  was  extrapolation,  may  not  l i m i t s to the  when  the  15 orders of maqni-  are widely s e p a r t e d ,  be very a c c u r a t e .  The  tly  designed to suppress "numerical n o i s e " ,  smoothness o f the p l o t t e d r o o t s u s u a l l y  overestiabove.  is  positive  for  a  r e a l k, then there i s an i n s t a b i l i t y a t t h a t wave number.  This i n s t a b i l i t y  can act as an a m p l i f i e r of a p r e e x i s t i n g  i n which case i t i s c a l l e d a c o n v e c t i v e or a m p l i f y i n q ty,  but  p e r t u r b a t i o n s are o f the form exp[ i (kz-tJt) ], consequen-  i f the imaginary part of the frequency  qiven  the  method of  mate the accuracy of the numbers i n the cases mentioned The  roots  or  it  can  an  instabili-  qrow away from the s t a r t i n q v a l u e , e i t h e r i n a  monotonic qrowth or i n ever i n c r e a s i n q called  wave,  absolute i n s t a b i l i t y .  oscillations,  which  A mathematical method of d i s -  t i n q u i s h i n q between the two types of i n s t a b i l i t y  based  on d e t e r -  mininq the behaviour of the wave as t-^oo, has been developed Dysthe  (1966),  Bers  (1975),  is  and Akhiezer and P o l o v i n  by  (1971).  53  They f i n d s e v e r a l c r i t e r i a f o r determining the type of ty,  the  easiest  stabili-  o f which t o apply i s t h a t i f the simultaneous  s o l u t i o n to D( ,k) = 0 and dD/dk=0, where D i s the d i s p e r s i o n  re-  l a t i o n p o l y n o m i a l , e x i s t s , and has an imaginary freguency g r e a t e r than z e r o , then the i n s t a b i l i t y i s a b s o l u t e . ry  and  s u f f i c i e n t c o n d i t i o n i n the approximation  infinite  atmosphere.  as  u) = u  (o ,ko)  to  e  the  two  equations  the  + A ( k - k e ) , where a i s a c o n s t a n t .  root  This imp-  2  0  l i e s t h a t an a b s o l u t e i n s t a b i l i t y qinary  of t->©«» i n an  The c r i t e r i o n means t h a t i n the neighbour-  hood of the s o l u t i o n varies  T h i s i s a necessa-  i s a saddle point of the  part of the frequency as a f u n c t i o n o f k.  ima-  The imaqinary  part c f the frequency w i l l be a t a maximum with r e s p e c t t o k  at  the s o l u t i o n ,  rate.  These two n o n l i n e a r  solved the  and t h i s freguency w i l l equations,  dominate the qicwth and  dD/dk=0,  wavenumbers  as  a startinq  part  of  point.  the  freguency  criminant  of  the  numerical In  order  the  c o e f f i c i e n t s of the two equations.  u n s u c c e s s f u l because o f the i m p o s s i b i l i t y  for  An attempt was made t o  f i n d common r o o t s to the two equations by c c n s t r u c t i n q  cient  were  s i m u l t a n e o u s l y with the a i d o f a computer r o u t i n e , using  l o c a l maximum of the imaqinary  real  D=0  real  of  dis-  T h i s was  retaining  suffi-  accuracy. to  understand  the  d i s p e r s i o n r e l a t i o n and the  p h y s i c a l o r i g i n of the r o o t s , a n a l y t i c e x p r e s s i o n s f o r the r o o t s will roots  be d e r i v e d f o r a number  simple  limiting  i n a complex s i t u a t i o n can be understood  of these s e v e r a l simple cases. been  of  derived  with  the  These  limiting  cases.  The  as s u p e r p o s i t i o n solutions  have  a i d c f numerical s o l u t i o n s , and u n l e s s  noted the c a l c u l a t e d r o o t s p l o t t e d came from a d i s p e r s i o n  rela-  ticn  with  coefficients  radiation f i e l d , 100  Km  s—*.  calculated  with a d e n s i t y of 1 0 The  from 11  cm ,  =0,  d /'/dT  2 . x 1 0 - i , dv/dz = .2x10~ , dn/dz = -2.1, 3  3  and  and a v e l o c i t y  -3  of  r e s u l t i n g e q u i l i b r i u m g u a n t i t i e s are i n cgs  u n i t s : T = 1.87x10* K;  = -0.3x10-1,  a gas i n an u n d i l u t e d  dgr/dn = -.15x10-*.  =  .46x10-*,  d^/dn  =  g rad = 1.18x10*, dgr/dT  I t i s found t h a t the  char-  a c t e r of the r o o t s changes l i t t l e with a v a r i a t i o n of the p h y s i c a l parameters around these values f o r t y p i c a l s t e l l a r  wind  con-  ditions. Case 1: Sound Haves I n An Atmosphere The  simplest  wave propogating with  case  which  has a non zero growth r a t e i s a  v e r t i c a l l y i n a s t a t i c , isothermal  no conduction o r r a d i a t i o n present.  atmosphere,  In t h i s case the  p e r s i o n r e l a t i o n as given i n the Appendix 3 reduces  D e f i n i n g H=n/(dn/dz) , t h i s has s o l u t i o n s o>=0 and  dis-  to  i n the l i m i t  of  l a r g e and s m a l l k the n o n t r i v i a l r o o t s become la.- * 5  00  to —» Cv  (22) This i s e s s e n t i a l l y the w e l l known s o l u t i o n of Lamb to the problem of wave propogation atmosphere.  But  (1945)  i n an i s o t h e r m a l , e x p o n e n t i a l  note that the value of H, the s c a l e height of  Fig.  15: P s e u d c I s o t h e r m a l  Static  atmosphere  Boots  56  the d e n s i t y g r a d i e n t , the i s o t h e r m a l termined  s c a l e height, but that the s c a l e height  by the v e l o c i t y gradient  equation. at  used i n t h e numerical c a l c u l a t i o n s was not  wavelength l i m i t  de-  conservation  (k-»a>) the  waves  move  a phase and qroup v e l o c i t y equal to the o r d i n a r y  sound  velo-  city.  I n the s h o r t  through the mass  was  Outward movinq waves  are  amplified  and  inward  moving  waves are damped at a r a t e such t h a t the momentum c a r r i e d i n the wave  i s kept constant.  ties.  These waves are not absolute  At lonq wavelengths  (k—>0) the r e a l  goes to a f i n i t e l i m i t , c a l l e d the and  instabili-  part c f the freguency  acoustic  cutoff  frequency,  the damping goes to zero. . T h i s means that these waves have  a phase v e l o c i t y going t o i n f i n i t y , but the group v e l o c i t y to  zero  and  no  energy i s prcpcgated.  goes  Physically this cutoff  r e s u l t s from the atmosphere as a whole moving with the wave  mo-  tion,  The  r a t h e r than a wave propogating away from the s o u r c e .  change over between the two l i m i t i n g s o l u t i o n s occurs f o r k order  H .  Figure  15.  the  _ 1  solution  In Figure  is illustrated  15, and a l l other  d i s p e r s i o n r e l a t i o n , the logarithm  imaginary against real  The  parts  of  the logarithm  part  a symbol  part i s negative.  the  i s , the  quently opposite each  i n the accompanying  graphs of the  roots  of  (base 10) of the r e a l and  wave freguency are s e p a r a t e l y  of the wave number. , On the graph  plotted of the  ^ o r O ) on the l i n e means that the r e a l  On the graph of the imaginary part  symbols i n d i c a t e s t h a t the wave i s unstable that  of  the  same  a t t h a t wave number, Note that  fre-  tfee two a c o u s t i c r o o t s have an i d e n t i c a l magnitude,  but  sign,  other.  imaginary p a r t i s p o s i t i v e .  so that i n the p l o t the two l i n e s l i e on top of  57  The which  p l o t s are done f o r k r a n g i n g  i s an u n r e a l i s t i c a l l y  The  10-  15  to  1  cm , -1  l a r g e range f o r the p h y s i c a l s i t u a -  t i o n , but i s done t o i l l u s t r a t e the roots.  from  asymptotic  limits  of  the  p h y s i c a l l y a c c e p t a b l e range of wave numbers i s f o r  wave numbers l e s s than the a wavelength  of  a  stellar  radius,  10-n  cm—i, to a wavecumber corresponding t o one mean free path,  about  10  - 2  There  cm—i. is  a  maximum freguency f o r which the s o l u t i o n s are  v a l i d , s e t by the longer time s c a l e , recombination or the tron i o n thermal e q u i l i b r i u m . o>t  &c  = .188  n  lk  The recombination freguency i s  (T/10* K ) - i / 2  and the e l e c t r o n i o n e q u i l i b r i u m  s-i,  (23)  frequency i s  UeL = 7x10* n „ • (T/10* K ) ~ / 3  The  elec-  2  s-»...  <2<4)  maximum frequency f o r which the c a l c u l a t i o n s are v a l i d  i s the minimum o f (Jr<.c and  The minimum freguency  of i n -  t e r e s t would be determined by the time f o r the complete  replace-  ment  of  the  star's  tJei. .  then  s t e l l a r wind envelope.  T h i s freguency i s  about 6x10-5 s - i .  Case 2: The E f f e c t Of Conduction Allowing conduction a f f e c t s roots. gives.  Taking  the  dominant  mostly  the  short  wavelength  terms i n the d i s p e r s i o n r e l a t i o n  58  For t h i s case the dominant terms c f the r o o t s f o r k  are,  f t v d which i s a h e a v i l y  damped  (25)  non  propagating  disturbance.  The  sound waves are given as  * which  are i s o t h e r m a l  1  (26)  /  sound waves, and always damped independent  of d i r e c t i o n c f propagation. the  *  The numerical  a n a l y t i c s o l u t i o n s only apply  s o l u t i o n shows  that  f o r k > 1 0 , and t h a t the slow - 3  root has a small r e a l p a r t at s h o r t wavelengths. Case 3: R a d i a t i o n E f f e c t s In the long be dominant.  wavelength l i m i t  The dominant  terms  we expect r a d i a t i o n e f f e c t s t o of  the  dispersion  relation  becc me w  *  {.,  f  +  c . R ] t  u>»  <b I i* r ^  + u {c2 %4£]  - ii: 1. r i*I  +  +  ^ 7 dt 7  (27) This  dispersion  r e l a t i o n has been d e r i v e d under the assumption  59  that  Tf  4"*The  ~H  ?  dominant term of one r o o t  X  C  C '  u  128)  i s f o r k->0  •  c  ct T *  * R.  <29) which e s s e n t i a l l y i s the thermal s t a b i l i t y c o n d i t i o n . of  qas  general and  with  case i f d J^dT were negative,  c c l l a p s e , and other  part of the gas  component depends on  t h e gas i s s t a b l e .  within a s t e l l a r with  a  ponent.  cool  The  '  =  +  rl-r  roots a r e shown i n F i g u r e  c i t y and a c c e l e r a t i o n .  '-  20.  hl-Z  'J  is  t (cool) =  discussed  outflow  R. / \ c|  7  K) com-  s u f f i c i e n t C VI region.  PI  ( 3 0 )  c|T I ,  the  discovery  and the numerical c a l c u -  instability.  later  Vjdv/dzj,  f c>  ?  R T  From Equation 30 we make  D e f i n i n g some b a s i c time s c a l e s as t(dynamic) =  10* to 1 0  16, f o r a gas with a nonzero v e l o -  d e c c e l e r a t i n g flows are u n s t a b l e ,  instability  appro-  two r o o t s a r e sound waves,  ,  l a t i o n f i n d s t h a t i t i s an absolute this  an  component  wculd be no need f o r a coronal  civ  that  The ex-  around 2x10* K) and hot (T around 1 0  dominant terms c f the other U  "The  two  The hot component may be able t o supply  atoms that there  cool  I f t h i s b i s t a b l e mode i s p o s s i b l e  wind, i t may l e a d t o a  (T  In a  may  parts may r i s e i n temperature.  p r i a t e heat source t o maintain a temperature of order where  parcel  d J^/dT <0 would probably tend t o c o l l a p s e .  i s t e n c e of the hot, low d e n s i t y  K,  A  An example  and i l l u s t r a t e d  of  i n Figure  Pig.  16: I s o t r o p i c  Badiation  Field  61  t (acoustic) = H/Cp where c = Jk2BTi x  The  c o n d i t i o n f o r the i n s t a b i l i t y  28)  is,  of  t (dynamic) x t ( c o o l ) << Note  that  the  conductive  deccelerating  flews  (Eg.  (t (acoustic) )  2  damping dominates the r o o t s f o r k >  10-*. Allowing plotted  a  radiation  i n Figure  17.  acceleration,  gives  the  as  The asymptotic l i m i t s a r e not changed by  the r a d i a t i o n a c c e l e r a t i o n , but t h e inward propagating wave i s unstable  roots  i n the range c f wavenumber  10  _ 1 1  acoustic  < k < 10~ . 7  The  r e s u l t i n g growth r a t e i s c l o s e t o 1200 seconds, but the i n s t a b i lity  i s only The  amplifying.  cooling  due t c c o l l i s i c n a l l y e x c i t e d l i n e s may be d i -  minished when the gas becomes o p t i c a l l y t h i c k i n lines.  dispersicn  relation  s i d e s the a m p l i f y i n g an  resonance  The e f f e c t of t h i s has been approximated by t u r n i n g the  l o s s rate o f f , but l e a v i n g the heating  is  the  in  roots  of  the  t h i s case are shown i n Figure  18.  Be-  instability  on.  The  from the r a d i a t i o n f o r c e  there  a d d i t i o n a l range o f i n s t a b i l i t y f o r both inward and c u t -  ward a c o u s t i c waves f o r 10~ < k < 1 0 . 7  -s  This  behaviour  results  from the term d jf/dn becoming s i g n f i c a n t . Figure d^/dt<0.  19  shows  the  effect  of  a  thermal i n s t a b i l i t y ,  The "slow" root has a r a p i d growth r a t e , which i s an  absolute  instability.  The  acoustic  roots  are  changed  only  slightly,  the a m p l i f i c a t i o n a c t i n g over a narrower range of  wa-  venumber, and not g u i t e as r a p i d l y . Figure  20 shows the pressure  which i s present  at 1 0  7  K.  dominated thermal i n s t a b i l i t y  In t h i s case the flow  speed i s sub-  L O  Fig.  G-  K  17: R a d i a t i o n  Force  Cn  6<4  sonic,  and the r a d i a t i o n f o r c e i s l e s s than 5% c f g r a v i t y .  instability waves  r e s u l t s when Eguation 28 i s v i o l a t e d .  have  an  absolute  instability  at long  The  The  acoustic  wavelengths.  The  growth r a t e i s p r o p o r t i o n a l t o the element abundance through the cooling rate. An s~  1  atmosphere of d e n s i t y  10  1 2  cm  and a v e l o c i t y of 100 Km  - 3  exceeds the maximum v e l o c i t y f o r an  The  roots  accelerating  when dv/dz i s negative are shown i n F i g u r e  i s an absolute  instability  at long  solution. 21.  Shis  wavelengths.  In summary the r o o t s o f the d i s p e r s i o n r e l a t i o n can be derstood  in  terms  c f combinations of the r o o t s which occur i n  simple p h y s i c a l cases. provides  a strong  For k> 10~* c m , -1  - 1 1  time  scale,  a  ^<^u R / ( d > ? / « ( T | .  slow  moving  - 1  wave,  instability  t(acoustic) » rates will  given  A  absolute the  dynamic time s c a l e .  be a b s o l u t e l y u n s t a b l e i f the v e l o c i t y  tive,  hot  gas,  instability  growth  rate  d//dt<0) which  The slow wave i s an  i f the d e r i v a t i v e d f/dT  the  cooling  compared t o the scund speed, with a  t (dynamic) then the a c o u s t i c by  the  There i s always a "thermal wave", t h a t  growth r a t e s e t by t h e thermal time s c a l e . absolute  d i v i d e d by the  $s.<aii  (dv/dz) ;  liuit,  time s e t by: the  e g u a l t o the s c a l e height  sound speed; the dynamic time s c a l e time s c a l e ,  always  i n the long wavelength  , the behaviour depends on the s h o r t e s t  acoustic  is,  the conduction  damping, e s p e c i a l l y t c the slow wave.  Thus i t can be concluded t h a t k< 1 0  un-  i s negative.  waves  have  If  growth  These a c o u s t i c waves gradient  is  nega-  with t ( a c o u s t i c ) < t (dynamic) w i l l have an a r i s i n g from the a c o u s t i c  i s about one hour, increases  as  n , 2  waves.  At  T=10  7  (one hour at 3x10* K where until  t (acoustic)  exceeds  LOG Fig.  20:  *  High T e m p e r a t u r e  Instability  Fig.  21:  Decceleraticn  I n s t a b i l i t y at n = 1 0  12  cur  3  68  Fig. 15 16 17 18 19  n 10*i 10*i 10i» 10i* 10*»  TABLE T 2x10* 2x10* 2x10* 2x10* 5x10s  5: THE DISPEBSICfr BELATION £ PLOTTED V dv/dz d /dT g Bemarks 0 0 0 0 pseudo i s o t h e r m a l 100 2x10-* 5x10-s 0 isotropic 100 2x10-* 5x10-s 1x10* standard case 100 2x10-* 5x10-s 1x10* no c o o l i n g 100 7x10-* -3x10-* 1x10* thermal  20  10**  1x10  100  21  10*2 2x10* 100  t(dynamic). amplifying  1200  The  5x10-5  8x10-8  270  -6x10-* - 3 x 1 0 - 3  high  5x103  f o r inward a c o u s t i c  a c t s to provide  waves.  for an o b s e r v a t i c n a l l y i n t e r e s t i n g  from  5x10  to  7  5x10** cm,  temperature  decceleration  r a d i a t i o n a c c e l e r a t i o n only  instability  c a t i o n acts lengths,  7  an  This a m p l i f i -  range  of  wave-  with growth times c f order  seconds. on the b a s i s of t h i s a n a l y s i s the o r i g i n a l CAK  " c o o l " atmo-  sphere i s s t a b l e only i f no waves are sent i n t o the  accelerating  wind from lower l a y e r s .  Otherwise the  provide  an  the gas,  but outwards with r e s p e c t  A  corona  a m p l i f i c a t i o n of the  with  a  temperature  always have an absolute instability instability  radiation force  inward moving t o the s t a r )  acts  (with respect acoustic  tc to  wave.  of s e v e r a l m i l l i o n degrees w i l l  instability,  e i t h e r the o r d i n a r y  of the r a d i a t i o n l o s s e s , or the o u t l i n e d above, which has  "high  thermal  temperature"  a growth time cf order an  hour. The et a l .  semi e m p i r i c a l model o f the (1978) has  the  i n a hot corona, and lerated  2one.  The  s i t u a t i o n would be corona  i s subject  wind heated  wind proposed by  with an i n i t i a l  Cassitelli  acceleration  then c o o l i n g i n the outer r a d i a t i v e l y accestability  a n a l y s i s leaves  expected  to  show  doubt t h a t t h i s  fluctuations.  to i n s t a b i l i t i e s which may  c r e a t i n g the shock waves to heat i t .  no  be  The  responsible  i n t o the wind where the l e n g t h  of 10  amplified.  t o 10**  cm  would be  for  Bemnants o f these f l u c t u a -  t i o n s would be c a r r i e d out 7  hot  scales  69  CHAPTER 6. , CONCLUSIONS The  program of o p t i c a l o b s e r v a t i o n s  s t e l l a r winds do observations  vary on time s c a l e s of one  were taken at s u f f i c i e n t l y  v a r i a t i o n s c f the t e l l u r i c of  ported  as v a r y i n g . Lambda Cephei, was  cant  stellar  l i n e s could  tions  solution  the  of one  an  upper  and  may  not be due  time  limit  •x'Cam, but  scale  of  be separated  A s t a r which has  Day  about to day  monitored  5x10  more. , These  from  This  on  & Ori.  signifi-  observation  the s i z e of confirmed i n  X  These v a r i a t i o n s  to f l u c t u a t i o n s w i t h i n the wind i t s e l f  region.  any  since  this  Causes o f the  long  the time  v a r i a t i o n i n c l u d e r o t a t i o n of the s t a r , i n t e r n a l o s c i l l a -  hence the The  s t a r , or a v a r i a t i o n of the the emergent  d r i v i n g f o r c e f o r the  cf Cen  reported  by  Schreier  That i s , the wind d e n s i t y v a r i e s s u f f i c i e n t l y  source i s o c c a s i o n a l l y smothered by the o p a c i t y of wind.  X-3  evidence f o r the suggested mechanism causing  term X-ray i n t e n s i t y v a r i a t i o n (1976)•  flux  In a d d i t i o n I have found  phases.  the  long  et  al.  that  the  stellar  that as the d e n s i t y i n the  the changing c h a r a c t e r  i n t e n s i t y at n o n - e c l i p s e  provides  the  changes, i t must be c o r r e l a t e d with the wind v e l o c i t y explain  and  wind.  a n a l y s i s of the X-ray o b s e r v a t i o n s  confirming  dips  re-  i s long enough to allow complete replacement of  t i o n s of the  to  varia-  o f t e n been r e -  null  cm  11  any  with a time  v a r i a b i l i t y was  not c o n c l u s i v e l y f o r  m a t e r i a l i n the l i n e formation scale  and  high r e s o l u t i o n that  seen i n the H ^ l i n e .  " b l o b s " i n the wind. Cep  lines.  day  hour over a p e r i o d of s i x hours but no  v a r i a t i o n was  puts  c o n c l u s i v e l y shows that  in  wind order  of the anomalous d i p s i n the Besides  these  semi-regular  the X-ray i n t e n s i t y shows i n t e n s i t y f l u c t u a t i o n s on a time  70  s c a l e i . of l e s s than one in  the  hour, which i s probably due  amount of mass being a c c r e t e d  of  the d e n s i t y  s i z e s c a l e s of 1Q*o The  t c 10*  and  rate  l i m i t the highest  i n the flow.  growth  rate,  of  holds f o r the thermal i n s t a b i l i t y  with r e s p e c t  t c 10  7  10  negative  K.  The  growth r a t e  to the abundances of the elements present.  between  stars  higher CMO  of  would expect  different  temperatures exceeding 10  s  In  would  a  grow on  on  moving  density  If this  differences  cooling  rate  t h i n gas.,  As a  a  shorter  s t a r s , and  An a m p l i f y i n g  time  gradient  defini-  instability  waves which i s u s u a l l y present i s the to the  this  wolf-Hayet s t a r s , which  abundances than OE  have higher t a s s l o s s r a t e s .  wave amplitude due  of  s c a l e height,  about  in  the  atmosphere.  of a c o u s t i c  The  waves i s an absolute  ins-  instability  t a b i l i t y , . The  growth r a t e i s j d v / d z ) , - 1  gas  seconds.  decceleration  2x10  for  simple growth of  atmosphere t h i s occurs on a time s c a l e of the  speed d i v i d e d by the  the  That i s , s t a r s with  K i n an o p t i c a l l y  have some bearing  appear to have higher CNO  acocstic  significant  composition.  r e s u l t the thermal i n s t a b i l i t y may  deriva-  r e l a t e d t o the c o o l i n g r a t e , which i s  or metal abundances would have a g r e a t e r  This  seconds,  This s i t u a t i o n arises i n  proportional  operates, one  wave-  which a r i s e s whenever  is  instability  directly  s  order  instability  tely  the  wind with  In the long  rate has a  to temperature.  temperature ranges of 3x10  scale.  is  cm.  1  the c o o l i n g r a t e minus the heating  for  The  t h e o r e t i c a l a n a l y s i s of the s t a b i l i t y o f a wind f i n d s a  usually  tive  accretion  v e l o c i t y i n the s t e l l a r  number of sources c f i n s t a b i l i t y length  changes  by the neutron s t a r ,  n a t u r a l source f o r the v a r i a t i o n i n the variation  to  3  usually  cf  order  1000  71  to  10*  seconds.  I f  t h e gas i s very h o t , g r e a t e r than  t h e r e i s an a b s o l u t e i n s t a b i l i t y s c a l e c f an h o u r . for  which  operates  cn  10  the  s  K,  time  The r a d i a t i o n f o r c e p r o v i d e s an a m p l i f i c a t i o n  wavelengths i n t h e range 10 -10 8  cm on t i m e s c a l e s o f 1200  1 1  seconds. As a r e s u l t o f t h i s  w o r k , a number o f  made f o r f u r t h e r i n v e s t i g a t i o n . sued  suggestions  Line v a r i a b i l i t y  i n order t o unravel the nature  These  observations  noise  segment  be  ob-  high  reso-  line.  o f t h e X - r a y d a t a s h o u l d be a n a l y z e d t o  c o n f i r m t h e model g i v e n portantly  must  or at a s u f f i c i e n t l y  l u t i o n t o niniffiize blending with the s t e l l a r longer  High r e -  s h o u l d e i t h e r be made i n s p e c t r a l  regions free of t e l l u r i c l i n e s ,  A  be  s h o u l d be p u r -  of the v a r i a t i o n .  s o l u t i o n o b s e r v a t i o n s , w i t h good s i g n a l t o tained.  can  f o r t h e a c c r e t i o n p r o c e s s , and s o r e  t o e s t i m a t e t h e d e n s i t y and v e l o c i t y  im-  as a f u n c t i o n o f  time. The to  t h e o r e t i c a l a n a l y s i s of i n s t a b i l i t i e s can  allow  a  vertical  be  extended  and h o r i z o n t a l wave v e c t o r , and a l l o w t h e  wave m o t i o n t c h a v e a h o r i z o n t a l a s w e l l a s a v e r t i c a l and  then  Besides  t o more g e n e r a l dynamical  waves,  such  X-rays.  a n a l y s i s t o be e x t e n d e d t o g u a s a r s purpose  of t h i s  of  i s observed  source would  spectra  allow  the  i s to acguire information  c f t h e mass l o s s i n t h e  a strong radiation f i e l d . t o be v a r i a b l e ,  This  vcrticity.  and n u c l e i o f g a l a x i e s .  whole s t u d y  on t h e f u n d a m e n t a l p h y s i c a l n a t u r e sence  allowing  generalizations, different  s h o u l d be a l l o w e d , p a r t i c u l a r l y  The  as  component  and t h a t  pre-  My t h e s i s i s t h a t t h e w i n d the  variability  on  s c a l e s c f a day o r l e s s c a n be a t t r i b u t e d t c i n s t a b i l i t i e s  time which  72  exist  in  the wind.  instabilities to  the  flow  established of  the  haviour.  I t i s suggested  that the presence  changes the fundamental dynamics of the of  the s t e l l a r  wind.  these  solutions  The l e n g t h and time s c a l e s  i n the a n a l y s i s w i l l allow the  flow to be attacked with  of  nonlinear  equations  a knowledge of t h e i r l o c a l  be-  73  BIELICGBAPHY 2.  Akhiezer, A. I. ana P o l o v i n , B. V., 1971, Sov , 14, 278.  3.  A l d r o v a n d i , M. V. and Pequignot, D., 1973, A s t r . Ag., 25, 137.  4.  A l d r o v a n d i , M. V. and Pequignot, D., 1976, A s t r . A j . , 47,  A  P J j s ^ Ospekii  321. 5.  Earlow, M. J . and Cohen, M. , 1977, A£.J.., 213, 737.  6.  E e a l s , C. S., 1929, J.N., 90, 202..  7.  E e a l s , C. S., 1951, Puh^DjjO, i x , 1.  8.  E e r s , A., 1975, Plasma P h y s i c s . Ed. D e i i t t , C. and Peyraud, J . . (Gordon and Ereach: New York), p. 121., Eerthcmieu, G., Provost, J . and Bocca, A., 1S75, A s t r . J j 3 . , 47, 413.  9. 10.  Eethe, H. A. and S a l p e t e r , E. E. 1957. The Quantum Mechanics o f One And Two E l e c t r o n Atoms, (Sprinqerv e r l a g : New York).  11.  Brucato, B. J. , 1971, M.N.,  12.,  Burgess, A. and Summers, H. P.,  1976, M.J., 174, 345.  13.  Burgess, A. and Summers, H.P.,  1969, A £ . J . ,  14.  Cannon, A. J . and P i c k e r i n g , E. C., 1918, Ann.t of Obs Harvard  Coll.  M  153, 435.  157, 1007. A  Q£  v o l s , 91-99.  15.  Cannon, C, J . and Thomas, B. N•, 1977, k£.J.,  16.  C a s s i n e l l i , J . P. and C a s t o r , J . I . , 1973, A D . J . , 179, 189,  17.  C a s s i n e l l i , J . P., Olson, G. I . , and S t a l i o , B., 1978,  18.  Castor, J . I . , Abbott,  AJE.J., JE.J.,  220, 195,  211, 910.  573.  D. C., and K l e i n , B. I. , 1975,  157.  19.  C a s t o r , J . I . , 1977, i n Colloquium Internationaux Eu CJBS N O J S . , 250. Mouvements Dans Les Atmospheres S t e l l a i r e s . ed. C a y r e l , B. and S t e i n b e r g , M•, (CNBS: P a r i s ) .  20.  C a s t o r , J . I . , 1978, a t IAU Symposium No. 8 3, Mass Loss And  74  21.  Chapman, E. E. and Henry, R. J . 8.,  22.  Chapman, E..D. and Henry,  23.  Conti,  P. S. and F r o s t ,  1971 Ap..J., 168, 169.  R. J . B., 1972 Ap.*?.,  173, 243.  S. A., 1974, A p . J . L e t t . ,  190,  L137. 24.  Conti,  P. S., 1978, A s t r .  Ap. , 63, 225.  25.  Cox, 0. E. and T u c k e r , B. H. , 1969, Ap,J.,  26.  Dupree, A. K., 1968, A s t r o p h v s . L e t t . .  27.  D y s t h e , K.  B., 1966, N u c l e a r  157, 1157.  1, 125.,  F u s i o n . 6, 215.  28. H e a r n , A. G., 1973, A s t r . .Ap., 23, 97. 29.. Fahlman, G. G., C a r l b e r g , fi. G., and B a l k e r , G. A. H., 1977, Ap.J. L e t t . , 217, L 3 5 . 30.  Field,  G. B. and S t e i g m a n , G.,  31.  Field,  G. B. , 1965, JLp.J., 1«2, 531.  32.  F l o w e r , D.,E. 1968, i n IAD Symposium NOj. 34., P l a n e t a r y N e b u l a e , e d . O s t e r b r o c k , D. E . and O ' D e l l , C. R. ( R e i d e l : D o r d r e c h t ) , p. 205.  33.  H e a r n , A. G., 1972, A s t r .  Ap.., 19, 417.,  34.  H e a r n , A. G., 1975, A s t r .  Ap.. , 40, 3 5 5 . ,  35.  Henry, B. J . W.  36.  H u t c h i n g s , J . B. , 1976, Aja. J . , 203 , 438.  37.  J a c k s o n , J . C. , 1S75, JS.J. , 172, 483.  38.  J o h n s o n , L. C. , 1972, Ap* J . ,  39.  Jordan, C ,  40.  K a t o , T., 1S76, Ajp.. J . Supji. , 30, 397.  41.  Kippenhahn, Ap.,  1S71, A j . J . ,  166, 59.  1970, A p . J . , 161, 1153.  174, 227.  1969, M.J. , 142, 501.  E., M e s t e l , L . , And P e r r y , J . J . , 1975, A s t r .  44, 123.  42.  Klein,  R. I . and C a s t o r , J . I . , 1978, A p . J . , 220, 902.  43.  K r o l i k , J . H.,  44.  lacy,  C. H.,  1977, P h y s . F l u i d s  20, 364.  1977, A p . J . , 212, 132.  75  45.  Iamb, H, , 1945,  46.  Earners, H. J , G..L. M. and Morton, D. C , Su££., 32, 715,  47.  lamers, H . J . 219,  Hydrody namies. (Dover: New  G, L. M.,  York). 19*76, A£.J.  and Snow, 1 . P., J r . , 1978,  A£.J.,  504.  48.  L c t z , W. , 1967, J f i . J . SupjS., 14,  49.  Lucy, L. B. And Solomon,  50. 51.  Lucy, L. B. , 1971, Ac.. J . , 163, 95. McWhirtler, E. 8. P., 1975, i n Atomic .And, M o l e c u l a r £Efi£J5se,s I n A s t r o p h y s i c s , ed. Hubbard, M. C. E, and Nussbaumer, H (Geneva O b s e r v a t o r y : Sauverny, S w i t z e r l a n d ) , p. 205.  52.  flestel, I . , Moore, D. W., 52, 203.  53.  Mewe, fi., 1972, A s t r . Ap_. , 20,215.  54.  Mihalas, D., 1972, Non-LTE Model Atmospheres f o r B and 0 S t a r s , NCAB-TN/STB-76, (National Center f o r Atmospheric Research: B c u l d e r ) .  55., Milne, E. A.,  P. M.,  207. 1970, j£.J.,  159,  879.,  and P e r r y , J . J . , 1976 A s t r *  1926, M. JJ., 85,  lj»,  813.  56.  Moore, C. E,, Minnaert, M. G. J . , and Houtgast, J . , 1966, The S o l a r Spectrum. NBS Monograph 6 1.  57.  Moore, C. E,, 1949, Atomic Energy l e v e l s , NBS C i r c u l a r  No.  467. 58.  Morton, D . C  and Smith, ». H.,  59.  Morton, D . C ,  60.  Mushotzky, R. F., Solomon,  1967, A j . J . ,  1973, Ajg.J. Su££. , 26, 333.  147, P. M.,  1017. And S t r i t t m a t t e r , P.  A.,  1972, Ap_. J . , 174, 7. 61.  Nelson, G. D.,  62.  Pounds, K. A., Cooke, 6. A., R i c k e t t s , M. J . , T u r n e r , M. J . , and E l v i s , M., 1975, M.N., 172, 473. Raymond, J . C , Cox, D. P., and Smith, B. H., 1976, J. • 204, 290. Bcsendahl, J..D., 1973, Afi.J., 182, 523.,  63. 64.  and Hearn, A. G.,  1978, A s t r . An.. , 65, 223.  76  65.  R y b i c k i , G. B. a n d Hammer, D. G.,  1978, A p . J . , 2 1 9 , 654.  66.  S c h r e i e r , E. J . , S c h w a r t z , K., G i a c c o n i , R. , F a b b i a n c , G., and M o r i n , J . , 1976, A j . J . ,  204, 5 3 9 .  67.  S e a t c n , M. J . 1958, E g V i JSod[ P h j ^ s . , 3 0 , 9 7 9 .  68.  S e a t c n , M. J . , 1 9 5 9 , M.N.,  69.  S i l k , J . And Brown, B. L., 1971, A j . J . ,  70.  Snow, T. P. J r . and M o r t o n ,  A  119, 8 1 .  E. C ,  163, 495.  1976, A p . J . Supja., 3 2 ,  429. 71.  Snow, T. P. J r . , 1 9 7 7 , A p . J . , 2 1 7 , 7 6 0 .  72.  S o b o l e v , V, V., 1 9 6 0 . M o v i n g E n v e l o p e s o f S t a r s * ( H a r v a r d U n i v e r s i t y P r e s s : Cambridge), S p i t z e r , L, J r . , 1962, P h y s i c s o J F u l l y . I o n i z e d G a s e s . 2nd e d . , ( I n t e r s c i e n c e : New Y o r k ) • S t e i g m a n , G., W e r n e r , M. 8., and G e l d o n , F.... B. , 1 9 7 1 , A p . J . , 168, 37 3.  73. 74. 75.  Summers, H. P., 1974, M.N.,  169, 6 6 3 .  76.  S u n y a e v , B. A. and V a i n s t e i n , L, A., 1968, A s t r o p h v s . ., 1, 193.  77.  Thomas, B. N., 1 9 7 3 , A s t r . Ajg., 2 9 , 2 9 7 .  78.  U n d e r b i l l , A. B. , 1 9 6 0 , i n JcJU o f SJbars A^d S ^ e l l a j c Systems, S t e l l a r Atmospheres. ed.,J. G r e e n s t e i n , (ti. .of C h i c a g o : C h i c a g o ) , p. 411. J r  79.  W a l k e r , G, A. H., B u c h o l z , V., F a h l m a n , G. G., G l a s p e y , J . , L a n e - W r i g h t , D., flochnacki, S., a n d C o n d a l , A., 1S76, P r o c , IAU C o l l o q u i u m 4 0 , e d . M. D u c h e s n e , ( r e i d e l : Dordrecht).  80.  H i e s e , W. L., S m i t h , M. W., NSBDS-NBS4.  81.  W i e s e , W.  and G l e n n o n ,  lett  B. M., 1 9 6 6 ,  L., S m i t h , M. «., and M i l e s , B. M.,  1969, NSBDS-  NBS22. / 82.  W i l s o n , B., 1962, J.O.S.B.T. 2.  83.  W r i g h t , A. E., and B a r l o w , M. J . , 1975, M.N.,  84.  Y c r k , D. G. , V i d a l - M a d j a r . A., L a u r e n t , C , 1S77, A p . J . L e t t . , 2 1 3 , 1 6 1 . .  477. 170, 41.  B o n n e t , fi. ,  77  APEENDIX 1. SUPEBSGNIC ACCBETTON T h i s appendix i s a r e p r i n t of the paper e n t i t l e d Effects  in  A§*rcjhxsical was  Supersonic  Accretion",  J o u r n a l Volume 220,  used i n Chapter I I I t c deduce  which  appeared  "Badiative in  the  p. 1041. The theory developed the  correlation  between  the  wind v e l o c i t y and wind d e n s i t y i n the observed i n t e n s i t y t r a n s i t i o n c f Cen X-3. /  THE ©  A S T R O P H Y S I C A L J O U R N A L , 2 2 0 : 1 0 4 1 - 1 0 5 0 , 1978  March  15  1978. The A m e r i c a n A s t r o n o m i c a l S o c i e t y . A l l r i g h t s r e s e r v e d . P r i n t e d i n U . S . A .  RADIATIVE EFFECTS I N SUPERSONIC ACCRETION R. G. CARLBERG Department of Geophysics and Astronomy, University of British Columbia  Received  1977 March  21; accepted  1977 September  22  ABSTRACT Supersonic gas flow onto a neutron star is investigated. There are two regimes of accretion flow, differentiated by whether the gas can cool significantly before it falls to the magnetosphere. If radiative losses are negligible, the captured gas falls inward adiabatically in a wide accretion column. If the radiative energy-loss time scale is less than the fall time, the gas will cool to some equilibrium temperature which determines the width of the wake. A n accreting neutron star generates sufficient luminosity that radiation heating may determine the temperature of the accretion column, provided the accretion column is optically thin. Gas crossing the shock beyond the critical radius forms an extended turbulent wake which gradually merges into the surrounding medium. As a specific example, the flow for the range of parameters suggested for the stellar wind X-ray binaries is considered. Subject headings: shock waves — stars: accretion — X-rays: binaries streamlines are taken to be coincident with particle trajectories i n a gravitational field; (2) the shockheated sheath where the incoming gas impinges on the accretion column; the transverse component of the velocity is rapidly halted, providing pressure to contain the accretion column; (3) the accretion column, in which gas falls inward, toward the accreting body; (4) a region of spherically symmetric flow which may exist near the accreting body; (5) the base of the accretion column; beyond this the flow is regulated by the physics of the magnetosphere around the accreting object; (6) the accreting body, where the kinetic energy of the gas is liberated at a surface shock; and (7) the far wake, several hundred times the length of the accretion column. The density contrast between the far wake and the surrounding medium gradually goes to zero. One major qualitative aspect of this model is that there is no bow shock standing off from the front of the body which is distinct from the tail shock. U n doubtedly there will be a preceding shock, but pressure waves generated there will not propagate very far in a transverse direction because the streamlines of the flow are bent in by gravitation. Consequently, the bow shock merges into the tail shock. In general, this will be the case for any body whose size is less than the "accretion radius" R = 2GM/V , where V is the free stream velocity. Calculations by Hunt (Eadie et al. 1975) indicate that part of the infalling column may " m i s s " the accreting body and force the leading shock forward. This occurs because small nonradial velocities increase toward the body, by conservation of angular momentum. This effect will be ignored. This model is to be applied to a neutron star orbiting a massive star with a strong stellar wind. For convenience, scaled variables will be used for the distance rv = rjRA: free stream density, w = H Q / 1 0 c m " ; free stream velocity, V = V j 10 cm s~ ; and  I. INTRODUCTION Recent observations of X-ray binaries, at both optical (Conti and Cowley 1975; Dachs 1976) and X-ray (Jones et al. 1973; Pounds et al. 1975; Eadie et al. 1975) wavelengths, show phase-dependent absorption of radiation. It has been suggested that this is caused by a wake trailing the compact object which emits the X-rays. Models of the wake based on the X-ray observations were put forward by Jackson (1975) and Eadie et al. (1975). The general problem of a gravitating body moving through a gas at a velocity much greater than the sound speed was first discussed by Hoyle and Lyttleton (1939). More recently, wakes were discussed by Davidson and Ostriker (1973), Illarionov and Sunyaev (1975), and M c C r a y and Hatchett (1975). These models are incomplete in that they lack a description of the gravitationally perturbed gas which is unbound, i.e., the far wake. Although most of these papers emphasize the importance of radiative effects, no clear analysis has been made of the variations in the flow of gas caused by radiative gains and losses. In this paper supersonic accretion onto a neutron star is considered. There are three basic physical parameters: the mass of the accreting body and the free stream velocity and density of the gas. The dynamics of the flow are essentially determined by the free stream velocity and the mass. The angular width of the accretion column depends on its temperature, which in turn is regulated by radiative cooling and heating and is sensitive to the gas density. The proposed description is worked out for linear motion, which is a good approximation for an accretion radius much smaller than the system dimensions. A schematic of the model is shown in Figure 1. The important regions are labeled: (1) the incoming supersonic gas; pressure forces can be neglected and  2  A  Q  0  11  u  3  3  B  1041  0  1  1042  Vol. 2 2 0  CARLBERG  F I G . 1 . — A schematic of supersonic accretion gas flow showing: (1) the incoming supersonic gas; (2) the shock-heated sheath; (3) the accretion column; (4) the possible spherically symmetric inflow at the bottom of the column; (5) the Alfven surface at the bottom of the column; (6) the accreting body; and (7) the far wake.  mass of the accreting body, m — M /M . Similarly, in later calculations, the temperature of the column will be represented as T = T/IO K. This form of notation will be used throughout. x  0  As a simplifying approximation, we take the case of (f> small and <j> « R /r, to obtain the relations, 2  A  6  6  II. D Y N A M I C S O F T H E G A S  FLOW  A gravitating body is placed in a uniform stream of gas moving at some velocity V . To the point where the gas crosses the tail shock, we assume that the streamlines of the flow can be found from particle dynamics, i.e., the flow is dominated by inertial forces. The velocity can be obtained from the equations of conservation of energy and angular momentum,  V* « V (RJry*, 0  and  0  Wr  2  + V, ) 2  GM  Wo  2  and  0)  V<t> — V • 0  where V and V# are, respectively, the radial and tangential components of the gas velocity relative to the accreting object. The trajectories are given by (Ruderman and Spiegel 1971),  d) The Sheath  The gas crossing the shock has a discontinuity in its motion described by the equations for the conservation of the total energy and of the normal components of mass flux and momentum. Assuming that a strong shock occurs (good for Mach numbers greater than ~3), the postshock density and temperature are (for a ratio of specific heats y = 5/3) P2  r  I 5* =  0 +C0S  & I *' +  Sin  (2)  where s is the impact parameter and <f> is the angle measured from the accretion axis. From these equations, Danby and Camm (1957) obtain the density n = n(r, </>) as + sin-  n = 2 sin 0/2  + sin - + sin 2 2  r A  + sin  2  (3)  = 4 , Pl  and  r = ^-^.  (5)  a  where R is the gas constant and 1 and 2 refer to the pre- and postshock conditions, respectively. V (x V^) is the component of velocity normal to the shock. An important point is that specific energy is conserved across a shock. If the gas is energetically unbound ahead of the shock, it will remain unbound behind the shock in the absence of cooling. On the other hand, if all the thermal energy is immediately lost, one finds that the gas is energetically bound for all radii less than R . The sheath is bordered by the shock on the outside and the inward-flowing gas of the accretion column on the inside. The sheath is a dynamically defined region where the gas slows to a stop, changes direction, and joins the accretion column. n  A  2  (4)  20  80 No. 3, 1978  The semiangle to the shock cone will be approximated by the semiangle of the accretion column. To this end we demonstrate that the thickness of the sheath is small for the case of a narrow shock cone. The radial-velocity component is approximately parallel to the shock and is continuous across the shock. In the limit of V = V , one easily finds that gas entering the shock sheath at r < R will travel to a maximum distance r given by r  0  0  r  A  (6) 0  before the radial velocity is brought to zero. The gas would then join the accretion column. An estimate of the sheath thickness can be obtained by equating the mass influx between r and r, (r x r , r « R ) -nn V r , to the mass flux through a cross section at distance r, H K 2 T 7 T < / > H ' , where vt' is the thickness of the sheath, n the postshock density at r, and <f> the angle to the shock. This gives the semiangular width of the sheath as 0  0  2  A  0  0  0  S  S  0  s  s  w r  (7)  ar  0  Thus the maximum width of the sheath is only a function of distance from the accreting object, and for r « R the sheath width will be negligible. The above calculation assumed laminar flow and no premature mixing of sheath gas into the column, whereas it is quite likely that the sheath is turbulent. The Reynolds number in the sheath is 4 . 1 x 1 0 / j r „ $ ~ K , indicating the possibility of turbulence. A turbulent sheath would come into equilibrium with the column more rapidly than laminar flow through mixing. As a consequence, a turbulent sheath would be even thinner than the limit set in equation ( 7 ) . A  4  2  1  1043  RADIATIVE EFFECTS  1  -  1  6  8  b) The Accretion Column The mass flux in the accretion column is simply dM/dt = irp V s , where s is the critical impact parameter, taken as the impact parameter of the streamline which would have a total energy of zero on the accretion axis. To allow for the thermal energy, a parameter (Z is introduced, such that the " true accretion radius" is equal to fiR . In principle, jS is determined once the physical parameters, the density, velocity, and accreting mass, are specified. The parameter jS will be taken as the ratio of the specific kinetic energy (%V ) to the specific enthalpy (5RT ) if (•JrKp < 5RT ), otherwise 0 = 1 , where T is the equilibrium temperature of the column at r„ = 1 . The accretion rate is then 2  0  0  c  V 2.13 x l O ^ " 4(2Y' Rp 2  7  0  <f>  2  c  1  Krad" . ( 1 0 ) 1  Pressure forces are unable to support the gas, and it falls inward toward the accreting object down the accretion column at a velocity v = (GM/r) . Using equation ( 1 0 ) and mass conservation, we find that the equilibrium accretion column density is, for fiR » r, 112  A  n  c  =  6 . 4 0  x  10  1  3  r - r ,a  6  t  3  ' « a  1  1  K8*/3-  1  cm"  3  ( 1 1 )  The assumption that the width of the column is maintained by gas pressure is justified by the required default of any stronger forces, turbulence in particular. One can do a pressure confinement calculation similar to the one above by assuming a fully turbulent accretion column. The internal pressure in the column would be generated by the turbulent velocity, which can be taken to be a fraction/ of the velocity of fall. Requiring that the opening angle of the column be less than, say, 1 radian, we find that f is restricted by  2V2 r  (12)  This implies that the turbulent velocity must become a smaller fraction of the fall velocity as it nears the neutron star; otherwise the turbulent pressure is impossible to contain. But the Reynolds number of the gas increases inward (except for adiabatic infall), and we would expect the turbulence, if present, to increase and disrupt the column. Therefore, if the column exists, it must be in laminar flow. There are several reasons to think that laminar flow can obtain in the column. The turbulence would probably originate in the "shear layer" between the sheath and the column, but the Reynolds number in the sheath decreases down the column. In addition, the gas is being strongly accelerated only in the radial direction, which does not provide a driving force for turbulence.  c  A  2  0  0  2  0  dM/dt =  From this one obtains a relation between the central temperature of the column and the opening angle,  0  3 . 6 5  x  1 0  1  6  / J H / M  2  K  8  -  3  g s" . 1  ( 8 )  For a column in equilibrium, the transverse momentum of the incoming gas must be balanced by thermal pressure in the column, Pi i  (9)  III. RADIATIVE EFFECTS  If the gas is unable to cool before joining the accretion column, the column gas will fall adiabatically and will resemble the accretion scenario found by Hunt ( 1 9 7 1 ) , i.e., a very wide accretion column trailing the accreting body. Note that Hunt's solutions were obtained with essentially zero pressure at the boundary of the accreting body, and that the accretion rate would probably be diminished by the nonzero base pressures of a magnetospheric shock above the neutron star. On the other hand, if the gas cools much faster than any time scale for movement, a cold, narrow, high-density column will be formed. In order to determine which regime prevails, we compare the time scales for radiative energy-loss mechanisms with the time scale for infall of the gas, which is the basic and only uniquely identifiable dynamic time scale of the  CARLBERG  1 0 4 4  problem. The fall time from the accretion radius is approximately,  '^ = 3 V ( G M )  =  2 5 0 R U 3 , 2 K R 3 / M S  -  °  3 )  Vol. 220  omitting the cooling due to forbidden and semiforbidden lines. The postshock cooling is assumed to be unaffected by any radiation present. Two assumptions are made for the density of the postshock gas in the sheath. The rightmost line is drawn for the minimum possible postshock density, 4n . This is an underestimate, since the density increases from its free stream value toward the accretion axis, by approximation (4). The angle to the shock decreases with the temperature by equation (10), and choosing the minimum temperature in the column as 10 K results in the cooling line on the left. Gas flows with densities and velocities in between these two cooling lines may be subject to an instability from the cooling to noncooling state and vice versa. If hot, uncooled gas mixes into the accretion column and expands it such that the shock moves outward, it will decrease the density of the incoming gas, by approximation (4). If the density drops sufficiently, the incoming gas may no longer cool and the column will expand to its uncooled state. Consequently we take the rightmost cooling line (labeled 4n ) as the effective cooling line. A point of interest is that, for gas crossing the shock at a distance of less than 3 x 10 cm, the postshock temperature is greater than 10 K, for which bremsstrahlung is the dominant cooling mechanism, until the gas is close enough (see eq. [25]) to be Compton cooled. The time scale for bremsstrahlung losses varies -l 1 2 with « r ' , which remains constant with distance in the sheath, whereas the dynamic time scale is decreasing as r ' . Consequently, even though gas entering the column at large radii may cool, lower down the gas in the sheath may remain hot. As shown in the Appendix, the sum of the pressure force for the postshock gas 0  a) Cooling Time Scales In the absence of any heating, the temperature of the gas is entirely dependent upon whether or not a significant amount of cooling can take place in the gas before it reaches the surface of the accreting body. In this section an estimate is made of the cooling time scale, which divides the density-velocity parameter space into regions of cooling and no cooling. In the following, all radiative time scales will be defined as 3kT divided by the appropriate heating or cooling rate, where k is Boltzmann's constant. When the gas crosses the shock, the ions get most of the thermal energy, since they have a much shorter mean free path than the electrons. The electron-ion equilibrium time (Spitzer 1962) in the sheath is, with n = 4« , a maximum of  4  0  0  r  eq  = 50.6^-^-^  s.  (14)  10  7  This time is compared with the fall time and is plotted in Figure 2. For the postshock gas, the equilibrium time decreases with density at the same rate as the cooling time and is always shorter than it. Therefore the postshock gas comes into collisional equilibrium and the electron and ion temperatures are assumed equal. The cooling from the postshock temperature can be taken from Figure 1 of Cox and Daltabuit (1971),  3  2  F I G . 2 . — T h e c o o l i n g d i a g r a m c o n s t r u c t e d with the free stream density a n d velocity. Solid lines, regions o f c o o l i n g f o r the m a x i m u m (IO K ) a n d m i n i m u m (4n ) densities. C o o l i n g occurs to the right, i.e., higher densities, of these lines. B e l o w the d a s h e d lines (t q < '/) the electron a n d ion temperatures are e q u a l . T h e hatched region is heated to the C o m p t o n e q u i l i b r i u m t e m p e r a t u r e a n d is certainly s u b s o n i c , whereas below the d o t - d a s h e d line (/W < 1) the M a c h n u m b e r o f the i n c o m i n g gas based o n the C o m p t o n h e a t i n g rate is less t h a n 1. 4  0  e  c  82  No. 3, 1978  RADIATIVE  and the gravitational force is still directed downward, but eventually the excess energy must be lost if the gas is to be gravitationally bound to the neutron star. One way to do this would be through turbulent mixing at the boundary between the upward-flowing sheath gas and the downward-flowing accretion column. This would decrease the effective cooling time for the lower sheath by diluting the hot gas with the cooler, denser gas of the column. If a mixing process is required in order to capture gas entering the column at small radii, it would imply that, if gas at the top of the column is unable to cool, then the accretion may become very inefficient.  1045  EFFECTS  The expression used for photoionization heating applies only if heating in a static gas would be able to attain this temperature and the absorption and scattering optical depth up the column are less than 1. The static condition, log £ > 2, is equivalent to V < \.02T r ~ , which is indicated in Figure 3, and is always satisfied in the cooling region. The photoionization absorption can be estimated from the equations of ionization balance and of optical depth, B  2l3  116  e  v  Le-  w ( D =^ p ^  G  and  b) Heating  dr  When the accreting body is a neutron star, the accretion luminosity may be sufficient to cause significant heating of the infalling gas. (An additional problem is that the incoming gas stream may be heated to a sufficiently high temperature that the assumption of supersonic flow is invalidated. This is considered in the Appendix.) Approximate rates for photoionization and Compton heating are derived and used to construct a heating diagram similar to the previous cooling diagram. Optical depths must also be considered, because radiative heating will be impossible if the optical depth up the column becomes too large. The accreting object is assumed to be a neutron star of radius 10 cm. The entire kinetic energy of the infalling gas is converted into radiation at the surface shock. The resulting luminosity is  (18)  dr  where n , n , and n are the number densities of electrons, ions, and ground-state absorbers, respectively. The integrals in the exact formulae have been approximated by quantities integrated over frequency, and it is assumed that one ionic species is doing the major part of the absorbing at any given temperature. In the neighborhood of 10 K, the major absorber is O VIII. Setting / as the fraction of atoms that are absorbing X-rays, we find a solution to the above equations similar to Mestel's (1954), e  t  a  6  = i * 4Ufa(T)  6  f n  (19)  e  The bottom of the column, r , has been assumed to be the magnetosphere at a distance near 10 cm from the center of the neutron star, and it is assumed that there is no significant opacity between the source of radiation and this lower boundary. This is consistent with the magnetospheric model of Arons and Lea (1976). Using the total recombination rate and 5 keV as an average photon energy, we find that the above integral becomes b  L = 4.70 x 10 /; K - w (^/10 cm)- ergss- . 36  3  11  3  6  1  (15)  1  8  An upper limit to the gas temperature in the column due to photoionization heating is required. The calculations of Hatchett, Buff, and McCray (1976). show that, for log £ greater than 2, where £ = L/nr , the CNO elements are completely ionized. The heating will then be limited by the total recombination rate to the ground state, so the limiting photoionization heating rate is n afx/3, where n will be taken to be the density in the column from equation (11), a is the recombination rate to all levels for completely ionized oxygen, / (=10 ) is the fractional abundance of oxygen, increased slightly to allow for some carbon and nitrogen, and x/3 is the average energy deposited per ionization for a v~ spectrum. (The spectrum may not be but all spectra deposit an average energy of order x-) The recombination rate used is the expression given by Allen (1973) for the total recombination rate lo 2 3 4 to the ground level, « = 3 x 10- Z r- ' . The resulting heating time is 2  e  e  8  1  0.9897V ' ln ( r / r ^ - n F 19  4  3  u  5 8  x (R /10 cm)(//10- )(Z/8) . 3  6  x  4  (20)  -3  l  f = 22.9T 'V' «ir >V j8 15  ph  2  4  1  1  6  x (//10- )-HZ/8)- s. 3  (16)  4  Comparing this with the fall time, we find that the temperature is limited by T  6  < 1.89«  4/15 n  x (Z/8)  V (//10- ) '  I6/15  15  .  3  4  For numerical estimates, the distance dependence of the optical depth will be ignored [In (r/r„) ~ 1]. This optical depth is meant to be useful only as an indication of how radiative heating is attenuated. As it turns out, this estimate of the photoionization optical depth is always less than the electron scattering optical depth for the range of parameters plotted. A more precise calculation is required to estimate the transmitted spectrum as a function of frequency. The other major source of heating in an X-ray illuminated gas is Compton scattering. The Compton heating rate is given by Buff and McCray (1974) as -  a-^kT  m <r e  L<7  r  4-rrr  2  (21)  15  (17)  where L is the source luminosity, e is a parameter which describes the effective temperature of the spectrum,  1046  Vol. 220  CARLBERG  6  LOG  7  DENSITY  FIG. 3 . — T h e density-velocity d i a g r a m f o r the p h o t o i o n i z a t i o n heated accretion c o l u m n , with a C o m p t o n heated base. T h e c o o l i n g line (solid) a n d the s u b s o n i c line (Mc < 1) are repeated f r o m F i g . 2. F o r l o g £ > 2 the a p p r o x i m a t i o n used for the heating rate w o u l d be v a l i d f o r a static gas. A s the flow parameters cross the T > T (dotted) line, the o p t i c a l d e p t h u p the sheath exceeds that o f the c o l u m n . S  and a describes the shape of the spectrum (a = 1.04 for a blackbody and i for an exponential spectrum). In principle, a and e are determined by the physical parameters n , V , M , and the radius of the accreting object. In view of the complexity of the calculation of the emitted spectrum, we chose to leave them as parameters. As typical values we chose a = \ and <r = 5 keV. This corresponds to a Compton equilibrium temperature {kT = ae) of 2.9 x 10 K. With this choice of parameters, the Compton heating time scale is 0  0  x  7  t = 1.20 x 10 r n -V*V '"~ /3"" 9  1  6  c  1  1  11  x CR /10°cm)- s,  (22)  1  JC  C  librium, or optically thick and cold. From equation (10) we see that the column is impossible to contain for V < 0.9, and the column will widen and may even become spherical at the bottom. For electron scattering the new effective base of the column is at a distance where the Compton heating and the fall time scales are equal, B  r = 1.39/; K - («/0.5)- ( /5 keV)~ 0 . (24) 2  8  s  > 0.4767' r ' )3- m- (i? /10 )- . (23) 1  6  a  1  2  6  CS  2  x  I)  1/2  ff  11  8  8  1  (25)  2  6  1  If the base of the column becomes optically thick, the radiation will be thermalized, so that Compton heating becomes negligible as a result of the e parameter's being reduced. As discussed by Felten and Rees (1972), spectrum alteration begins when the optical depth T* = (3T T ) exceeds 1, where r and T are the free-free and electron scattering optical depths. The calculation indicates that the optical depth at a photon energy of 5 keV is always much less than 1 as long as the flow is supersonic. As one gets closer to the source of radiation, the time scale for Compton heating decreases faster than the fall time. The bottom of the accretion column may be heated to Compton equilibrium, even though the regions higher up may be in photoionization equiFF  2  £  2  c  "iiV -  2  r = 9.18« K 7V (/- /10 cm)" ' .  f  2  6  8  The spherically infalling material below this has a negligible contribution to the optical depth, because the density is reduced by the much greater column angle. The electron scattering optical depth along the column is  or t < t for c  11  c s  The actual line plotted on Figure 3 for the electron scattering opacity assumes that the effective base of the column is at the Compton heated distance of equation (25) and that the column temperature is determined by the photoionization heating temperature of equation (17). The range of photoionization heating is thus extended by the reduction of column opacity. If one computes the Alfven radius from B /8n = l/2py , one finds that in some cases the Alfven radius exceeds the Compton heated radius and will determine the effective base of the flow. But these cases turn out to be in the region of the density-velocity diagram above the cooling line and hence of little interest to the heating calculation. The column is effectively optically thin to the sideways loss of radiation because the sideways optical 2  2  No.  1047  RADIATIVE EFFECTS  3, 1978  depth of the column is dominated by electron scattering and is always less than 1 for the range of parameters plotted. The higher-energy X-rays will be attenuated by K shell absorption by elements with ionization potentials greater than the CNO elements. For a spectrum with a typical photon energy of 5 keV, the K shell cross sections of Daltabuit and Cox (1972) and the abundances of Allen (1973) suggest that the dominant K shell absorber will be silicon. The calculations of Hatchett et al. indicate that a typical silicon atom, for log £ > 2, will have several electrons left, and therefore will have a cross section of order 10" cm , which is relatively independent of temperature and radiation flux. Combining this with a fractional abundance of 3 x 10~ , the effective cross section at the absorption edge is only 4.5 times the electron scattering cross section. Photons below the edge will be less affected, primarily interacting with only lower-abundance magnesium, and for those above, the cross section decreases approximately as E~ , until another edge, due to low-abundance sulfur, is encountered. In general we expect that K absorption will be of the same magnitude as the electron scattering. Similarly, the K shell photoionization heating rate is different from Compton heating only by a multiplicative factor of order 1, and will be ignored. 19  2  heating dominates, the temperature rises to the Compton equilibrium value, and the column expands so that the infall becomes almost spherical. It is of particular interest to compare the electron scattering opacity up the column with that up the edge of the sheath in the postshock gas. The opacity up the sheath is  Evaluating this integral, and choosing (somewhat arbitrarily) the maximum extent of the column to be the distance at which the density has dropped to 4/3n , i.e., r = R l(2<j> ), gives 0  2  A  m  5  3  r  = 2.26«nK r - , 2  s  (27)  2  8  6  whereas the electron scattering opacity up a column with a Compton heated base is 7.79V^TQ' . A.S a result we find that the opacity up the sheath is greater than up the column if « K " > 3.45, a result which is independent of the column temperature. This line has been included in Figure 3. If the stellar wind in which, the neutron star is embedded has velocity and density variations, this analysis predicts potentially observable effects. The most obvious is that, if the line-of-sight optical depth is constant, the X-ray luminosity responds to variations in n V ~ (eq. [16]) on times of variation longer than about two fall times, or 500V ~ s. This variation reflects the local structure of the wind for regions of size greater than 2R (5 x 1 0 K ~ cm). Another expected effect is that, as n V ~ increases, the optical depth up the sheath will exceed that up the column. Thus, if the X-ray source is occulted by the accretion column, the X-ray absorption would change from a single dip (T > r ) to a double dip (T < T ). In addition, Jackson's calculations (1975) indicate that, if the gas fails to cool, the absorption up the sheath always dominates. 2  S  U  B  3  c) Accretion Scenarios  0  0  3  B  In Figure 3 a box has been drawn which encloses the suggested range of wind densities and velocities for stellar wind X-ray sources Cen X-3 and 3U 1700 — 37. Only a small part of this region is subsonic and beyond the description given here. For a given density and velocity, it is possible to qualitatively describe the flow. If the density and velocity parameters of the free stream lie above the cooling line, the accretion may be less efficient as pressure forces in the hot gas of the sheath become more important. This would be reflected in a diminished luminosity. In general, flows with parameters above the cooling line would broadly resemble the scenario found by Hunt (1971). Captured gas falls inward with its temperature rising adiabatically. Below the cooling line, the gas temperature drops to some equilibrium value and falls down the accretion column. Although the K absorption edges will alter the spectrum somewhat, the line above which the electron scattering opacity exceeds 1 is almost coincident with the cooling line, so that heating of the most distant parts of the accretion column will be possible below the cooling line. Typical maximum temperatures for V — 1 are T = 1 at n — 0.1 and T — 3.5 at /in = 10. The column semiangles are 2° and 9°, respectively. The gas will remain at the equilibrium temperature specified by the local radiation field, since all radiation time scales are more rapid than the fall time. Near the source Compton e  e  e  n  10  2  8  A  2  0  c  0  s  C  s  IV. THE FAR WAKE  The Reynolds number of the gas flow is extremely high (V R jv = l O ^ - / ? ! ^ - ' / ; ! ! . ) , and the far wake is expected to be turbulent. Turbulence in supersonic flows is not well understood, but experimental studies of supersonic wakes (Demetriades 1968) indicate that a phenomenological theory as outlined by Townsend (1976) provides a reasonable description of supersonic far wakes. Unfortunately, the dynamics of laboratory wakes are not dominated by a gravitational field, and therefore the applicability of the description to this case must be carefully considered. The subsonic theory is based on the observation that the wake remains self-similar with respect to a characteristic velocity and length scale. This is combined with the momentum equation from which all small terms have been dropped. The axisymmetric far wake is found to be self-similar with respect to the 12  0  A  1  1  5  2  1048  Vol. 220  CARLBERG  half-width, /, and the turbulent velocity scale, u, denned by I  \  (±\ (_r_Y' 113  to the gas pressure at the center gives the central density »„ =  3  x  1.06  10  1 1  n  ro "%- ' '-i2~ ' . 1  1 1  1  3  1  (30)  6  Note that this density is greater than that which would be found by using the external static pressure by a factor of ^f- = 376.07-4- J V ' r 3  1  where the so-called momentum radius R (the radius such that the drag force is 1 /2pV TrR ) has been taken to be R . R is the turbulent Reynolds number, observed by Demetriades to be 12.8. The width scale of the wake implies that the small angle approximations for the density and transverse velocity apply for the exterior supersonic flow. Hence the effective exterior pressure will be the transverse momentum flux, which varies with distance along the wake. But the equations of momentum used to derive the length and velocity scale assumed that there was no pressure gradient in the free stream. For the gravitational wake, an order-of-magnitude estimate of the pressure gradient term in the small angle approximation-finds that it is of order u /l(R /l), whereas the retained terms in the momentum equation are of order u jl. Consequently, for R « I, the pressure gradient term can again be dropped. Using the half-width of equation (28), we find that R /1 is of order (RJr) . Thus the equations are consistent for r » R , but the crucial observation that a gravitational wake is selfsimilar is unavailable. Experiments also indicate that the flow may not be self-similar for distances of several tens of the momentum radius, but the deviation of the turbulent velocity scale from the self-similar value is a factor of 2 or less. For sufficiently low Reynolds numbers, part of the wake may be in laminar flow, ln this case the velocity defect on the axis is u/V = 3/2R /r, and the halfwidth varies as (rR ) (Lamb 1924). The Reynolds number grows with distance, and the flow will eventually become turbulent. With the extreme Reynolds numbers present here, the wake is expected to become turbulent within the sheath of the accretion column. If the gas in the wake has no energy losses, i.e., 5RT + \v constant, the temperature on the axis is found to be  5 / 6  i a  •  (31)  M  2  2  0  A  M  T  The temperature estimate and density estimate of equations (30) and (31) assume that the wake is isoenergetic, but at these temperatures and densities, radiation cooling can be significant. Time scales of interest are the cooling time (where A is the cooling coefficient) 12.4« " K 1  1 X  5 / 3 8  /- -  1 / 2  1 2  (10-  ergs cm s"7A) s,  2 2  (32)  3  turbulent dissipation of kinetic energy time scale, 3RT/(u /l) = 3.04 x 1 0 K 4  3  1/3 8  m- ' r 2  3  5 12  ' s,  (33)  3  2  A  and the turbulent time scale, 2.40 x 10 K - / - m ' s .  2  3  A  113  A  A  0  A  xlz  A  2  n  ^  =  = 3.55 x }0 V m r - ' e  2l3  e  2l3  2  l2  3  K, (29)  where T is the temperature of the gas external to the wake. This temperature implies that the turbulent velocity scale is subsonic. The temperature becomes equal to 2T at R^Volc^, where R is the Bondi radius, 10 (10 K./T) cm, and c is the sound speed at Tm. Experimentally it is observed (McCarthy and Kubota 1964) that the pressure is approximately constant across the wake. Equating pv' at the wake boundary K  m  !4  B  4  x  2  2  (34)  3  12  These time scales imply that, for large distances, cooling removes most of the thermal energy from the wake. If the sound speed within the wake drops below the turbulent velocity scale, the turbulence would then become supersonic, leading to shocks which rapidly heat the gas, but the shocks would occur on the basic turbulence time scale, and would not be able to reheat the bulk of the gas. One might speculate that the temperature would decline to the minimum of 10 K, but with an extremely clumpy-distribution. If cooling is complete, the simple model used, which does not consider the energy budget, may break down completely. Its value lies in the fact that, as the gas cools, the Reynolds number becomes even greater, and the dynamics of the gas flow in the far wake are almost certainly dominated by turbulence. One can combine the density and width to show a column density across the wake of sufficient size to produce optical absorption of radiation from the primary. That is n 2l = 2 x 10 c m for an optical depth in the wake of one at Ha, assuming the lower level is populated by recombinations at 10 K and depopulated by radiative transitions. This would be possible whenever the wake was silhouetted against the primary star. But these simple considerations fall well short of the ability to reproduce line profiles as seen by Conti and Cowley (1975). The wake will remain cold in the presence of an X-ray source for L/nr < 10, or distances from the X-ray source of r > 3L37 ' /;!! . The absorption cross section for X-rays by a cold gas of cosmic abundances is about 1 0 ( £ ' / k e V ) - cm . Again the column densities are adequate for X-ray absorption, but the absorption would be very sensitive to the inclination of the wake 4  33  2  T-T  1  8  - 5  4  2  1 2  1 2  -22  112  3  2  86  No. 3, 1978  RADIATIVE EFFECTS  with respect to the X-ray star, since the far wake is very narrow. V. CONCLUSIONS  1049  ture, causing the pressure to rise sufficiently that the base of the column will spread to a broad inflow. It is predicted that the electron scattering will cause the X-ray light curve absorptions to change from single dips to double dips as n V ~ increases, if the parameters are in the cooling region. The gas that is gravitationally perturbed but does not become bound to the neutron star forms the far wake. The high velocity and low viscosity indicate that the far wake is almost certainly turbulent. An extension of the similarity description of supersonic wakes experimentally studied provides the temperature and density in the wake. But the cooling time of the gas in the wake is then found to be less than the basic turbulence time scale, which may mean that whole description is invalid. In spite of this, we suggest that the far wake is composed of a hot gas entering the wake and denser clumps of cold gas, a description which is marginally consistent with the "wake" observations of Conti and Cowley (1975). The model outlined here is intended to be useful for providing qualitative insight into the physics of supersonic accretion. The numerical quantities employed are expected to be accurate to a factor of 3 or so, and should provide basic regimes which can be further explored with a numerical model. 2  The supersonic accretion of gas onto a neutron star has been described, working from the basic model as shown in Figure 1; the main features are the sheath and the accretion column. The angular width of the column, a measurable quantity in the X-ray light curve, is found to depend on the ratio of V to the column temperature, and therefore yields information about the local wind velocity provided the column temperature can be specified. An accurate estimate of the temperature would require a hydrodynamic calculation including radiation transfer, but upper limits to the temperature can be obtained by estimating the relevant heating and cooling rates. !The most important consideration in determining the thermal state of the gas is whether or not the gas can cool before it falls all the way down the accretion column. The cooling line of Figure 2 separates the flow into two main regimes. If the postshock gas in the sheath is unable to cool, it will fall inward adiabatically in a wide accretion column, with the accretion efficiency (the /? factor) reduced by the thermal pressure. Below the cooling line of Figure 2, the gas will cool to an equilibrium value determined by the radiation field. In the region of the density and velocity parameters which apply to the stellar wind X-ray binaries, this means that the upper part of the accretion column will be photoionization heated to temperatures of order 10° K. The base of the column will be heated to the Compton equilibrium tempera2  0  0  G. G. Fahlman provided invaluable advice and criticism, and read several rough drafts of this paper during the course of this research. General encouragement and useful discussions were provided by members of the UBC Institute of Astronomy and Space Science and the Dominion Astrophysical Observatory.  APPENDIX In the case of a heated gas, the thermal pressure forces may become large enough to destroy the assumption that the flow is dominated by inertial forces. In this Appendix the region of validity of the supersonic description of accretion is examined. The incoming free stream may be heated such that the Mach number, M = V /{2yRT) ' , becomes less than I. The maximum temperature which can be produced by Compton heating is 2.9 x 10 (a/0.5) x (e/5 keV). This temperature can be attained for « F " > 13.8, which is shown as the crosshatched area in Figure 2. This gives only the area for which subsonic flow is guaranteed in the presence of Compton heating, but what is really wanted is a line on which the Mach number is equal to 1. If only Compton heating is considered, we find that (nonequilibrium) temperatures are produced such that the Mach number is less than 1 for ' ? 8 ~ > 17.2. This line (M < 1) is shown in Figure 2. If photoionization heating is included, the subsonic region is increased very slightly at low velocities. We conclude that most of the box of Figure 3 is indeed in supersonic flow. The deviation of streamlines from particle trajectories will depend on the ratio of pressure forces to inertial forces. As a worst case we assume that the gas 1  7  Z  8  -  n  C  /  1^ dp /GM  _ RT  /GM  0  p 8r  (Al)  SRT  2  0  U  is fully Compton heated. The ratio of radial pressure force to gravitational force for gas outside the shock is  4  This implies that the net force is outward for r > 5R , where the thermal radius is R = GM/SRT = 3.19 x 10 7' cm. Similarly, in the transverse direction, the ratio of pressure to the momentum flux is T  T  10  0  _1  7  r 5%.  (A2)  In the shock-heated sheath, the ratio of radial pressure force to gravitational force is 9/16, which will act only to reduce the effective mass of the gravitating object in the sheath. In the column the ratio of pressure forces to gravitation is, for a constant temperature,  3r/5R . T  In general the pressure forces can be safely ignored, even in the presence of strong heating, provided that we remain in the area of validity of the supersonic flow assumption.  1050  CARLBERG REFERENCES  A l l e n , C . W . 1973, Astrophysical Quantities (3d e d . ; L o n d o n : A t h l o n e Press). A r o n s , J . , a n d L e a , S. M . 1976, Ap. J., 210, 792. B o n d i , H . , a n d H o y l e , F. 1944, M.N.R.A.S., 104, 273. Buff, J . , a n d M c C r a y , R. 1974, Ap. J., 189, 147. C o n t i , P., a n d C o w l e y , A . P. 1975. Ap. J., 200, 133. C o x , D . P., a n d D a l t a b u i t , E . 1 9 7 i , Ap. J., 167, 113. D a c h s , J . 1976, Astr. Ap., 47, 19. D a l t a b u i t , E., a n d C o x , D. P. 1972, Ap. J., Ill, 855. D a n b y , J . M . A . , a n d C a m m , G . L., 1957, M.N.R.A.S.,  111,  50. D a v i d s o n , K., a n d O s t r i k e r , J . P. 1973, Ap. J., 179, 585. D e m e t r i a d e s , A . 1968, AIAA J., 6, 432. Eadie, G., Peacock, A., Pounds, K. A., Watson, M . , Jackson, J . C , a n d H u n t , R. 1975, M.N.R.A.S. Short Comm., Ill, 35p. F e l t e n , J . E., a n d Rees, M . J . 1972, Astr. Ap., 17, 226. H a t c h e t t , S., Buff, J . , a n d M c C r a y , R. 1976, Ap. J., 206, 847. H o y l e , F., a n d L y t t l e t o n , R. A . 1939, Proc. Cambridge Phil. Soc., 35, 405.  H u n t , R. 1971, M.N.R.A.S., 154, 141. I l l a r i o n o v , A . F., a n d S u n y a e v , R. A . 1975, Astr. Ap., 39, 185. J a c k s o n , J . C . 1975, M.N.R.A.S., 483. Jones, C , F o r m a n , W., T a n a n b a u m , H . , Schreier, E., G u r s k y , H . , K e l l o g g , E., a n d G i a c c o n i , R. 1973, Ap. J. (Letters), 181, L 4 3 . L a m b , H . 1924, Hydrodynamics (5th e d . ; C a m b r i d g e : C a m bridge U n i v e r s i t y Press). M c C a r t h y , J . F., a n d K u b o t a , T . 1964, AIAA J., 2, 629. M c C r a y , R., a n d H a t c h e t t , S. 1975, Ap. J., 199, 196. M e s t e l , L. 1954, M.N.R.A.S., 114, 437. P o u n d s , K . A . , C o o k e , B. A . , R i c k e t t s , M . J . , T u r n e r , M . J . , a n d E l v i s , M . 1975, M.N.R.A.S., 172, 473. R u d e r m a n , M . A . , a n d Spiegel, E . A . 1971, Ap. J., 165, 1. Spitzer, L . 1962, Physics of Fully Ionized Gases ( N e w Y o r k : Interscience). T o w n s e n d , A . A . 1976, 77;e Structure of Turbulent Shear Flow (2d e d . ; C a m b r i d g e : C a m b r i d g e U n i v e r s i t y Press).  112,  j  R. G . CARLBERG: Department of Geophysics and Astronomy, University of British Columbia, Vancouver, B.C., Canada, V6T 1W5  88  APPENDIX 2. GAS  PHYSICS  P hot c i o n i z a t ion The  simplest  process  to  d e s c r i b e i s the  r a t e which i s independent of the gas and  i s simply  given  photoionization  temperature  and  density,  by hi/  where  tf"cj  i s the  zation level J  was  p h o t o i o n i z a t i o n c r o s s - s e c t i o n of atom i , i o n i -  j . For these c a l c u l a t i o n s the mean r a d i a t i o n  taken simply  allowing  for  as the f l u x energing  geometrical  LTE  50,000 K,  in  terms  make  numerical  of the emergent f l u x ,  d i l u t i o n , H= 1/2 ( 1 - ( 1 - ( r ^ / r ) ) / ) , 2  heating  l  2  and  calculations.  assumed to vary as  the  flux  Since  the  (r*/r) , 2  the  0  to the  surface  (fTy).  VI  i o n i s of p a r t i c u l a r importance the  t o i o n i z a t i o n r a t e s of a l l i o n s up to comparable  ionization  phopo-  (IP of 0 V i s 113.9eV) were i n c l u d e d . p h o t o i o n i z a t i o n c r o s s s e c t i o n s used i n the c a l c u l a t i o n s  are given below. The  Salpeter  as  To  i s the s t e l l a r r a d i u s . For the computations i l l u s t r a t e d  of the s t a r , 4<J>/ = 2  The  Non-  F , whereas the mean i n t e n s i t y  here, a l l done a t a r a d i a t i o n f i e l d corresponding  The  the  l o g g = 4 model. Mihalas gives the r a d i a t i o n f i e l d  t h i s change, the mean i n t e n s i t y was  tentials  computations  (1972), s p e c i f i c a l l y  i s needed f o r the i o n i z a t i o n and  where r  field  from a model atmosphere  d i l u t i o n . The  used a model computed by Hihalas  4 J  ( A l l . 1)  Hydrogen (1957).  cross s e c t i o n s used are given i n below. cross  section  was  taken  from  Bethe  and  89  2 * r r * AO* l  ^  77c)  e^p  (-  ^  <*><c°*r\)  ( A l l . 2) where  «C i s the f i n e s t r u c t u r e c o n s t a n t a.  i s the Bohr r a d i u s  a  CH. i s the Hydberg  Z i s the i o n charge The Helium I c r o s s s e c t i o n was obtained from Brown  (1971).  The formula quoted by him was m u l t i p l i e d by 16 t o agree with h i s numerical  values,  and  a  f a c t o r of 2  was i n c l u d e d  p o n e n t i a l f a c t o r to reduce t o t h e Hydrogen  formula.  i n the exThe  cross  section i s  (All.3) where  *  ^  and are  l/j  =  W/>  k 1^  w i t  h  a l l ^ ' s r e p l a c e d by ji's. The c o n s t a n t s  *=2. 182846 0 = 1.188914 Z 2 r  Z =1 b  k=( (hV-24. 587e¥)/13.59 8eV) V  2  The Helium I I c r o s s s e c t i o n i s t h e same as the p h c t c i c r i z a ticn  c r o s s s e c t i o n of Hydrogen but with Z=2 everywhere.  90  For the remaining i o n s the c r o s s s e c t i o n s have been lated  by  various  authors  calcu-  using the p r i n c i p l e s of guantum me-  c h a n i c s , and then making a f i t t o a standard polynomial to resent  the  data  as  rep-  a f u n c t i o n of i n c i d e n t photon energy.  forms f o r the polynomial  are used  Two  here, one due to Seatcn (1958)  ( A l l . 4) where i ^ , i s the t h r e s h o l d freguency and the other c o n s t a n t s fitting  parameters.  The  are  other form i s a s l i g h t l y more g e n e r a l  polynomial due t o Chapman and Henry (1971)  </»-*<)fej"*"V(i*-v»^p 7. i  *•<.»)*  (All.5)  Icr  TABLE 6: PHOTOICNIZiTICN CBOSS SECTION PARAMETERS s otft Beference ev 10- cm* l8  C C C C  H70 H70 SB 71 SE71  11.26 24.383 47.887 64.492  12. 19 4.6 1.84 0.713  2.0 3.0 2. 6 2.2  2.7  N I N II N III N IV N V  14.534 29.601 47.448 77.472 97.89  11.42 6.65 2.06 1.08 0.48  2.0 3.0 1.62 3.0 2.0  4.287 2.86 3.0 2.6 1.0  —  0 I II III IV V VI  13.618 16.943 18.635 35.117 54.943 77.413 113.90 138,2  2.94 3.85 2.26 7.32 3.65 1.27 0.78 0.36  1.0 1.5 1.5 2.5 3.0 3.0 3.0 2. 1  2.661 4.378 4.311 3.837 2.014 0.831 2.6 1.0  --—  H70 H70 H70 F68 F68  Ne I Ne I I  21.564 40.962  5. 35 4, 16  1.0 1. 5  3.769 2.717  —— —  H70 H70  0 0 0 0 0  3. 17 1.95 3. 0  --  I II III IV  —  --  — — —  H70 H70 H70 F68 F6 8  —  H70  —  91  44.166 47.874 Ne I I I 63.45 68.53 71. 16 97.11 Me IV Ne V 126.21 Ne VI 157.93  2.71 0.52 1.89 2.50 1.48 3.11 1.40 0.49  1.5 1.5 2.0 2.5 2.5 3.0 3.0 3.0  2.148 2.126 2.277  —  H70  2.225 1.963 1.471 1.145  ----  H70 H70 H70  Mg Mg Mg Mg Mg  I II III IV V  9.92 3.416 5.2 3,83 2.53  1.8 1.0 2.65 2.0 2.3  2.3 2.0 2.65 1.0 1.0  — — — —  I s o To S i I I I I s o Io S i IV S58 S58 S58  Si  I  7.37 8. 151 Si I I 16.345 S i I I I 33.492 S i IV 45.141  •12.32 •25.17 2.65 2.48 0.854  3 5 3.0 1.8 1.0  6.459 4,420 0.6 2.3 2.0  5.142 8.943 -— —  SB71 SB71 SB71  S I  12.62 19.08 12.70 8.2 • .35 • .24 0.29 0.62 0.214  3.0 2. 5 3.0 1.5 2.0 2.0 2.0 1.8 1.0  21.595  3.062  CH72  7.646 15.035 80.143 109.31 141.27  10;360 12.206 13.40 8 23.33 33.46 34.83 47.30 72.68 88.05  S II S III S IV S V S VI  * means t h a t weights  the  cross  The  references  H70: Henry S58:  Seatcn  SB71: S i l k  coded  along  above a r e :  1958 1971  CH71  Chapman and Henry  1971  CH71  Chapman and Henry  1972  F68:  Flower  1968.  1.159 1.6-95 10.056 16.427 6.837 2.3 2.0  weighted  CH71  5.635 4.734 -2. 236 -3.278 0.592 4.459 -—  by  CH72 CH72 CH72 I s o To S i I I I I s o To S i IV  the  statistical  t r a n s i t i o n s . The a b b r e v i a t i o n  an i s o e l e c t r o n i c s e g u e n c e .  1970  and Brown  0.135  section  of the f i n e s t r u c t u r e  means e x t r a p o l a t i o n  2.346  Iso  92  The Jecombination The  Bate  recombination  r a t e f o r Hydrogen was  e x p r e s s i o n given by Johnson  calculated  usirg  (1972) which has a c o r r e c t i o n  b u i l t i n allowing for f i n i t e  d e n s i t y . The  an  factor  rate to l e v e l n i s •z  S  (c»,n) = D < I * / k T ) 3 / 2 e x p ( W k T ) £  g^(n)x-^'  c-o  E Above the l e v e l n  w  <x I*/kT)  ( A l l . 6)  0  the p o p u l a t i o n s can be assumed to be i n e g u i -  l i b r i u m with the continuum, t h a t i s the p o p u l a t i o n s Saha  e g u i l i b r i u o . The value of n  sion y\  given by Jordan Z. n  b  Vie  i s calculated  are  as  from an  in  expres-  (1969)  1 7  ( A l l . 7) The  value of x, i s d e f i n e d t o be  5. 197x10-** cm . 2  and  the g  The  1-(n/n„) . 2  functions E «"  a r e  t  n  e  The  constant  exponential i n t e g r a l s ,  (n) are Gaunt f a c t o r c o e f f i c i e n t s , determined  i n the f o l l o w i n g  is  as shown  table.  TABLE 7: GAONI FACTOBS n=1  n=2  n>2  g (n)  1. 1330  1.0785  0. 9935 + . 25 28n~  g, (JJ)  -0.4059  -0.2319  -n~» {.6282-. 5598n~*• . 52 99n-*)  g^(n)  .07014  . 02947  n~* (.3887-1. 181n-*+1.470n-z)  0  The  values i n t h i s t a b l e were taken  , 1296n~^  from Johnson  In order to o b t a i n the t o t a l recombination t i o n s to the l e v e l s n=1  cf  interest  have  rate,  recombina-  to 9 summed t o g e t h e r .  Computations of the recombination ions  (1972).  been  r a t e f o r a l l of the other  made by A l d r o v a n d i and  Peguignot  93  (1973), with e r r a t a i n A l d r o v a n d i and Peguignot  (1976). The data  i s provided i n the form of f i t s t o simple f u n c t i o n s . The  radia-  t i v e r a t e i s given by *= r  *r»el <T/10* K)~\.  and the d i e l e c t r o n i c recombination  r a t e by  A ^ T - 3 / e x p (-T /T)|; l4Bc(i€Xp (-T,/T) 1 .  (AIT.9)  v a r i o u s c o n s t a n t s used are given i n the accompanying  table.  oL^-= The  (All.8)  2  c  The range of v a l i d i t y o f t h e f i t s Tcrit  g i v e s the temperature  are  Tmax/1000  <  T  above which d i e l e c t r o n i c  <  Tmax.  recombina-  t i o n i s important.  ATOM HE C C C C C N N N N N N 0 C 0 0 0 G  0 NE NE NE NE NE NE NE NE NE MG MG MG  I I II III IV V I II III IV V VI I II III IV V VI VII I II III IV V VI VII VIII IX I II III  TABLE 8: RECOMBINATION FIT CONSTANTS ABAC ADI ETA TMAX TCBIT TO 4. 3E-13 4.7E- 13 2.3E- 12 3.21- 12 7.5E- 12 1.71- 11 4. 1E-13 2.2E- 12 5.0E- 12 6.5E- 12 1.5E- 11 2.91- 11 3. 1E-13 2.0E- 12 5. 1E-12 9.6E- 12 1.2E- 11 2. 3E-11 4. 1E- 11 2.2E- 13 1. 5E-12 4.41- 12 9. 1E- 12 1.51- 11 2.3E- 11 2.8E- 11 5.0E- 11 8.6E- 11 1.4E- 13 8. 8E-13 3.5E- 12  .672 . 624 ,645 .770 .817 .721 .608 .639 .676 .743 .8 50 .750 .678 .646 .666 .670 .779 . 802 .742 .759 .693 .675 .668 .684 .704 .771 .832 .769 .855 .838 .734  1E5 514 1E5 3E5 1E6 3E6 1E5 115 3E5 315 3E6 117 5E4 215 5E5 116 6E5 3E6 1E7 115 1E5 215 3E5 615 1E6 1E6 6E6 317 3E4 115 3E5  5. 0E4 1.2E4 1.2E4 1. 1E4 4.4E5 7. 0E5 1.8E4 1. 8E4 2.4E4 1. 5E4 6.8E5 1.0E6 2.7E4 2. 2E4 2.4E4 2. 5E4 1.6E4 1.0E6 1.5E6 3. 0E4 3. 3E4 3. 3E4 3.5E4 3. 6E4 3.6E4 2. 9E4 1. 5E6 3. 816 4.0E3 7. 4E4 6.6E4  1. 9E-3 6. 9E- 4 7.0E- 3 3. 8E-3 4. 8E-2 4, 8E-2 5. 2E-4 1. 7E-3 1. 2E-2 5. 5E- 3 7.6E- 2 6. 6E- 2 1.4E- 3 1. 4E-3 2. 8E-3 1.7E- 2 7. 1E-3 1. 1E-1 8. 6E-2 1. 3E-3 3. 1E-3 7. 5E- 3 5.7E- 3 1.0E- 2 4. 0E-2 i . 1E- 2 1.8E-1 1. 3E-1 1. 7E-•3 3. 5E-3 3. 9E-•3  4.7E5 1. 1E5 1.515 9. 1E4 3.4E6 4. 1E6 1.3E5 1.415 1.8E5 1. 1E5 4.7E6 5.4E6 1.7E5 1.7E5 1.815 2.2E5 1.3E5 6.2E6 7.016 3.115 2.915 2.6ES 2.415 2.4E5 2.9E5 1.7E5 9. 8 16 1.1 E7 5.114 6.1E5 4.415  BDI T1 0.3 9.414 3.0 4.914 0. 5 2. 315 2.0 3.7E5 0.2 5. 115 0.2 7.6E5 3. 8 4.814 4.1 6.8E4 1.4 3. 8E5 3.0 5.915 0.2 7. 215 0.2 9.8E5 2.5 1.3E5 3.3 5.8E4 6. 0 9. 114 2.0 5.9E5 3.2 8.0E5 C.2 9.5E5 0. 2 1.316 1.9 1.5E5 0.6 1.715 0.7 4.515 4. 3 1.715 4.8 3.515 1.6 1. 116 5.C 1 .3E6 0. 2 1.416 0.2 2.6E6 0.0 0.0 0.0 0.0 3.0 4. 1E5  9a  MG MG MG MG MG MG MG MG SI SI SI SI SI SI SI SI SI SI SI SI SI S s s s s s s s s s s s s s s  IV V VI VII VIII IX X XI I II III IV V VI VII V III IX X XI XII XIII I II III IV V VI VII VIII IX X XI XII XIII XIV XV  7.71- 12 1. 4E-11 2.3E- 11 3.2E- 11 4.6E- 11 5.8E- 11 9. 1E-11 1. SB- 10 'S. 9 E- 13 1. OE- 12 3.7E- 12 5. 5E-12 1.21- 11 2. 1E-11 3.0E- 11 4.3E- 11 5.8E- 11 7.7E- 11 1.2E- 10 1. 5E- 10 2. 1E- 10 4.1E- 13 1.8E- 12 2.7E- 12 5.7E- 12 1.2E- 11 1.7E- 11 2.7E- 11 4.0E- 11 5.5E- 11 7. 4E-11 9.2E- 11 1. 4E- 10 1.7E- 10 2. 5E-10 3.3E- 10  .718 .716 .695 .6 91 .711 .804 . 830 .779 .601 .786 .693 .821 .735 .716 .702 .6 88 .703 .714 .855 .831 .765 .630 .6 86 .745 .755 .701 .849 .733 .696 .711 .716 .714 .7 55 .832 .852 .783  A number o f s m a l l approximations dielectronic The based field  515 1E6 1E6 1E6 2E6 3E6 117 5E7 3E4 1E5 2E5 3E5 615 1E6 1E6 2E6 2E6 3E6 117 3E7 517 3E4 115 2E5 3E5 5E5 1E6 1E6 216 2E6 316 5E6 6E6 1E7 118 2E8  5. 5E4 4.4E4 4. 5E4 4.5E4 5. 014 3.4E4 2. 4E6 4.0E6 1. 114 1. 1E4 1. 1E4 1.7E5 9.514 8. 0E4 7. 4E4 6. 8E4 6.6E4 6.5E4 4. 514 3.7E6 6. 316 2.2E4 1. 2E4 1.4E4 1.5E4 1.4E4 2.9E5 1.3E5 1. 115 9. 0E4 9.0E4 9.0E4 8. 3E4 6.0E4 5. 0E6 9. 0E6  in  order  to  smooth  3.2 3. 2 6.7 4.4 3.5 10.0 0.2 0.2 CG 0.0 0.0 0.0 10.0 4.0 8. 0 6. 3 6.0 5.0 10.5 0. 2 0.2 0. 0 2.5 6.0 0,0 0.0 0.0 22.0 6.4 13.0 6. 8 6. 3 4.1 12.0 0.2 0. 2  8.715 1.016 5.4E5 3.615 1.616 2. 116 2.416 3.5E6 0.0 0.0 0.0 0.0 1.0E6 1.3E6 1.7E6 6.015 1.116 2.516 2.816 3. 1E6 4.4E6 0.0 8. 8E4 1.515 0.0 0.0 0.0 1. 8 E6 2.0E6 2. 316 1.216 1.316 3.416 3.616 4.616 5.516  limits  of  the  t h e t u r n on t r a n s i t i o n f o r  recombination.  the assumption  present,  environment  3.9E5 3.4E5 3. 1E5 3.115 3.6E5 2.115 1.4E7 1.517 1.1E5 1.215 1.0 E5 1.216 5.5E5 4.915 4.2E5 3.8E5 3.7E5 4.2E5 2.5E5 1.917 2.0E7 1. 115 1.215 1.315 1.8E5 1.515 1.9E6 6.7E5 5.9E5 5.515 4.7E5 4.215 5.0E5 3.015 2.4E7 2.517  c h a n g e s have been made i n t h e  d i e l e c t r o n i c recombination  on  9. 3E-3 1.5E- 2 1.2E- 2 1. 4E-2 3. 8E-2 1.4E- 2 2. 6E-1 1.7E- 1 6. 2E-3 1.4E- 2 1. 1E-2 1. 4E-2 7. 8E-3 1.6E- 2 2. 3E-2 1..1E-2 1. 1E-2 4. 8E-2 1. 8E-2 3.4E- 1 2. 1E-1 7.3E- 5 4. 9E-3 9. 1E-3 4. 3E-2 2. 5E-2 3. 1E-2 1. 3E-2 2. 1E-2 3. 5E-2 3. OE- 2 3. 1E-2 6. 3E-2 2. 3E-2 4.2E- 1 2. 5E-1  rates  computed  o f a low d e n s i t y gas w i t h  whereas t h e e n v e l o p e o f a s t e l l a r  of moderately will  effect  high  density  and  above no  radiation  wind s t a r  strong  were  i s an  radiation  field  which  the r a t e . D i e l e c t r o n i c recombination  occurs  when a f r e e e l e c t r o n e x c i t e s a bound e l e c t r o n t c a  higher  95  energy l e v e l , thereby energy  a l l o w i n g the f r e e e l e c t r o n t o l o s e  enough  t o become bound i n t o a very high guantum l e v e l . The atom  then can s t a b i l i z e by a s e r i e s of cascades  of the two  electrons  to the ground s t a t e . S c h e m a t i c a l l y t h i s process i s S(n)  + !  +e-5*J  + l  -'in',n")  A**-* (n,n«») + h ^ A * * - i (n,n »•) +h v  —> *  1  where  in  general  n* = n+1  t  and n'*>>n, n  I  f  s u f f i c i e n t l y dense or i f the r a d i a t i o n f i e l d electron  i n the high l y i n g guantum l e v e l n*  lisicnally  or r a d i a t i v e l y i o n i z e d  was  devised  t r o n i c recombination  to  strong enough, 1  the  can be e i t h e r c c l -  out o f the atom before i t has  time t o s t a b i l i z e by photoemission. factcr  the gas becomes  A rough e m p i r i c a l c o r r e c t i o n  allow f o r t h i s decrease  i n the d i e l e c -  rate.  The p r i n c i p a l guantum number  of the s t a t e a t which h a l f the  captured e l e c t r o n s are s t a b i l i z e d by cascades  to ground and h a l f  are r e i o n i z e d i s given by, 1(collisions)=<1.4x10 2 T / /ne) / l s  6  l  2  l  1 ( r a d i a t i v e ) -Z (3 flUn (1) / (HkT  r < w  Dupree (1968)  7  j)) V  2  Sunyaev and V a i n s t e i n (1968) where W i s the g e o m e t r i c a l  dilution  factor  field  blackbcdy  of temperature Trad*  approximated  by  a  of  the  numbers can be c a l i b r a t e d a g a i n s t the depression of bination  rate  calculated  radiation  the  These recom-  by Summers (1974). I t was found  that  data i s roughly f i t t e d by the m u l t i p l i c a t i v e f a c t o r f , such t h a t ^=t  *^(n=0,W=0), where f i s f = exp[-2. 303* (. C15*a2+,092*a) ]  where  a =  12.55-7*log10(1).  96  That i s , the adjusted recombination r a t e i s found by m u l t i p l y i n g the  value found from the f i t s given by A l d r o v a n d i and  Peguignot  times the f f a c t o r given above. In the  addition  to  this correction  to the d i e l e c t r o n i c r a t e ,  semicoronal approximation o f Wilson  (1962) has been used  to  add t c the r a d i a t i v e r a t e . T h i s a l l o w s f o r some recombinaticn t o upper  levels, 1.8X10-A» "&j(kT)-a/* Y c j  =  where  Xcjl ) ~ ^ / l 1  bination  makes  2  ( A l l . 10)  ( c o l l i s i o n s ) . In a d d i t i o n three body  significant  and i s simply approximated  recom-  c o n t r i b u t i o n s a t low temperatures,  by (Burgess and Summers 1976)  ^^=1,16xl0-«J3fV fle,  (All.11)  2  where J i s the charge o f the i o n . ,  Collisional The  Ionization  rate  of  taker from Johnson  c o l l i s i o n a l i o n i z a t i o n f o r Hydrogen was a l s o (1972) as  (All.12) where "  ^  Zfftr  vL  = %I^/kT  z' |  v  t  o  i*3  = x (I^/kT+r^) 0  (t) = E (t)-2E, ( t ) * E ^ ( t ) , 0  V  *  97  and the Gaunt f a c t o r s and x in  Hydrogen.  are as f o r the  recombination  Only i o n i z a t i o n from the ground  c o n s i d e r e d , so r|=0.45 and b =  s t a t e n=1  rate  will  be  -0.603.  (  A l l other i o n s have c c l l i s i o n a l  i o n i z a t i o n r a t e s based  an approximation i n v e s t i g a t e d i n d e t a i l by l o t z  upon  (1967). A s l i g h t  m o d i f i c a t i o n to the o r i g i n a l formula has been made by McWhirtier (1975) t o allow f o r the decrease of the i o n i z a t i o n r a t e at temperatures.  The r a t e i s given  high  by  z.  (All.13) where  s  goes from  1 to 2 i n the c a l c u l a t i o n s here, n (s)  number of e l e c t r o n s i n the s u b s h e l l , and  %••{ 1)  is  the  i s the normal  J  ionization  p o t e n t i a l as given by A l l e n  (1973),  y '(2) i s ")(''(1) t  plus the e x c i t a t i o n energy of the lowest e x c i t e d new  ion  guration -1  of  1s 2s 2p 2  can  2  2  or  the  2  tentials values  52315 cm-*  to C I I I 2s2p. The ( i n eV) and were  obtained  by removing can  take  by removing  2  c t r o n s . The energy  2  ionization  i o n i z e t o C I I I 1s 2s2p,  2s  in  the  the  an  electronic  confi-  proceed with the a d d i t i o n c f 1S6659  of energy t o C I I I 1 s 2 s  electron,  III  level  with one of the i n n e r s h e l l e l e c t r o n s removed. For i n -  stance, the i o n i z a t i o n of C I I which has  cm  t  the one  outer  196659*52315 c a r  shell and  -1  one of the two s s h e l l  ele-  i s simply the energy t o go from attached t a b l e g i v e s i o n i z a t i o n  number  of  C po-  subshell  e l e c t r o n s . , The  from the t a b l e s of l o t z  (1967) and Moore  (1949) * TABLE 9: IONIZATION POTENTIAL AND ATOM IP1 N1 IP2 N2 H I 13.598 1 HE I 2 4.587 2  SHELL ELECTBON POPULATIONS  98  HE C C c  c c  c N N N N N N  N 0 0 0 0 0 0 0 0 NE NE NE NE NE NE NE ME NE NE MG MG MG MG MG MG MG MG MG MG MG MG  SI SI SI SI SI SI SI SI SI SI SI SI  II I II III IV V VI I II III IV V VI VII I II III IV V VI VII VIII I II III IV V VI VII VIII IX X I II III IV V VI VII VIII IX X XI XII I II III IV V VI VII VIII IX X XI XII  54.416 11.260 24.383 47.887 64.492 392.08 489.98 14.5 34 29.601 47.448 77.472 S7.89 552.06 667.03 13.618 35. 117 54.934 77.413 113.90 138.12 739.32 871.39 21. 564 40.962 63.45 97.11 126.21 157.93 207.26 239.09 1195.8 1362.2 7.646 15.035 80.143 109.31 141.27 186.51 224. 95 265.92 328.0 367.5 176 1.8 196 3. 8. 151 16.345 33.492 45. 141 166.77 20 5.08 246.49 303. 16 351. 1 401. 4 476. 1 523.  1 2 16.6 1 30.9 2 323. 1 342. 2 1 3 20.3 2 36.7 1 55.8 2 469. 1 492. 2 1 4 28.5 3 42. 6 2 63.8 1 87.6 2 642. 1 66 9. 2 1 6 48.5 5 66.4 4 86. 2 3 108. 2 139. 1 172. 21072. 11106. 2 1 2 60.420 1 67.809 6 118.768 C 14 4.42 -/ 4 17 2. 0 1 3 201.22 2 241.14 1 283.38 21680.4 11719.8 2 1 2 13.616 1 22.870 2 137,709 1 149. 358 6 217. 170 5 250.48 4 285.26 3 321.76 2 371. 2 1 422.4 22340.8 12388.  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  2 6 6 2 2 2 2 2 2 2 2 2 2 6 6 2 2 2  2 2 2 2 2  99  SI XIII 2438. SI XIV 2673. S I 10.360 s II 23. 33 s III 34.83 IV s 47.30 72.68 S V VI s 68.05 s VII 280.01 s VIII 3 28.33 s 379. 1 IX s X 447. 1 S XI 504.7 s XII 565. s XIII 652. XIV s 707. s XV 3224. , XVI s 3494.  2 1 4 3 2 1 2 1 6 5 4 3 2 1 2 1 2 1  20.204 33.747 43.737 57.60 24 3.31 258.68 342.45 352.24 402.8 46S.0 551. 2 621.  2 2 2 2 6 6 2 2 2 2 2 2  Again, f o l l o w i n g S i l s o n  (1962) we make a s m a l l a d d i t i o n  to  the i o n i z a t i o n r a t e a l l o w i n g f o r high d e n s i t y e f f e c t s c f i o n i z a t i o n s out of upper l e v e l s , = 4.8x10-« I-*/ exp {-Vy/kTJ/OCiyiz { c o l l i s i o n s ) ) . 2  ( A l l . 14)  Charge Exchange In order to i n c r e a s e the g e n e r a l usefulness of t h i s program the charge exchange r a t e s of H+ • 0 ^ H+ + N ^  were  included  Steigman  using  0+ + H N+ + H  expressions  e x a c t l y as given by F i e l d and  (1971) and Steigman, e t a l .  (1971). Since the tempera-  t u r e s here are u s u a l l y i n excess of 10* K, the charge r a t e i s at almost constant  and at i t s maximum. ,  exchange  100  The  Heating  Bate  A l l h e a t i n g i s due t c energy  gain by p h o t o i o n i z a t i o n , which  i s simply given by  (All.15)  The  t o t a l gain i s  "J  ( A l l . 16)  J  Cjgcl in.g Bates The  emission  of radiation  i s c a l c u l a t e d under the assump-  t i o n the medium i s o p t i c a l l y t h i n . The c o o l i n g due to bremsstrahlung i s (Cox and Tucker -A = 8  where  1969). ,  2.29x10-2* 1-1/2 n ^ / n  ( A l l . 17)  2  n » i s the number d e n s i t y of Hydrogen. T h i s l o s s mechanism  dominates f o r temperatures The  i n excess of 1 0  r a d i a t i v e recombination -A'Jj = * Cj ( V,;j., + kT) X -j A  energy  7  K.  loss rate i s  r  t  c  r- o . 0 7 / 3 + ± J U V  *• O.bH  Lt~''* 1  ( A l l . 18) where  0= X^/kT. /  The  correction  frcm the a n a l y s i s of Seatcn  f a c t o r i n b r a c k e t s was d e r i v e d  (19 59) f o r the recombination  i n Hydrogen. I t r e p r e s e n t s the c o r r e c t i o n t o the combination  process  radiative  re-  r a t e r e g u i r e d t o convert i t t o the energy r a t e , ac-  counting f o r the p r e f e r e n t i a l capture of slow  electrons.  The l o s s r a t e due t o d i e l e c t r o n i c recombinations  was  esti-  101  mated as, =^7}<{^ j + 6 E ^ ) X -  -A^ The for  y  H  t J  recombination r a d i a t i o n  A- .  (All.19)  t  i s the dominant l o s s  mechanism  temperatures o f 2x10* K and l e s s . .The energy d i f f e r e n c e  AErj  i s taken as the lowest energy permitted t r a n s i t i o n t c the ground state. Between 2x10* and 1 0 l i s i c n a l excitation rate  7  K t h e dominant l o s s mechanism i s c c l -  of l i n e s . In p r i n c i p l e a c a l c u l a t i o n of t h i s  r e g u i r e s t h e c o l l i s i o n a l cross s e c t i o n  p a r t i c u l a r t r a n s i t i o n as a f u n c t i o n  for excitation  of incident  electron  With the a i d of the Milne r e l a t i o n , which r e l a t e s the cross  section  tc  the  of a  energy. collision  i n v e r s e process of p h o t c a b s c r p t i c n , the  l o s s rate can be approximated as (Mewe 1972) -A* = 1 . 7 x 1 0 - 3 T ~ V  2  f. " I g U r f t] ( U e x p i - A l t j ( l ) / k T ) A  where g i s a gaunt f a c t o r , f J t  tion,  and 4E 'j (1) t  is  X ' (J  ( A l l . 20)  i s the f value f o r the t r a n s i -  the energy of the emitted photon. The g  f a c t o r has been c a l c u l a t e d and  (1)  c  by Mewe (1972) f o r many  transitions  given a simple e x t e n s i o n by Kate (1976) t o c o v e r a l l t r a n s i -  tions.  They  both  use  the same f i t t i n g f u n c t i o n  f o r the Gaunt  factor, g= where and  A* (By-Cy +D) exp (y) E, <y)*Cy  y=AE tj'(l) /kT,  and A, E, c , D are constants given by Mewe  Rato, which a r e l i s t e d  first  exponential  i n the accompanying t a b l e ,  integral.  t r a n s i t i o n number (G ID), number  ( A l l . 21)  2  E , i s the  The c o n s t a n t s are i d e n t i f i e d by a  which  is  matched  tc  a  transition  o f a l l the l i n e s used i n the c a l c u l a t i o n . For the a c t u a l  computation the complete l i n e l i s t multiplet  given was reduced by t a k i n g  average over f i n e s t r u c t u r e  l e v e l s . In  the  Table  a 10  102  the  A,  B, C, and D correspond  t c the constants f o r the f i t t i n g  f u n c t i o n . When a value o f 9 9 . 0 i s entered t h e constant becomes a simple f u n c t i o n as given by Mewe..  A  0 . 13 0.04 0.20 0.25 0.27 0.28 0.2 9 0.05 0.05 0.02 0.2 0.02 0.3 0. 0. 0. 0.  TABLE E  0. 99. 99. 99. 99. 0 . 13 0 . 11 0. 1 0.09 99. 0.54 0.43 0.35 0.3 0.0 5 0.2 -0.17 -0.04 -0.3 -0.3 -0.2 -0.2 0 . 15 0.6 0.59  -0.12 0.04 0.06 0.04 0.03 0.02 0.02 -0.04 0.01 0.02 0.05 0. 0.05 0. 0. 0. 0. 0. 0. 99. 99. 99. 0. 0. 0. 0. 0. 99. -0.25 - 0 . 19 -0.15 -0.12 0.2 0 . 15 0.25 0.2 0.4 0.5 0.3 0.5 0. 0. 0.21  0.27 0.33  0.08 0.05  0,  1 0 : THE CONSTANTS FOE THE LINE GAD NT FACTCJB C 0  0.13 0.02 0. 0. , 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 07 0. 1 0.2 0.2 0.04 0. 3 99. 99. -0. 2 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. , 0. 0. 0. . 0. 0. 0 . 04 0. o.  0.28 0 . 28 0.28 0 . 28 0.28 0 . 28 0.28 0. . 0. 0 . 28 0.28 0. 0.28 0. 0. 0. 0. 0. 0. 0 . 28 0.28 0. 28 0. 0. , 0. 0. 0. 0. 0. , 0. 0. 0. 0.28 0 . 28 0.28 0 . 28 0.28 0.28 0.28 0 . 28 0.28 0.28 0.28  0.28 0 . 28  G1  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46  10 3  0.36 0.37 0.38  0.04 0.03 0.03  0. 0. 0.  0. 28 0. 28 0. 28  The f o l l o w i n g t a b l e lation  of  gives the l i n e l i s t  the r a d i a t i o n a c c e l e r a t i o n  l i n e s were taken (1973),  47 48 49  Morton  from  tables  and the c o o l i n g  compiled  ( p r i v a t e communication,  and Morton 1976), Kate  used i n t h e  by  Morton  r a t e . The and  but mentioned  (1976), Miese, e t a£-.  calcu-  i n Lamers  (1966), and S i e s e ,  S i £2•. (1969). I n the t a b l e the l i n e i s i d e n t i f i e d by atom i o n i z a t i o n species,  usually  o r i g i n , t h e wavelength transition,  a  Smith  and  with a remark about the m u l t i p l e t of  i s given i n Angstroms, the f value o f t h e  number i d e n t i f y i n g which s e t c f c o n s t a n t s a r e t o  be used t o c a l c u l a t e the Gaunt f a c t o r , the atomic number and the ion s p e c i e s are g i v e n .  TABLE 11: LINES USED FOB THE CALCULATIONS ATOM LAMDA F VALUE G ID 2 1 9 HI 920.960 7 0. 16C5E-02 1 8 HI 923.150 7 0.2216E-02 1 7 HI 926.220 7 0. 3 183E-02 1 6 HI 930.740 7 0.4814E-02 1 H I 937.803 5 6 0.7800E-02 1 4 H I 949.743 0. 1394E-01 5 1 H I 972.537 3 4 0. 2899E-01 1 H I 1025.722 2 0.7910E-01 3 1 H I 1 1215.670 0.4162E+C0 1 HE I 10 507.058 0.2093E-02 13 2 HE I 9 507.718 0. 2748E-02 13 2 8 HE I 508.643 0.3991E-02 13 2 7 HE I 509.998 0. 5931E-02 13 2 HE I 6 512.098 13 0.8480E-02 2 HE I 515.617 0. 1531E-01 13 5 2 4 HE I 522.213 0.3017E-01 13 2 HE I 3 537.030 0. 7342E-01 11 2 HE I 2 584.334 0.2763E+00 10 2 HE I I 10 229.736 0. 1201E-02 7 2 HE I I 9 230,139 7 0. 1605E-02 2 HE I I 8 230.686 7 0.2216E-02 2 BE I I 7 231.454 7 0.3183E-02 2 HE I I 6 232.584 7 0. 4814E-02 2 5 234.34 7 HE I I 6 2 C.7799E-02 c HE I I 4 237.331 0. 1394E-01 2 3 HE I I 243,027 4 0.2899E-01 ~j 2  J  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2  104  BE I I HE I I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I C I CII CII CII CII CII CII CII CII CII C II C II C II C II C II  2 1 31AUTO 31AUTO 31 AUTO 9 9 9 9 9 9 7 7 7 7 7 6 5 5 5 5 5 5 4 4 4 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 B1 10 10 9 9 3 3 3 3 2 2 1 1 1  256.317 303.786 945. 191 945.338 945.579 1260.736 1260.927 1260.SS6 1261. 122 1261.426 1261.552 1277.245 1277.282 1277.513 1277.550 1277.723 1277.95 4 1279.229 1279.890 1280.135 1280.333 1280.404 1280.597 1280.847 1328.833 1329. C86 1329. 100 1329.123 1329.578 1329.600 1560.310 1560.683 1560.708 1561.341 1561.367 1561. 438 1656.266 1656.928 1657.008 1657.380 1657.907 1658.122 43. 200 687^050 687.350 858.090 858.550 903.62 0 903.960 904.140 904.480 1036.337 1037.018 1334.532 1335.662 1335.708  C.7 912E-01 0.4162E+00 0.2730E+00 0.2730E+00 0.2720E+00 0.3790E-01 0. 1260E-01 0.9480E-02 0. 1580E-01 0.9480E-02 0. 2840E-01 C.8S70E-01 0.6730E-01 0.2240E-01 0.7S30E-01 0.1350E-01 0. 8S70E-C3 0.3810E-02 C. 6400E-02 0.2020E-01 0. 1510E-01 0.5040E-02 0.6720E-02 C.5040E-02 0. 3920E-01 0. 1310E-01 0. 1630E-01 C.S800E-02 0. 2940E-G1 C.9800E-02 C. 6 100E-01 C.6C80E-01 0. 2020E-01 0.1210E-01 0.8100E-03 C.6800E-01 0.5660E-01 0.1360E+00 0.1020E+00 C.3400E-01 0. 45 30E-C1 0.3390E-01 0.3800E+00 0.2700E+00 0.2300E+00 0.4600E-01 0. 4600E-G1 0.1700E+00 0.3400E+00 0.43COE+00 0.8400E-01 0.1250E+00 0. 1250E+C0 0.1180E+00 0. 1180E-01 0.1060E+00  3 1 42 42 42 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 42 41 41 41 41 42 42 42 42 42 42 42 42 42  2 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6  2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2  105  CI 11 BE1 C I I I 3.09 C I I I 3.08 C I I I 3.07 C I I I 3.03 C III 3 C I I I 2.03 C III 2 C III 1 CIV 4 C IV 3 C IV 3 C IV 2 C IV 2 C IV 1 C IV 1 CV HE1 CV HE2 CV HE3 CV HE4 H5 CVI H4 CVI CVI H3 CVI H2 CVI H1 N I 2 N I 2 N I 2 N I 1 N I 1 N I 1 N i l 1310 Nil 9 Nil 9 Nil 9 Nil 9 9 Nil Nil 3 Nil 3 Nil 3 Nil 7 Nil 7 Nil 7 7 Nil Nil 7 Nil 7 N II 2 N II 2 N II 2 N II 2 N II 2 N II 2 N II 1 N II 1 N II 1 N II 1  42.510 270.324 274.051 280.043 291.326 310.170 322. 574 386.203 977.026 222.790 244. 907 244.907 312.422 312.453 1548.202 1550.774 32.800 33.430 34.970 40.270 26,000 26.400 27.000 28.500 33.700 1134.165 1134.415 1134.980 1199.549 1200.224 1200.711 529.680 533.500 533.570 533.640 533.720 533.880 644.620 644. 820 645.160 671.010 671.390 671.390 671.620 671.770 671.990 915.612 915.962 916.012 916.020 916.701 916.710 1083.990 1084.562 1084.580 1085.529  0.5660E+00 0.3287E-02 0.3378E-02 0.3527E-02 0.3817E-02 0.16C1E-01 0. 4680E-02 0.2549E+00 0.6740E+00 C.2630E-01 0. 1S87E-01 0.3975E-01 0. 1335E + 00 0.6673E-01 0.1940E+00 0.S700E-01 C.2800E-01 C.5600E-01 0.1460E+00 0.6940E+00 0. 8000E-02 0.14 00E-01 0.2900E-01 0.7900E-01 0.4160E+00 0.1340E-01 0. 2680E-G1 0.4020E-01 0. 1330E + 00 0.8850E-01 0. 4420E-01 0.820OE-01 0.2600E+00 0.1900E+00 0. 6500E-C1 0.2200E+00 0. 3900E-01 0.2300E+00 0.2300E+00 0.2300E+00 0. 3700E-01 C.6700E-01 0.8900E-G1 0.2200E-01 0. 3000E-01 0.2200E-01 0. 1490E+00 0.4950E-01 0.6190E-01 0.3710E-01 0. 1110E + 00 0.3710E-01 0.1010E+00 0.2520E-01 0.7550E-01 0.1010E-02  42 41 41 41 41 41 41 41 42 41 22 22 21 21 20 20 13 13 11 9 48 47 46 45 43 42 42 42 41 41 41 41 41 41 41 41 41 42 42 42 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42  6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7  3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  106  H N  N N N N N N N N N N M H  K N N N K  13  N N 8 N »  N N N M N N N N N K N N N N S N N N N N N N N N N N N N N N  II II III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III  1 1 AUTO AUTO AUTO 7.25 7.25 7. 25 7. 15 7.15 7. 15 7.12 7. 12 7. 12 7. 10 7.10 7. 10 7.08 7.08 7.08 7.07 7.07 7.06 7.06 7.06 7.05 7.05 7.04 7.04 7.04 7.03 7. 03 7.02 7.02 7.01 7.01 7.01 7 6 6 6 6 5.01 5.01 5 5 5 4 4 3 3 3 3 2 2 1  1085.546 1085.701 246.206 246.249 246.311 262.184 262. 233 262.289 268.347 268.473 268.473 270.073 270.200 270.201 272.523 272.653 272.654 276.193 276.326 276.326 278.436 278.572 282.070 282.209 282.209 285.855 286.000 292.447 292.595 292.596 299.661 299.818 305.761 305.920 311.550 311.636 311.721 314.877 323.436 323.493 323.6 20 323.675 332.140 332.333 374.204 374.441 374.449 451.869 452.226 684.996 685.513 685.816 686.335 763.340 764.357 989.790  0. 1510E-01 0. 8450E-01 0. 1515E-02 0. 1363E-02 0. 15 15E-03 0.1718E-02 0. 15 46E-02 0.1718E-03 0. 1801E-02 0.1800E-03 0. 1620E-02 0.1824E-02 0. 1823E-03 0. 1641E-02 0. 18 57E-02 0.1857E-03 0.1671E-02 0.1908E-02 0. 19C7E-03 0. 1716E-02 0.3878E-03 0. 3876E-03 0. 2481E-01 0.2480E-02 0. 2232E-01 0.4088E-03 0.4086E-03 0.4666E-01 0. 4149E-G1 0.4655E-02 0.4492E-03 0.4490E-03 0. 4677E-C3 0.4674E-03 0.2427E-02 0.2183E-02 0. 2426E-G3 0. 1091E-01 0. 5235E-03 0.1047E-02 0. 13C8E-02 0.2615E-03 0.7063E-02 0.7G46E-02 0.2918E+00 0.2625E*00 0.2916E-01 0.2381E-01 0. 2379E-01 0. 1207E+00 0.2412E+00 0.3013E*00 0.6022E-01 0.5664E-01 0.5657E-01 0.1C70E+00  42 42 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 42 42 42 42 42 41  7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7  2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  107  N III 1 N III 1 N IV 2 N IV 1 NV LI1 NV 3 KV LI2 NV 2 KV LI5 NV 2 N V 1 N V 1 NVI KE1 NVI HE2 NVI HE3 NVI HE4 NVII H5 NVII H4 NVII H3 NVII H2 NVII H1 OI J39 CI H5 0 I 5 5 0 I 0 I 5 5 0 I 0 I 5 0 I 5 0 I 4 0 I 4 0 I 4 0 I 4 0 I 4 4 0 I 0 I 2 0 I 2 0 I 2 CII 10 on 10 on 10 2 on CII 2 on 2 on 1 on 1 CII 1 0 III 0 III 0 III 0 III 0 III 0 III 0 III 0 III 0 III  991.514 991.579 247.205 765.148 148.000 162.560 162.560 209.270 209. 280 209.330 1238.821 1242.804 23.300 23.770 24.900 28.79 0 19. 100 19.400 19.800 20.900 24.800 811.370 878.450 988.581 988.655 988.773 990.127 990.204 990.801 1025.762 1025.762 1025.762 1027.431 1027.431 1028. 157 1302.169 1304.858 1306.02 9 429.910 430.040 430.170 539.080 539.540 539.850 832.750 833.320 834.460 228.834 228.893 228. S88 240.979 241.000 241.000 241.000 241.037 248.468  0. 1060E-01 0.S580E-01 0.5497E+00 0.5451E+00 0. 3000E-01 G.6690E-01 0. 67G0E-01 0.1570E+00 0.2360E+00 0.7840E-01 0.1520E+00 0.7570E-01 0. 2800E-01 0.5600E-01 0. 1460E+00 0.6940E+00 0. 8000E-02 0. 1400E-01 0. 2900E-01 C.7900E-01 0.4160E+00 C.7700E-02 0.3700E-01 0.5100E-03 0. 7640E-02 0.4280E-01 0. 1270E-01 0.3810E-01 0.5C80E-01 0.62 00E-01 0. 1110E-01 0.7380E-03 0.5530E-01 0.1840E-01 0. 7360E-01 0.4860E-01 0. 4850E-C1 0.4850E-01 0. 5400E-01 0. 1100E+00 0.16C0E+00 0.5600E-01 0. 3700E-01 0.1900E-01 0.7000E-01 0.15G0E+00 0.2100E+00 0.8128E-02 0.7SS6E-02 0.7962E-02 0. 2523E-01 0.3366E-03 0. 84C9E-02 0.3364E-01 0. 2825E-01 0.1533E-01  41 41 41 42 22 22 22 21 21 21 41 41 13 13 11 9 48 47 46 45 43 41 41 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 42 42 42 41 41 41 41 41 41 41 41 41  7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8  3 3 4 4 5 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3  108  0 0 0 0 G 0 0 0 G 0  c  0 G 0 0 0 G 0 0 0  c  0  c 0 G 0 0 0 0 0  c  0 G 0 0 0 0 0 C  0  c  0 0 0 C 0 G 0 0 0 G 0 0 0 G 0  III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III  6 6 6 6 6 5 6 5 5 5 5 5  248.538 248.574 248.618 248.693 255.000 255.044 255. 113 255.158 255. 188 255.302 262.000 262.700 262.700 262.729 262.882 262.900 263.692 263.728 263.768 263.818 263.818 263.903 264. 257 264.317 264. 329 264.338 264.471 264.480 266.843 266. S67 266.985 267.030 267.050 267. 188 275.281 275.336 275.513 280.116 280.234 280.265 280.328 280.412 280.483 303.411 303.460 303.515 303.621 303.693 303.769 303.799 305.596 305.656 305.703 305.836 305.879 308.306  0.6129E-01 0.4596E-01 0. 5147E-01 0.9188E-02 0. 15 46E-G2 0. 1932E-02 0. 4636E-02 0.3476E-02 0. 1159E-02 0. 1158E-02 0. 5868E-01 0.1951E-01 0. 1463E-01 0. 2438E-01 0.4386E-01 0.1462E-01 0. 1549E+G0 0. 10C7E+00 0. 2254E-01 0.5768E-02 0.1002E+00 0.1384E-03 0.2291E-01 C.7636E-02 0. 5793E-02 0.9613E-02 0.5768E-02 0. 1742E-01 0.S355E-02 0.6310E-01 0. 47C8E-01 0.5141E-01 0. 1559E-G1 0.6325E-03 0. 3020E-01 0.2971E-01 0. 2958E-C1 0.5406E-02 0. 13 18E-01 0.9573E-02 0. 3257E-02 0. 4394E-02 0.3 244E-02 0.1383E+00 0.4611E-01 0.3457E-01 0. 5761E-01 0.3455E-01 0. 3521E + 00 0.1036E+00 0.4167E+00 0.3125E+00 0.1041E+00 0.6245E-01 0.4166E-02 0.1995E-02  41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41  8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  109  0 III 4 0 III 4 0 III 4 0 III 4 0 III 4 0 III 4 0 III 3 0 III 3 0 III 3 0 III 2 0 III 2 0 III 2 0 III 2 0 III 2 C III 2 0 III 1 G III 1 0 III 1 C III 1 0 III 1 C III 1 01V B2 01V E3 0 IV 5 5 G IV 0 IV 5 0 IV 4 0 IV 4 0 IV 3 0 IV 3 0 IV 3 0 IV 3 0 IV 2 0 IV 2 0 IV 1 0 IV 1 0 IV 1 ov 2 0 V 1 OVI LI1 OVI LI2 OVI 2 OVI 2 0 VI 1 0 VI 1 OVI I HE 1 OVI I HE2 OVII HE3 OVII HE4 OVIII H5 OVII I H4 OVIII H3 OVIII H2 OVIII H1 Nil 2 NEI 1  373.605 374.005 374.075 374.165 374.331 374.436 507. 391 507.683 508.182 702.332 70 2. 822 702.891 702.899 703.645 703.850 832.927 833.701 833.742 835.055 835.096 835.292 195.860 203.000 238.360 238.571 238.580 279.631 279.933 553.330 554.075 554.514 555.261 608.398 609.829 787.711 7 90.109 790. 199 172.160 629.730 104.810 115. 800 150.080 150. 120 1031.945 10 37.627 17. 420 17.770 18.630 21.600 14.600 14.820 15.200 16.000 19.000 735.890 743.700  0. 2573E-01 G.6171E-01 0. 4627E-01 0. 1542E-01 0. 2055E-01 0.1541E-01 0.1387E+00 0.1387E+00 0. 1385E+00 0. 14C4E+00 0. 4676E-01 0.3507E-01 0.5844E-01 0.3502E-01 0. 1051E+00 0. 1049E+00 0.2621E-01 0.7863E-01 0. 1048E-02 0.1570E-01 0. 8791E-01 C.96 00E-01 0. 1730E+00 0.4977E+01 0. 4 47 6E + 01 G.4973E+00 0.3560E-01 0.3556E-01 0.94 32E-01 0.1884E+00 0.2353E+G0 0.4700E-01 0. 7062E-01 0.7046E-01 0. S345E-01 0.9317E-02 0. 8384E-01 G.59C0E+00 0.4405E+00 0.3200E-01 0.7300E-01 0. 1750E+00 0. 6740E-01 0. 1300E+00 0. 6480E-01 0.2800E-01 0. 5600E-01 0.1460E+00 0.6940E+00 0.80C0E-02 0. 1400E-01 0.2900E-01 0. 7900 E-01 0.4160E+00 0. 1620E+C0 0.1180E-01  41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 22 22 21 21 20 20 13 13 11  c  48 47 46 45 43 35 35  8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 7 7 7 7 8 8 8 8 8 1 1  110  NE I I HE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I NE I I 4 NE I I 4 4 NE I I 3 NE I I NE I I 3 NE I I 3 NE I I 3 NE I I 1 NE I I 1 NEIIIM13 NEIIIM11 NEIIIM11 NE I I I 5 NE I I I 5 NE I I I 5 NE I I I 5 NE I I I 5 NE I I I 5 NE I I I 4 NE I I I 4 NE I I I 4 NE I I I 4 NE I I I 4 NE I I I 4  324. 567 324.570 325.393 326. 519 326.542 326.787 327.250 327.262 327.355 327.626 328.090 328. 102 329.773 330.214 330.626 330.658 330.790 331.069 331.515 352.24 7 352.956 353.215 353.935 354.S62 355.454 355.948 356.441 356.541 356.800 357.536 361.433 362.455 405.846 405.854 407. 138 445.040 446.226 446.590 447.815 460.728 462.391 227.400 227.620 229.060 251.120 251.129 251. 134 251.540 251.549 251.720 267.047 267.070 267.500 267.512 267.530 267.710  0. 1066E-02 0.1091E-01 0. 1256E-01 0.4988E-02 0. 3962E-01 0.2443E-01 0. 1018E-01 0.5104E-01 0. 4666E-01 0.2393E-Q1 0. 4355E-C1 0.5OC0E+0O 0. 5177E-02 0.2177E-02 0.2685E-01 0.9726E-02 0. 3784E-G1 0.8491E-02 0.2393E-01 0.2805E-02 0. 1374E-G1 0.1094E-01 0. 5 236E-02 Q.1066E-01 0.5469E-02 0. 4775E-01 0. 2084E-01 0.7726E-02 0. 30C6E-01 0.2685E-01 0. 1577E-01 0.1694E-01 0. 1251E-01 0.1126E+00 0.1247E+00 0.1723E-01 0. 8590E-01 0.6867E-01 0. 3424E-01 0.3300E+00 0.3288E+G0 0.5500E-01 0.1200E+00 0.9600E-01 0.1858E+01 0.3317E+00 0.2213E-01 0. 1656E+01 0.5519E+00 0.2206E+01 0. 8576E-02 0.2573E-01 0. 1142E-01 0.8561E-02 0. 1427E-01 0. 3422E-01  41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 42 41 41 41 41 41 41 41 41 41 41 4 1 41 41 41 41  10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  111  NE I I I 3 NE I I I 3 NE I I I 3 NE I I I 3 NE I I I 3 NE I I I 3 NE I I I 2 NE I I I 2 NE I I I 2 NE I I I 1 NE I I I 1 NE I I I 1 NE I I I 1 NE I I I 1 NE I I I 1 NEIV N7 NEIV Ni NEIV M7 NE IV 1 NE IV 1 NE IV 1 NEV C9 M8 KEV NEV m NEV C5 NE V 3 NE V 3 NE V 3 NE V 2 NE V 2 NE V 2 NE V 2 NE V 2 NE V 2 1 NE V 1 NE V 1 NE V 1 NE V 1 NE V 1 NE V NEVI E1 N EVI B2 SEVI E3 NEVI M9 NEVI M8 NE VI NE V I NE VI NE VI NE VI NE VI NE VI NE VI NE VI NEVII BE1 NE V I I  283. 125 283.150 283.170 283.647 283.660 283.870 313.050 313.680 313.920 488.100 488.870 489.500 489.640 490.310 491.050 148.800 172.600 208.€3 0 541.127 54 2.073 543.891 118.80 0 142.610 143.320 173.900 357.S50 358.480 359.390 480.410 4 81.280 481.360 481.367 482.990 482.990 568.420 569.760 56 9.830 572.03 0 572.110 572.340 14.100 98.000 111. 100 122.620 138.550 399. 820 403.260 410.140 410.930 433.180 435.650 558.590 562.710 562.800 13.920 465.221  0. 5632E-03 C.8440E-02 0. 4726E-01 0.1404E-01 0.4212E-01 0.5612E-01 0. 3977E-01 0.3969E-01 0.3966E-01 0.4108E-01 0. 5469E-01 0.1229E*00 0. 4095E-01 0.1636E+00 0.6806E-01 0.6100E+00 0.5400E+00 C.95C0E-01 0. 2S58E-G1 0.59C5E-01 0. 8829E-01 G.2500E+00 0. 2000E+00 0.61G0E+0O 0.8600E-01 0. 1795E-01 0. 1792E-01 0.1788E-01 0. 1522E*00 0.5G63E-01 0. 63 27E-01 0.3796E-01 0. 1135E + 00 0.3784E-01 0. 9 259E-C1 0. 2309E-01 0.6927E-01 C.9994E-03 0. 1380E-01 C.7724E-01 C.49C0E+00 0.1020E+00 0. 1750E+C0 C.5400E+00 0.2 900E-01 0.4909E-01 0. 2434E-01 0.S787E-01 0.1194E+00 0.5273E-01 0.5243E-01 0.8388E-01 0. 6 3 28E-02 0.7493E-01 0.67C0E+00 0. 3748E+00  41 41 41 41 41 41 41 41 41 42 42 42 42 42 42 41 41 41 42 42 42 41 41 4 1 41 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 42 42 42 42 42 42 42 42 42 41 42  10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7  112  NEVIII LI1 NEVIII 2 NEVIII LI2 NEVIII 1 NEVIII 1 NE IX HE1 NE IX HE2 NE IX HE3 NE I X HE4 NE X 6 NE X 5 NE X 4 NE X 3 NE X 2 MG I MG I 2 MG I 1 MG I I MG I I MG I I MG I I MG I I 1 MG I I 1 MGIII NE1 MGIII NE2 MGIII 5 4 MGIII MGIII 3 MG I I I MG I I I 1 F1 MGIV MGIV F5 MGIV F2 MGIV F3 MG IV MG IV MGV 07 MGV 02 MGV 01 MGV 03 MGV 05 MGV 04 MG V MG V MG V MG V MG V MG V N7 MGVI MGVI N1 MGVI N3 MG VI MG VI MG VI MGVII C9 AG V I I C1  60.810 88.130 98.000 770.400 780.320 10.800 11.000 11.560 13.440 9.370 9. 490 9.720 10.250 12.150 1827.940 2025.824 2852.127 1025.968 1026. 113 1239.925 1240.395 2795.528 2802.704 171.500 182. 500 186.510 187.190 188.530 231.730 234.258 120.000 130.000 147.000 181.000 320.994 323.307 103.900 114.030 121.600 132.500 137.800 146.500 351.089 352.202 353.094 353.300 354.223 355.326 80.100 95.500 111.600 399.289 400.676 403.315 68.100 77.100  0. 3300E-01 0. 2980E+00 0. 8000E-01 0.1020E+00 0. 5020E-01 0.2800E-01 0. 5600E-01 0.1490E+00 0.7230E+00 0.8000E-02 0. 1400E-01 0.2900E-01 0.7900E-01 0.4160E+00 0.5260E-01 0.1610E+00 0.1900E+G1 0. 1480E-02 0.7 400E-03 0.9680E-03 0.4840E-03 0.5920E+00 0.2950E+00 0.1000E+00 0.8G00E-02 0.27COE+00 0. 1600E + 00 0.4000E-02 0.2101E+00 0.1111E-01 0.2500E+00 0.1340E+00 0. 1500E + 01 0.3200E*00 0.1348E+00 0.1339E+00 0.1200E+00 0.1800E+00 0.3000E+00 0.1340E+00 0. 4800E-01 0.2900E-01 0. 5643E-01 C.7500E-01 0. 1683E+00 0.56C7E-01 0.2237E+00 0.9293E-01 0.27C0E+0O 0.5000E«-00 0.7200E-01 0.4383E-01 0. 87 35E-01 0. 1302E+00 0.2400E+00 0.1000E+00  22 4 1 22 42 42 13 13 1 1 9 48 47 46 45 43 41 4 1 42 41 41 41 41 42 42 34 36 33 33 33 35 35 41 41 41 41 42 42 41 41 41 41 41 41 42 42 42 42 42 42 41 41 41 42 42 42 41 41  10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12  8 8 8 8 8 9 9 9 9 10 10 10 10 10 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 7 7  113  MGVII C2 MGVII C3 MGVII C4 MGVII C5 MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MG V I I MGVIII B1 MGVIII B3 MGVIII B2 MGVIII11 MGVIII11 MGVIII1 1 MGVIII10 MGVIII10 MGVIII 3 MGVIII 3 MGVIII 3 MGVIII 3 MGVIII 2 MGVIII 2 MGVIII 1 MGVIII 1 MGVIII 1 MGIX BE1 MGIX 6 MGIX 2 MGX LI1 MGX 112 MGX LI5 MG XI HE4 MG XI HE3 MG X I HE2 MG X I HE 1 MGXII 6 MGXII 5 MGXII 4 MGXII 3 MGXII 2 SI I 41.12AO SI I 41.12AU SI I 41. 12AU SI I 10 SI I 10  78.400 83.96 0 84.020 95.300 276.145 277.007 278.406 363.770 365.230 365.267 365.270 367.679 367.701 429.134 431.220 431.318 434.615 434.710 434.923 9.470 64. 500 69.000 74.850 75.030 75.040 82.590 82.820 311.780 313.730 315.020 317.010 335.250 339.010 430.470 436.680 436.730 9. 380 62.750 368.070 41.000 44. 050 57.89C 7.310 7.470 7.8 50 9.160 6. 510 6. 590 6.750 7. 120 8. 440 1255.276 1256.490 1258.795 1845.520 1847.473  0.4500E-01 0.2100E+00 0.61G0E+00 0.5900E-01 0.1201E+00 0.1197E+00 0. 1191E+00 0. 1115E+00 0. 4628E-01 0.2777E-01 0. 37C2E-01 0.8276E-01 0. 27 58E-01 0.1040E+00 0.2588E-C1 0.7762E-01 0. 1028E-02 0.1540E-01 0. 8621E-01 0.5100E+00 0. 1670E+00 0.1070E+00 0.6100E+00 0.5500E+00 0.6100E-01 0.2420E-01 0. 2400E-01 0.6800E-01 0.1400E+00 0.1700E+00 0. 340OE-01 0.4500E-01 0. 4500E-01 0.8800E-01 0.87C0E-02 0.7800E-01 0.7000E+00 G.5800E+00 0.3 140E+00 0.3500E-01 0.8500E-01 0.3200E+00 0. 2770E-01 0.5690E-01 0.1520E+00 C.7450E+00 0.8000E-02 0.1400E-01 0. 2900E-01 G.7900E-01 0.4160E+00 0.2200E+00 0.2200E+00 0.2200E+00 0. 1520E+G0 0. 1140E+00  41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 41 42 22 22 21 13 13 13 13 48 47 46 45 43 41 41 41 41 41  12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 14 14 14 14 14  7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 •8 8 8 8 8 8 9 9 9 10 10 10 11 11 11 11 12 12 12 12 12 1 1 1 1 1  114  SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI  I I I I I I I I I I I I I I I I I I I I I I II II II II II II II II II II II II II II II II II II II III III IV IV IV IV If IV IV IV V V V V V  10 10 10 10 7 7 7 7 7 7 3 3 3 3 3 3 1 1 1 1 1 1 6 6 6 5,01 5.01 5 5 5 5 4 4 4 3 3 2 2 1 1 1 11 2 2.02 2.02 2.01 2.01 2 2 1 1 NE 1 NE2 5 4 3  1848.150 1850.672 1852.472 1853.152 1977.579 1979.206 1980.618 1983.232 1986.364 1988.994 2207.S78 2210.894 2211. 744 2216.669 2218.057 2218.915 2506.897 2514.316 2516. 112 2519.202 2524. 108 2528.509 989.867 992,675 992.690 1020.699 1023.693 1190.418 1193.284 1194.496 1197.389 1260.418 1264. 730 1265.023 1304,369 1309.274 1526.719 1533.445 1808.003 1816.921 1817.445 566.610 1206.510 327.13 7 327. 181 361.560 361.659 457.818 458. 155 1393.755 1402.769 85.200 90.500 96.430 97.140 98.200  0. 3800E-01 0. 1280E+00 0. 2 280E-01 0.1520E-02 0.2110E-G1 0.1G40E-01 0. 7770E-02 0.1290E-01 0.7750E-02 0.2320E-01 0. 5890E-01 0.4420E-01 0. 1470E-01 0.4930E-01 0.8800E-02 0.5870E-03 0. 6 520E-G1 0.1560E*00 0.1170E+00 G.3890E-01 O.E180E-01 0.3880E-01 0.2440E+00 0.2190E+00 0. 24 3OE-01 0.4820E-01 0. 4800E-01 C.6500E+00 0. 1300E+01 0.1620E+01 0.3230E+00 O.S590E+00 0.6600E+00 0.9560E-01 0. 1470E + 00 0.1470E+00 0. 7640E-01 0.7600E-01 0. 3710E-02 0.3320E-02 0.3690E-G3 0.4600E-01 0. 1660E + 01 0.4886E-02 0. 2449E-02 0.9527E-02 0. 47751-02 0.2201E-01 0.1100E-01 0.5280E+00 0.2620E+00 0.2700E+00 0. 1000E-01 0. 2000E+00 C.8400E+00 0.3800E-02  41 41 41 4 1 41 41 41 41 41 41 42 42 42 42 42 42 41 41 41 41 41 4 1 41 41 41 41 41 42 42 42 42 42 42 42 42 42 41 41 42 42 42 41 42 41 41 41 41 41 41 42 42 34 36 35 35 35  14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5  SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI  V 2 V 1 VI F5 VI F1 VI F2 VI F3 VI VI V I I 07 V I I 01 V I I 02 V I I 03 V I I 05 V I I 04 VII VII VII VII VII VII VII 1 VII 1 VII 1 V I I I N7 V I I I U3 C9 IX IX C1 IX C3 IX C4 IX C5 3 IX IX 3 IX 3 IX 2 IX 2 IX 2 IX 2 IX 2 IX 2 1 IX 1 IX 1 IX 1 IX 1 IX 1 IX X B1 X E2 B3 X B9 X X 4 4 X X 4 4 X X 2 X 2 X 1  117.860 118.970 69.200 70.000 83.000 99.400 246.001 249.125 60.800 68.000 69.660 79.500 81.900 85.600 272. 641 274.175 275.352 27 5.665 276.839 278.445 314,310 316.200 319,830 50.000 69.600 44.200 52.800 55.100 55.300 61.600 223.720 225.03 0 227.000 290.630 292. 830 292.830 292.830 296,190 296.190 341.950 345.010 345.100 349.670 349.770 439.960 6. 850 39.000 47.540 54.900 253.810 256.580 258.390 261.270 272.000 277.270 347.430  0. 1900E+00 0.2100E-01 0. 2 100E+00 0.2500E+00 0.1500E+01 0.9000E+00 0. 1133E+00 0.1119E+00 0.1400E+00 G.44C0E+00 0.2100E+00 0.2600E-01 0. 4 300E-01 0.2600E-01 0.3448E-01 0.4571E-01 0. 1024E + 00 0.3410E-01 0.1358E+00 0.5627E-01 0. 3900E-01 0.7400E-01 0. 1100E+00 0.3100E+00 0. 5500E- 01 0.2300E+00 0. 2000E-01 0.23 00E+00 G.6300E+00 Q.5700E-01 0. 1000E+00 0.9900E-01 0.9900E-01 0.9200E-01 0. 2300E-01 0.3000E-01 0. 3800 E-01 C.6800E-01 0. 2300E-01 0.6500E-01 0.2100E-01 0.6200E-01 0. 6300E-03 0. 1300E-01 0.6900E-01 0.5400E+00 0. 1100E+00 0. 1430E+00 0. 2400E-01 C.6000E-01 0. 12COE+00 0.1500E*00 0. 2900E-01 0.3700E-01 0. 3600E-01 0.7400E-01  35 35 41 41 41 41 42 42 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 41 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 41 41 42 42 42 42 42 42 42  14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14  5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10  SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI SI S S s s s s s s s s s s s s £  s s s <;  s s s s s s s s s s s s s s s s s s s  X 1 X 1 XI BE1 XI 2 XII LI1 XII LI 2 XII XII XIII HE1 XIII HE2 XIIJ HE3 X I I I HE4 XIV 6 XIV 5 XIV 4 XIV 3 XIV 2 I 9 I 9 I 9 I 9 I 9 I I 9 I I I 8 I 8 I 8 I 8 I 8 I 8 I 6 I 6 I 6 I 5 I 5 5 I I 5 I 5 I 5 I 3 I 3 I 3 I 3 I 3 I 3 I 2 I 2 I 2 II 1 II 1 II 1 III III III  356.070 356.070 6.780 303.580 28.500 31.000 499.399 520.684 5. 290 5.410 5. 680 6.650 4.780 4. 840 4. 9 60 5.230 6.200 1295.661 1296. 174 1302.344 1302. £ 6 5 1303.114 1303.420 1305.885 1310.210 1313.250 1316.570 1316.610 1316.620 1323.521 1323.530 1326.635 1401. 541 1409.368 1412.899 1425.065 1425.229 1425.240 1433.328 1433.328 1437.005 1474.005 1474.390 1474.56 9 1483.036 1483.232 1487.149 1807.341 1820.361 1826.261 1250.586 1253.812 1259.520 484.194 484.580 484.892  0. 6600E-01 0.7300E-02 0.7200E+00 0.2640E+00 0. 3700E-01 C.8800E-01 •0. 7 2941-01 0.3498E-01 0.28COE-01 0.5700E-01 0. 15C0E + 00 C.7500E+00 0. 8000E-02 0.1400E-01 0. 2900E-01 0.7900E-01 0.4160E+00 0.1C80E+00 0.3610E-01 0.6000E-01 0. 3600E-01 0.4790E-01 0. 1630E-01 0.1440E+00 0. 1620E-01 0.1610E-01 0. 3450E-01 0.6150E-02 0.4110E-03 0. 3060E-01 0. 1020E-01 0.4C70E-01 0.1580E-01 0.1570E-01 0. 1570E-01 0.1810E+00 0.3220E-01 0.2150E-02 0. 1600E + 00 0.5340E-01 0.2130E+00 0.7820E-01 0. 1400E-01 0.9320E-03 0.6940E-G1 0.2310E-01 0.S230E-01 0.1120E+00 0.1110E+00 0.1110E+00 0.5 350E-02 0.1C70E-01 0. 1590E-G1 0.4074E-01 0.3111E-01 0.8568E-02  42 42 42 42 22 22 42 42 13 13 11 9 48 47 46 45 43 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 42 42 42 42 42 41 41 41 41 41 41 41 41 41 42 42 42 41 41 41  14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16  10 10 11 11 12 12 12 12 13 13 13 13 14 14 14 14 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3  117  s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s  III III III III III III III III III III III III III III III III III III III III III III III III III III III III III IV Iv IV IV IV IV IV IV IV IV IV IV IV IV IV IV V  7 7 7 7 7 6 6 7 6 6 6 6 5 5 5 5 5 5 4 4 1 1 1 1 1 1 5 5 4 4 c 4 3 s £ 3 3 s 3 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s SVI 73 SVI 2 s VI 2 SVI 2 s V I I HE1 s V I I NE2 SVII 5 SVII 4 2 SVII SVII 1  485.220 485.640 486. 154 677.750 678.460 679.110 680.690 680.950 680.979 681.500 681.587 682.883 683.070 683.470 685.350 698.730 700.150 700.184 700.290 702.780 702.820 724.290 7 25.852 1190.206 1194.061 1194.457 1200.970 1201.730 1202. 132 551.170 554.070 657.34 0 661.420 661.471 744.920 748.400 750.230 753.760 809.690 815.970 9 33.382 944.517 1062.672 1072.992 1073.522 786.480 191.510 248.980 249.270 249.270 52.000 54.800 60. 160 60.800 72. 020 72.660  0. 3 476E-C1 0.4179E-02 0. 2296E-G3 0. 9644E+00 0.7225E+00 0.24G6E+00 0.8066E+00 0.1440E*00 0. 5593E-01 0.1341E+00 0.S5S7E-02 0.3346E-01 0, 4 460E-01 0.1003E+00 0. 3334E-01 0.7406E-02 0.3080E-02 0.1848E-02 0. 2463E-G2 0.5523E-02 0. 1841E-02 0.4677E+00 0.47C8E+00 0.2240E-01 0. 1670E-01 G.5570E-02 0. 1860E-01 0.3320E-02 0. 2220E-03 0. S5C7E-01 0. 9 4571-01 0.9106E+00 0.81451+00 0.9049E-01 0. 3155E+C0 0.6295E+00 0.8278E+00 0. 1730E+00 0.1514E+00 0.1502E+00 0.4260E+00 0.2100E+00 0. 3770E-01 0.3360E-01 0.3730E-02 0.1263E+01 0. 2800E-01 0.4710E-01 0. 25C6E-01 0.2440E-01 0. 4200E+00 0. 1000E-01 0.16C0E+0O 0.1400E+01 0.1700E+00 0.3600E-01  41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 42 42 42 42 42 41 41 41 41 41 41 41 41 4 1 41 41 41 41 42 42 42 42 41 41 42 41 34 36 41 41 41 41  16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 6 6 6 6 7 7 7 7 7 7  S V I I I F5 S V I I I F1 S V I I I F2 S V I I I F3 S V I I I E4 07 S IX S IX 02 S IX 01 05 S IX S IX 03 S IX 04 S IX 06 N7 S X S X N1 S X N3 N6 S X C9 S XI S XI C3 C4 S XI S XI C5 S XI C6 C7 S XI S XII Bl S X I I B2 S X I I B3 S X I I B10 S X I I B11 S X I I I BE 1 S X I I I EE13 S XIV LI1 S XIV 112 £ XIV 115 S VI 2 S XIV S XIV SI X I I I HE 1 SI X I I I HE2 SI X I I I HE3 SI X I I I a£4 SI XVI 6 SI XVI 5 SI XVI 4 SI XVI 3 SI XVI 2  45. 300 46.COO 53.000 63.300 199.900 41.000 47.400 49.200 54. 100 54.200 56. 100 224.750 35. 500 42.500 47.700 257.100 31.000 39.300 39. 300 41.000 188.600 247.000 5. 180 27.800 33.300 221.000 227.200 5. 130 256.680 21.000 23.050 30.430 248.990 417.640 445.694 4. 010 4. 100 4.300 5.040 9.3 70 9. 490 9.720 10.250 12.150  0.26C0E+0O 0.2500E+00 0. 1500E+01 0.6000E-01 C.9600E-01 0.2300E+00 0.2300E+00 0.8G00E+00 0. 40 00E-01 0.2400E-01 0.23 00 E-01 0.1600E+00 0.32OOE+0O 0.1700E+00 0. 4800E-01 0.1900E+00 0.2100E+00 0.2100E+00 C.6100E+00 0.3500E-01 0. 8600E-01 0. 8400E-01 0.55G0E+00 0.1120E+00 0.1190E+00 0.1600E+00 0. 2900E-G1 C.7300E+00 0.2500E+00 0.3800E-01 0. 9000E-G1 0.3500E+00 0. 4775E-01 0.5573E-02 0. 2611E-02 0.2800E-01 0.57C0E-01 0.1500E+00 0.7500E+00 C.8000E-02 0. 1400E-G1 0.2900E-01 C.7900E-01 0.4160E+00  41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 42 41 41 41 41 42 42 22 22 21 41 42 42 13 13 11 9 48 47 46 45 43  16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16  8 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 11 11 11 11 11 11 12 12 12 12 12 13 13 14 14 14 14 14 14 15 15 15 15 16 16 16 16 16  119  APPENDIX 3 , THE The  LINEARIZED EQUATIONS AND  l i n e a r i z e d e q u a t i o n s and  THE  DISPERSION BELATION  the d i s p e r s i o n  relation  used  here were d e r i v e d with the a i d of a program c a l l e d BEDUCE a v a i l able  from the UBC  was i t s a b i l i t y cf the  Computing Centre. Of p a r t i c u l a r i n t e r e s t  to allow the a l g e b r a i c d e f i n i t i o n  of  functions  form (AIII.22)  Q(x,t)=Q +v{dQ/dz)+&q, e x p [ i <kz-<4fc) J. B  The  term v (dQ/dz) appears  frame  because the a n a l y s i s i s  moving along at the gas speed.  f o r the d e n s i t y , temperature, such of  done  their  and v e l o c i t y , and  d e n s i t y and temperature  are  e v a l u a t e d , and  other g u a n t i t i e s in  t i n the  matrix  the terms of f i r s t  r e s u l t s e v e n t u a l l y must be expressed  of  t i e s . The  coefficients determinant  relation.  In  order i n as  S are given  i n the form  of a  times a v e c t o r of p e r t u r b a t i o n g u a n t i -  of t h i s matrix w i l l  give  the  dispersion  order to reduce the number of m u l t i p l i c a t i o n s i n -  volved i n the e v a l u a t i o n of the determinant,  the c o e f f i c i e n t s of  k have been combined t o g e t h e r as much as p o s s i b l e . T h i s i s  the motivation f o r the form ing l i n e a r i z e d eguations their  eguation  energy;  the term  w,  and  k,  of  of  a  of the eguations below. The  origin,  m,p,  c; and the l i n e a r i z e d  series  result-  have c o e f f i c i e n t s which are l a b e l l e d  v. The  by  and e, f o r mass momentum and  being m u l t i p l i e d l a b e l l e d by  b e l l e d by n, T, and form  terms  conservation  c o l l e c t e d . T h i s g i v e s the set of l i n e a r i z e d eguations below. The  the  d e r i v a t i v e s . , Then the v a r i o u s  p a r t i a l d e r i v a t i v e s with r e s p e c t to z and eguations  in  These d e f i n i t i o n s are made  as the c o o l i n g r a t e have p e r t u r b a t i o n s expressed  and  here  the  coefficient,  g u a n t i t y being m u l t i p l i e d l a -  equations have been  written  in  the  of terms which when summed together must be  120  e q u a l t o z e r o . The as j u s t dv.  the "numerator"  The  "*"  i s  ponentiation. statements  with respect to z  c f the d e r i v a t i v e ,  the m u l t i p l i c a t i o n  The r e s u l t s  because  BEEUCE , and the  derivatives  this  i t i s how  are a b b r e v i a t e d  i . e . dv/dz  i s essentially are input  to  s i g n and ** r e p r e s e n t s ex-  are p r e s e n t e d i n the form  they  goes  how  t o the  of  EGBIBAN  they are output program  which  from does  n u m e r i c a l computations.. , The  mass  linearized  eguations are:  conservation,  n1* <-i*w+dv) +t1*C + v 1 * <i*k*n0+dn)=0. Momentum  conservation,  n1* (i.*.k.*pkn*pcn) +t1* <i*k*pkt+pct) + v1* (-i*w+dvg)=0. find e n e r g y  conservation,  n1*<-i*w*ewn+i*k*ekn+ecn +t1*  (-i*w*ewt+i*k*ekt-k**2*ccnkap*ect)  + v 1 * (-i*w*ewv+i*k*efcv+ecv) =0. In divided  the f o l l o w i n g by  t h e nuaber  vc1=1.-v0/c  and r h o c v i s t h e mass  density  density.  p k n = k b o l t z / ( n 0 * r h o c v ) * (dnedn*t0+t0) p c n = k b c l t z / ( n 0 * * 2 * r h o c v ) * (-dn*t0+nO*dnedn*dt -neO*dt-(dnedn*dn+dnedt*dt)*t0)-vc1*dgrdn p k t = k b o l t z / ( n 0 * r h o c v ) * (neO + d n e d t * t 0 * n 0 ) p c t = k b c l t z / < n O * r h o c v ) * (dn*dnedt*dt+dnedt*dt+dnedn*dn) - v c 1 * d g r d t dvg=dv+gradO/c  121  ewn=(dedn*nO+eO)*rhocv ekn=-dkdn*dt e c n = r h o c v * (dv*hO+ dv*nG*dhdn+)-d2t*dkdn-vc1*dgdn+dldn •rhocv*(vC*(dedn*dn+dedt*dt)+dedn*vO*dn) ewt=dedt*nO*rhocv e k t = - d k d t * d t + (-dkdt*dt-dkdn*dn) e c t = d l d t - v d *dgdt* (dhdt*dv*nO)*rhocv-dkdt*d2t +rhocv*dedt*vO*dn ew v=0 ekv=hO*nO*rhccv ecv=Gn*hO+nO*(dhdt*dt+dhdn*dn)) *rhocv+gO/c • r h o c v * (nG*vO*dv)  In  order  ncntrivial zero  t h a t t h e above s e t o f a l g e b r a i c e g u a t i o n s  solution  the matrix of the c o e f f i c i e n t s  d e t e r m i n a n t , which  l o w s . The  dispersion  gives  relation  D{ ,k) = i £ w  where t h e c o e f f i c i e n t s  u j  m  the d i s p e r s i o n i s computed  - i r * - i  -ecn*pct*dn-dvg*ect*dv+ pct*ecv*dv+ pcn*dn*ect c i d (2,1) = -ecn*pct*nO-ecn*pkt*dn-ekn*pct*dn -dvg*ekt*dv+pct*ekv*dv+pkt*ecv*dv+pcn*dn*ekt  crd(3,1) = ecn*pkt*nO  relation  ( c r d ( n , m ) * i * c i d (n,m) ) • •  c r d and c i d a r e :  a  must have a as  fro a the constants  c r d (1,1) =  +pcn*nO*ect+pkn*dn*ect  have  folby, (1)  122  *€kn*pct*nO*-ekn*pkt*dn+dvg* (-conkap) * d v - p k t * e k v * d v •pcn*dn*ccnkap -pcn*nO*ekt-pkn*dn*ekt-pkn*nO*ect cid(4,1) = -pkt*ekv+(-conkap)*dv)+ ekn* pkt*nOpcn*nO*(-conkap)-pkn*dn*(-conkap)-pkn*nO*ekt crd(5,1) = pkn*nO* (-conkap) cid(1,2) = ewn*pct*dn+dvg*ect*dvg*ewt*dv-pct*ecv-pct*ewv*dv -pcn*dn*e¥t*ect*dv c r d (2,2) = -ewn*pct*nO-ewn*pkt*dn - d v g * € k t + p c t * e k v * p k t * e c v + p k t * e w w*d v*pcn*nO*ewt + pkn * d n * e w t - e k t * d v c i d (3 ,2) = -€wn*pkt*nO-dvg*(-ccnkap)+pkt*ekv •pkn*nC*e«t-(-conkap)*dv c r d (1,3) = dvg*ewt-pct*ewv+ect«-ewt*d v c i d (2,3) = -pkt*ewv+ekt crd(3,3)=conkap c i d (1,4)=-ewt The placed will  form o f t h e s e c o e f f i c i e n t s by  i s such t h a t  t h e n e g a t i v e o f i t s complex  be t h e n e g a t i v e  i f  k  is  conjugate the root  of t h e complex c o n j u g a t e  of  the  refound  original  123  root.  This  solution  be  b e h a v i o u r i s demanded i n o r d e r recovered  \  independent  c f the  that the signs of  same and  physical k.  124  APPENDIX 4. THE HAJOB COMPUTER PBCGBAMS This  appendix  describes  the  major computer programs f o r  actually  performing the numerical  written  i n t h e FOBTBAN l a n g u a g e . The p h o t o i o n i z a t i o n  tions  and t h e r e s u l t i n g i o n i z a t i o n and h e a t i n g  lated  by  t h e program  p e n d i x 2 as i n p u t . ling The are  rates zero  the  PHOTION  using  They  rates  the tables  cross are  given  The i o n i z a t i o n b a l a n c e and h e a t i n g  a r e c a l c u l a t e d by, HCMAIN w i t h s u b r o u t i n e  order  physical guantities  and  radiation  are a l l sec-  calcu-  i n t h e apand  coo-  S0BCHEAT.  acceleration  worked o u t by, COEF. The c o e f f i c i e n t s o f t h e d i s p e r s i o n r e -  lation tion  computations.  a r e done i n COCALC, and t h e r o o t s  i n DISPEB. The f l o w a i d o f t h e comments.  of the d i s p e r s i o n  c f t h e programs c a n  be  followed  relawith  125  PEOGBAM PHOT ION C C  PHOTOIONIZATION  AND HEATING  BATES  DIMENSION INDEX (16,9) INTEGEB IZED{9) BEAT NJNU (100) , DELNU{100) ,PHOT ("76) ,PHEAT(76) INTEGEB NOO (76),NUF (76) BE AX. SIGMA(100,76) , INU (100) , FLUX {100) ,DELE (100) LOGICAL VEBBOS COMMON /A/ INDEX,IZED,DEN,T,VEBBOS,NFEEQ,N NOT COMMON /HELIDM/ ALP HA,BITA,A2,B2,ZF,ZB,BEN 1,ZF2 $ ,AEZ3,BZB31,AZB31 COMMON /PH/ PHOT,PHFAT,NJNU,DELNU,SIGMA EQUIVALENCE (DELNU (1) ,DELE (1) ) KAMELIST / P P / VEBBOS DO 20 IJ=1,76 DO 20 IN= 1,100 SIGMA(IN,IJ) = 0.  20 C C UNIT 1 HAS STELLAS RADIATION FLUXES AND FBEQUENCIES C UNIT TBC HAS PHOTOIONIZATION IDCES AND STELLAB C FLUX FEEQUENCIES C BEAD (1) NFBEQ BEAT (1) ENU,FLUX,NJNU,DELE BEAD (2,9774) (NUO ( I J ) , N U F ( I J ) , I J = 1 , 7 6 ) 9774 FOBMAT (214) C C CALCULATE TOTAL FLUX C FTGT=0. DO 22 IN=2,NFBEQ FTOT=FTOT + . 5* (FLUX (IN-1) +FLUX(IN) ) *DELE (IN) 22 CONTINUE VEBBOS=,TBUE. SBITE (6, 9775) 9775 FOBMAT ('1') REAL(5,PP) HBITE(6,PP) C C GO THBCUGH ALL ATOMS ( I ) C AN £ ALL IONS OF ATOMS (J) C DO 10000 1=1,9 II=IZED (I) DO 10001 J = 1 , I I IJ=INDEX(J,I) NUNOT=NUO(IJ) NUI NF= NUF (I J) NUINF1=NUINF-1 NUNCT1=NUN0T+1 PHOT(IJ)=0.0 EHEAT ( I J ) = 0 . 0 C  126  C EEINCH TO COEBECT ATOM C C ATOMS ABE I D E N T I F I E D £Y EBANCH LABEL C COBBESPONDS TO 2 OF ATOM C GO TO (1,2,6,7,8,10,12,14,16) ,1 1 XIP=13.598 ZADJ=1.0 GO TO 9910 2 I F ( J . E Q . 2 ) GO TO 202 ALPHA= 2. 182846 BETA=1.188914 A2=4.7648166 B2=1. 4135164 DEN1=0.567759716 AB23=139.8332 ZF=1. ZB=2.  202  ZF2=1. , BZB31=0.03083696 AZB31=0.01366421 XIP=24.587 GO TO 9920 ZADJ=0.25 XIP=54. 416 GO TO 9910  C C ALL FOLLOWING CALCULATIONS ABE IDENTIFIED BY THE C Z OF THE ATOM AND THE J OF T E E ION C EG 601 IS C I C EG 1204 IS MG I I I C 6 GO TO (601,602,603,604,605,606),J 601 SIGNOT=12. 19 FZEBC=11.26 A=3.317 S=2.0 GO TO 9930 602 SIGNOT=4.60 FZEBO=24.383 A=1.95 S=3.0 GC TC 9930 603 SIGNOT-1.84 FZERC=47.887 A=3.0 S=2.6 GO TO 9930 604 SIGNGT=0.713 FZEBO=64.492 A=2.7 S=2.2 GO TC 9930 605 GO TO 99 80 606 GO TO 9980 7 G O T O (701 ,702,703 ,704,7G5,706,707) , J  127  701  702  703  704  705  706 707 8 801  811 802  803  804  805  SIGNOT=11.42 FZEHO=14.534 8=4.287 S=2.0 GO TO 9930 SIGNOT=6.65 S=3.0 A=2.86 IZEBO=29.601 GO TO 9930 SIGNCT=2.G6 A=3.0 S=1.626 FZEBO=47.448 GO TC 9930 SIGNOT-1.08 A=2.6 S=3.0 FZEBG=77.472 GO TO 9930 SIGNGT=0.48 S=2.0 A= 1.0 FZEB0=97.89 GO TO 9930 CONTINUE GO TO 9980 GO TO (801,802,803,804,805,606,9980,9980),J BO 811 IN= NUNOT,NUINF SIGMA ( I N , I J ) =2.94*SEATGN (END (IN) , 13. 6 18,2. 661 ,1.0) I F (ENU (IN) . I T . 16.943) GO TO 611 SIGMA (IN,IJ)=SIGMA ( I N , I J ) + 3 . 65*SEATON (ENU(IN) , $ 16.943,4.378,1.5) I F (ERU (IN) . LT. 18. 635) GO TO 811 SIGMA (IN,IJ)=SIGHA (IN,I J) +2.26*SEATON (ENU(IN) , $ 18.635,4. 311, 1. 5) CONTINUE GO TO 999 SIGNOT=7.32 S=2.5 A=3.837 FZEEC=35.117 GO TO 9930 SIGNCT=3.65 S=3.0 A=2.014 FZEBO=5 4.943 GO TO 9930 SIGNOT=1.27 S=3.0 A=0,831 FZEB0=77,413 GO TO 9930 SIGNCT=0.78 S=3.0 A= 2.6  806  10 1001  1002  1012 1003  1013 1004  1005  FZEB0=113.90 GO TC 9930 SIGNOT=0.36 S=2. 1 A=1.0 FZEB0=138. 12 GO TO 9930 G O T O (1001, 1002, 1003, 1004, 1005, 1006) J GO TO 99 80 SIGNOT=5.35 S-1.0 A=3.769 FZEEO=2 1.564 GG TO 9930 DO 1012 IN=NUNOT,NUINF SIG MA(IN,IJ)=4. 16*SEATON(ENU ( I N ) , 40.962,2. 717,1.5) I F (ENU (IN) . LT. 44. 1 66) GG TO 1012 SIGMA(IN,IJ) =SIGMA (1N,IJ) +2.71*SEATON (ERU (IN) , $ 44. 166,2. 148, 1.5) I F ( E N U ( I N ) .LT. 47. 874) GG TO 1012 S I G 8 A ( I N , I J ) = S I G M A ( I N , I J ) +0.52*SEATON (ENU(IN) , $ 47.874,2.126,1.5) CCNTINUE GO TO 999 DO 1013 IN=NUNOT,NUINF SIGMA(IN,IJ)=1.80*SEATGN(ENU (IN),63.45,2.277,2.0) I F (ENU (IN) . LT. 68.53) GO TO 1013 SIGMA (IN,IJ)=SIGMA ( I N , I J ) + 2. 50*SEATON (ENU(IN) , $ 68.53,2.346,2.5) I F (ERU (IN) . LT. 71. 16) GO TO 1013 SIGMA(IN,IJ)=SIGMA ( I N , I J ) +1.48*SEATON(ENU(IN) , $ ,71. 16,2.225,2. 5) CCNTINUE GO TO 999 SIGNOT=3.11 JZEBC=97.11 A=1.963 S=3.0 GO TO 9930 SIGNOT=1.40 FZERO=126.21 A= 1.471 S=3.0 GO TO 9930 SIGNOT=0.49 FZE80=157.93 A=1.145 S=3.0 GO TO 9 930 G O T O (1201,1202,1203,1205),J GO TO 9980 SIGNOT=9.92 A=2.3 S=1.8 FZERO=7.646 GG TO 9930 #  0  1006  12 1201  1202  1203  1204  1205  14 1401  1411 1402  1403  1404  16 1601  SIGNOT=3.416 A=2.0 S=1.0 FZEEC=15.0 35 GO TO 9930 SIGNOT=5.2 A=2.65 S=2.0 FZEBO=80.143 GO TO 9930 SIGNOT=3.83 A= 1. 0 £=2.0 FZEEC=109. 31 GO TO 9930 SIGNGT=2.53 A=1.0 S=2.3 FZEBO=141.27 GO TO 9930 GO TO (1401,1402,1403,1404),*] GO TO 9980 DO 1411 IN=NUNOT,NUINF SIGMA(IN,IJ)=12.32*CHAHEN(ENU(IN),7.370,6.459, $ 5.142,3.) I F ( E N U ( I N ) .LT.8. 151) G O T O 1411 SIGMA (IN,IJ)=SIGMA(IN,IJ)+25.18*CHAHEN(END(IN) $ 8. 151,4.420, $ 8.934,5.) CONTINUE GG TC 999 SIGNOT=2.65 A=0.6 S=3.0 FZEBO=16.345 GO TO 9930 SIGNGT=2.48 A=2.3 S=1.8 FZEBO=33.492 GG TO 9930 SIGNOT=0.854 A=2.0 S-1.0 FZEIO=45. 141 GO TO 9930 GO TO(1601, 1602, 1603, 1604, 1605, 1606) , J GO TO 99 80 DO 1611 IN=NUNOT,NUINF SIGMA (IN , I J ) = 12.62*CHAHEN (E NO (IN) ,10.360, $ 21. 595,3.062,3. 0) I F (ENU (IN) .LT. 12.206) GO TO 1611 SIGMA(IN,IJ)=SIGMA(IN,IJ)+19.08*CHAHEN (ENU(IN) $ 12.206,0. 135,5.635, $ 2.5) I F (ENU (IN) . LT. 13. 40 6) GO TO 1611  13 0  SIGMA (IN,IJ)=SIGMA (IN ,1 J) +1 2 .70*CHAHEN (END (IN) , 13.408, 1. 159,4.7113, 3.0) CONTINUE GO TO 999 SIGNOI=8.20 FZEBC-23.33 A=1.695 E=-2.236 S=-1. 5 GO TC 9940 DO 1631 IN=NUNOT,NUINF SIGMA(IN,IJ)=. 350*CHABEN(ENU(IN) , 33.46, 10. 056, $ -3.276,2.0) I F (ENU(IN).LT.34.83) GO TO 1631 SIGMA (IN,IJ)=SIGMA ( I N , I J ) •. 244*CHAHEN (ENU(IN) , $ 34.83,18.427, % 0.592,2.0) CONTINUE GO TG 999 SIGNOT-0.29 FZEEC=47.30 A=6.837 E=4.459 S=2.0 GG TO 9940 SIGNOI=0.62 A=2.3 S=1.8 FZEBC=72. 68 GO TO 99 30 SIGNOT=0.214 A=2.0 S=1.0 FZEBO=88.0 5 GC TO 9930 % $  1611 1602  1603  1631 1604  1605  1606  C C NOW THAT CONSTANTS ABE SET UP C IN THE RELEVANT FORMULA C CALCULATE THE CBOSS SECTION AT TBE C INTEGFBEQ FREQUENCIES (UNIT 2) C 9910 DO 9911 IN=NUNOT,NUINF SIGMA (IN,IJ)=ZADJ*HS1G (ENU (IN) ,XIP) 9911 CONTINUE GC TC 999 9920 DO 9921 IN=NUNOT,NUINF 9921 SIGMA (IN,IJ)=HEISIG(ENU (IN) ,XIP) GO TO 999 9930 I F (NUNOT. GE, NUINF) G O T O 9960 DO 993 1 IN=NUNOT,NUINF 9931 SIGMA(IN,IJ)=SIGNOT*SIATON(ENU(IN) ,FZEBO,A,S) GO TO 999 9940 DO 9941 IN=NUNOT,NUINF 9941 SIGMA(IN,IJ)=SIGNOT*CHAHEN(ENU (IN) , FZEBC,SIGNOT , A , B,S ) GG TO 999  131  9980  SIGMA(NFREQ ,IJ)=-1 .0 GG TO 9981 999 DO 998 INU=NUNOT1,NUINF PHINT=. 5* (NJNU (INU-1) *SIGM A < INU- 1, IJ) *NJNU (INU) $ *SIGMA(INU,IJ)) f H O T ( I J ) =PHINT*EEL E (INU) + PHCT (IJ) PHINT=.5*(FLUX(INU-1)*S1GMA(INU-1,IJ)• $ FLUX (INU) *SIGMA (INU,IJ) ) PHEAT ( I J ) = PHINT*BELI (INU) + P E E A T ( I J ) 998 CONTINUE C FLUX HAS UNITS ERG CM-2 S-1 ( I V ) - 1 C NJNU HAS UNITS # CM-2 S-1 (EV)-1 C PHOT HAS UNITS # S-1 C P HEAT HAS UNITS EEG S-1 PHEAT (IJ) = PBEAT (IJ) 9981 B R I I E ( 6 , 9 7 7 1 ) II,J,ENU(NUNOT),ENU(NUINE) $ , PHOT ( I J ) , PHEAT (I J) 9771 FORMAT(* 0ION•,213,* FREQUENCIES*,2F12.3, $ • IONIZATION, HEATING RAT IS*,2E15.4) IF(VERBOS) WRITE (6,9773) I I , J 9773 FORMAT{'OCRGSSECTIONS FOR ION (Z,N)=',2I3) IF(VEBBOS) flBITE(6,9772) (SIGMA (IN , IJ) , $ IN=1,NFREQ) 9772 FOBMAT(1X,10E12.3) 10G0 1 CONTINUE 10000 CCNTINUE WRITE (7) PHOT,PHEAT,SIGMA,FTOT C C FLUX HAS UNITS ERG CM-2 S-2 (EV)-1 C NJNU HAS UNITS # CM-2 S-1 (EV)-1 C PHOT HAS UNITS # S-1 C PHFET HAS UNITS ERG S-1 C STOP END ELGCK DATA COMMON /A/ INDEX,IZED ,DEN,T,VERBOS,LAST,NNOT INTEGER IZED (9) DATA IZED /1,2,6,7,8,10,12,14,16/ DIMENSION INDEX (16,9) DATA INDEX /1,15*0,2,3,14*0 ,4 ,5 ,6 ,7,8 ,9,10*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,8*0, $ 2 5,26,27,28,29,30,31,3 2,33,34,6*0, $ 35,36,37,38,39,40,41,4 2,43,44,45,46,4*0, $ 47,48,49,50,51,£2,53,54,55,56,57,58,59,60,2*0, $ 61,62,63,64,65,66,67,68,69, I 70,71,72,73,74,75,76/ END  REAL FUNCTION  HSIG(E,XIP) C C FOE CALCULATING THE HYDROGEN C CECSS SECTION C I F (ABS ( E - X I P ) . L T . 0.0001) GG TO 1 ETA1=SQRT(E/XIP-1.0)  1  ETA=1./ETA1 HSIG=3.442Q4E-16*(XIE/E)**4. $ *EXP(-4.*ETA*ATSN (E TA 1) ) / $ (1.-EXP (-6.238185 + ETA) ) BETUBN HSIG=6.30432E-18 BETUBN END BEAL FUNCTION H E I S I G ( E , X I P )  C C HELIUM I CBOSS SECTION C COMMON /HELIUM/ A L P H A , B E T A , A 2 , B 2 , Z F , Z £ , B E N 1 , Z F 2 $ ,A EZ3,EZB31,AZ E31 BK2=(E-XIP) /13. 598 I F ( B K 2 . L E . 0 . 0 ) GO TO 1 BK=SQBT(BK2) FEXP=-6.283185*ZF/BK ALPHAI=(2.*ALPHA-ZF)*EXP <FEXP*ATAN(BK/ALPHA)) I *(BK2 + A 2 ) * * ( - 3 . ) BETAI=(2.*BETA-ZF)*EXP(FEXP+ATAN(BK/BETA)) $ *(BK2*B2) **<-3.) DFE=2730.667*E*ZF*AEZ3* (BK2+ZF2) * $ (ALPHAI*BZB31+BETAI*AZB31)**2 $ (1. -EXP (FEXP) ) *DEN1 HEISIG=8.067291E-18*DFE BETUBN 1 I F (XIP. GT. 24. 587) GO TO 2 HEISIG=8.334E-18 BETUBN 2 I F (XIP.GT.392.08) GO TO 3 BEISIG=4.7113E-19 BETUBN 3 I F (XIP. GT. 552. 06) GO TO 4 HEISIG=3.316E-19 BETUBN 4 I F ( X I P . G T . 7 3 9 . 3 2 ) 60 TO 5 HEISIG=2.46E-19 BETUBN 5 8BITE (6,1000) E,XIP 1000 FOBMAT (' HEISIG PBOBLEMS • ,2F15.4) BETUBN END BEAT FUNCTION SEATON (F, IZEBO , A, S) C C SEATCN CBOSS SECTION FOBMULA C EN= FZEBC/F SEATON=1.0E-18*FN** (+S) *(A+ (1.-A) *FN) BETUBN END BEAL FUNCTION CHAHEN (F, FZEBO,A, B, S) C C CHAPMAN AND BENBY CBOSS SECTION FOBMULA C FN= FZEBO/F  CHAHEN=A+{B-2.*A)*FK+ (1.+A-B)*FN*FN CHABEN=1.E-18*FN**S*CHAHFN RETURN END  134  PBGGBAM HCMAIN C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  PASAKETEBS DEN:TOTAL DENSITY Ts TEMPEBATUBE F J : COVEBSION FBOM FIBST TO 2EBOTH MOMENT BADIATION FIELD =1 FOB UNIDIRECTIONAL =2 FOB A HEM ISP HEBE NIT: NUMBER OF ITERATIONS IN ION FRACTIO N LOOP VEBBGS: OUTPUT ALL CAICULATED QUATITIES AT END OF NIT LOOP ULTRA: OUTPUT DITTO EVERY CYCLE TEBSE=.TRUE. FABUND: MULTIPLY ALL ABUNDANCES Z>2 BY T HIS NUMEEB FE: GUESS AT ELECTRON DENSITY WF: DILUTION FACTOR FOB B AEI AT ION FIELD NLINE: NUMBER OF LINES IN CCCIING CALCUL ATION WLINE: WRITE INDIVIDUAL LINE CCCIING AND HEATING TCL: TOLERANCE FCB CONVERGENCE OF BNOT,DENE,EQUILIBRIUM TEMPERATURE EGUIM: TRUE FOB FOBCING BALANCE OF HEATING AND COOLING BATES CHABGX TBUE FOB CHABGI EXCHANGE H-N, H-0 CALCULATIONS NELMNT: # OF ELEMENTS STARTING WITH H IN IONIZATION CA LCULATICN USEFUL SOMETIMES IN EQUIM FOB PBELIMINAEY ESTIMATE DMAX: MAX FRACTIONAL CHANGE ALLOWED IN DELTA T, PBEVENTS WILD OSCILLATIONS DIELEC: FALSE TURNS ALL DIELECTBONIC RECOMBINATION OFF MI-OOP: NUMEEB OF ITEBATIONS ALLOWED IN CONVEBGENCE TO T I F EQOIM IS CN TBAD: BACIATION TEMP EBATUBE OE PHOTON SOURCE WFTBAD IS APPROXIMATELY A BRIGHTNESS TEMPEBATUBE FUDGE: TBUE FOB BE DUCT ION OF DIELECTBONIC B ECO MB IN A HO N WITH DENSITY FUDGE FACTOR CALCULATION IS DESIBED THEEEB: THREE BODY RECOMBINATION DXENDT TRUE FOB COMPUTING EEBIVATIVES IN N AND T SEBIES IS (T,N) , (T* { 1+-DERDEL) , N) ) . <T ,N* < 1 + -DEBDEL) ) OUTPUT FALSE I F NO OUTPUT OF QUANTITIES TO UNIT 7 CNGAB CHANGE OF CNO ELEMENTS FSCM SCLAE VALUES TSERIE TBUE IF SERIES OF TEMPERATURES TO EE CALCULATED DSEE1E TBUE FOB A DENSITY SEBIES SEEINC SEBIES STARTS FROM INPUT DENSITY AND TEMFEBATUEE AND INCBEASE LOGABITHMICALLY EY 10 TO SERINC SERENC MAX VALUE OF N OR T WTNE WEIGHT GIVEN TO OLD VALUE CF ELECTRON DENSITY IN CONVERGENCE OF IONIZATION EQUATIONS. DV FOR ESTIMATE OF EFFECTS OF OPACITY ON LINE COOLING  135  C TAUMAX GREATER THAN ZERO TO TURN ON CALCULATION C LOGICAL VERBOS,SEMICO,ULTRA,WLINE,EQUIM,CNVG,FIRST, $ CHABGX LOGICAL YERBO,WLIN,DIELEC,F UDGE,THREEB,TERSE,NOW AST $ ,QUIT,DXDNDT LOGICAL 0 UTPUT , TSERI E , DSERI E , BOTH DE,FSEE REAL SIGHA(100,76) ,PHOT (76) ,PEEAT(76) REAL P P H G T ( 7 6 ) , P P H E A T ( 7 6 ) , CPHEAT(76) REAL RATIO (16) , REL ( 16) ,X (17, 9) REAL TOPIN( 16) ,TOPOUT(16) REAL HLCCOL (9) REAL LOWLIN (76) INTEGER INDEX (16,9) INTEGER IZED (9) EEAL ABUND (9) REAL CHIT (76) REAL IP1 (76) ,IP2(76) ,CS(76) INTEGER NUM1 (76) ,NUM2(76) BEAL ARAC(76) , ETA(76) ,TMAX(76) , TCRIT (76) , ADI (76) , $ TO(76),BDI(76) , $ T1 (76) ,BREC(76) , BflEC (76) ,UREC(76) BEAL AREC{76) ,SLTE (76) REAL LINLOS,LRRAD,LBREMS,PHEET REAL AG(49) ,BG(49) ,CG(49) ,DG(a9) REAL LCOOL(76) ,ELINE(407) , F I (407) BEAL LCLX (76) INTEGER IIND(407) , JIND (407) , I DENT (407) COMMON /A/ INDEX,IZID,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBO,LAST COMMON /RECG/ RREC,IREC,UREC,ARAD,ETA,TMAX,TCRIT, $ ADI,TO,BDI,T1 COMMON / C I C N / IP2,NUM1,NUM2,CS,SLTE COMMON /COLREC/ IP1,CHIT,RN NOT COMMON / L I N E / LCOOL,ELINE,FL,IDENT,IIND,JIND,NLINE COMMON /GFACT/ AG,BG,CG,DG COMMON /CCNTRO/ SEMICO,ULTRA COMMON /CFUDJ/ FUDJ,RNOT,FUDGE COMMON /THICK/ X,ABUND,DV,TAUMAX NAMELIST /PARAM/ DEN,T,FJ ,NITP,VERBOS,FABUND,FE,WF, $ NLINE,SEMICO $ ,ULTRA,WLINE,TOL,EQUIM,CHARGX,NELMNT,DMAX,DIEIEC $ ,NLOCP,TRAD $ ,FUDGE,THREEB,TER SE,D XDNDT,DERDEL,OUTPUT,C NOAB $ ,TSEBIE, DSERIE, S EBINC, EOT H EE, SER END, WTN E $ ,DV,TAUMAX C C SET UP DEFAULTS C REWIND 1 REWIND 2 ABUND (1) =1.0 ABUND(2)=8.5E-2 ABUND(3)=3.3E-4 ABUND (4) = 9. 1E-5 ABUND (5) =6.6E-4  136  A BUND (6)=8.3E-5 ABUND(7)=2.6E-5 ABUNB (8)=3.3E-5 AB0ND(9) = 1.6E-5 NLOGf-15 SEMICO=.TRUE. ULTRA=. FALSE. FUDGE=.TRUE. THBEEB=. TSUE, TERSE=.FALSE. NGWAST=. FALSE. TBAB=50000. FJ=1. NITP=10 VEBEOS=.TBUE. WLINE=.TEUE. FABUND=1. CNOAB=1. FE= 1.002 WTNE= 1. WF=1.0 WFvJ0LD=-1. N1INE-407 EQUIM=.FALSE. EMAX=.25 BIELEC=.TRUE. NELEKT=9 CHARGX=.TBUE. T0L=1.E-03 DXDNDT=.FALSE. DEEDEL=.01 OUTPUT=.TRUE. ESEE=.TRUE. X (1, 1) =0. TAUEAX=0. DV=0. C  C C C C C C C C C C  TSEBIE=.FALSE. CSEE3E=. FALSE. BOTHDE=.FALSE. SEBINC=.1 SEBEND=0. BEAD IN I AT A UNIT 1 HAS PHOTOIONIZATION DATA CALCULATED BY PHOTION ASSUMED TC EE GF THE ICBM: FIRST MOMENT OF RADIATION FIELD*QDANTITIES UNIT 2 HAS THE CGNSTANTS REQUIRE £ FOR RECOMBINATION, I ONIZATION AND IINECOOLING (LINES AND EXCITATION G FACTOR) LOWLIN HAS LOWEST DELTA ENERGY LINES FCR IONS HITH D IELECTBGNIC BECCM BEAE (1) PHOT,PHEAT,SIGMA,FTGT BEAD (2) ABAD,ETA,TMAX, TCRIT ,ADI,10,BDI ,T1 BEAD (2) IP1,NUM1,IP2, NUM2 READ (2) ELINE,FL,IDENT,IIND,JIND  READ (2) AG,BG,CG,DG BEAD (2) L01LIN C C SET DP OF I N I T I A L CONDITIONS FOB MULTIPLE LOOPS C ITDER=0 10000 CONTINUE QUIT-,FALSE. DIFOLD=0. DIFF=0. CNVG=.FALSE. ICLOOP=0 I F (IT DER. EQ, 5} DEN= DEN/ (1. - DEB DEL) IF{DXDNDT.AND. I I D E E . I E . 4 ) GO TO 10100 ITDF B=0 IF(TSERIE.OR,DSERIE) GO TO 3001 BEAD (5,PARAM, END=10001) LAST-INDEX(IZED(NELMKT),NELMNT) IF(EQUIM) DXDNDT=. FALSE. IF(DXDNDT) EQDIM=,FALSE. I F {. NOT, EQUIM) CNVG=.TRUE, I F (ULTRA) VERBOS=.TBUE. I F (TEBSE) VEBBOS=. FALSE. I F (VEBBOS) « L I N E = . T B U E . I F (TEBSE) SLINE=. FALSE. I F ( N I T P . L T . 2 ) NITP=2 I E (X (1 , 1) . NE.O. ) G O T O 200U X (1, 1) =0. X (2,1) = 0. X(1,4)=0. X(2,4) = 0. X (1,5) =0. X (2,5)=0. EUO=0. EDC=0. BUN=0. EDN=0, C C TE P. PEE ATUBE OB DENSITY SERIES LOGIC C 2004 I F (TSEBIE. OB. DSEBIE) SEBINC= 10. **SEBINC IF(.NOT.(TSERIE.OR.DSEBIE)) GC TO 3004 IF(.NOT.EOTBDE) GO TO 3002 EQUIM=.FALSE. DXDNDT-. TRUE. , CUTEUT=.TRUE. 3001 IF(.NOT.BOTHDE) GO TO 3002 EQUIM=.NOT.EQUIM EXDNDT=.NOT.DXDNDT CUTPUT=.NOT.OUTPUT 3002 IF(DXDNDT.AND.ITDER.NE.O) GC TO 3004 I F (DSERIE) GO TO 3003 IF(.NOT.TSERIE) GO TO 3004 I F (. NOT. FS ER. AND. {. NOT. EOTH DE. OR. EQUIM) ) T=T*SERINC IF(T.GT.SEREND) GO TO 10001 GO TO 3004  138  3003  CONTINUE I F (. NOT. FSEB. AND. (. NGT. EOTH BE.OB. EQUIM) ) DEN=DEN* $ SEEINC I F (DEN. GT. SEBENB) G O T O 10001 3004 CONTINUE I F (EQUIM) GO TO 10 101 10100 CONTINUE I E (.NOT. DXDNDT) GO TO 10101 C C E E E I V A T I V E CALCUIATICN LOGIC C ITDEB=ITDEB*1 IF(ITDER.EQ.1) GO TO 10101 I F (ITDEB. EQ. 2) T=T* (1. + DEB DEL) I F (ITDEB. EQ. 3) T=T* (1.-DEBDEL) / (1 . + DEBDEL) IF(ITDEB.EQ.4) GO TG 10111 I F (ITDEB. EQ. 5) EEN=EEN* ( 1. - DERDEL) / (1.+ DEBDEL) GO TO 10101 10111 T=T/ (1. -DEBDEL) DEN=DEN*(1. + DERDEL) 10101 CONTINUE IF (EQUIM) NIT=MIN0 (5,NITP) I F (. NOT, EQUIM) NIT=NITP FIBSI-.TBUE. TCLD=0. I F (.NOT,TERSE.OR..NOT,NGWAST) WRITE (6,1008) 1008 FOBHAT ( 1 *) I F (. NOT. TERSE. OB. . NOT. NGSAST) WRITE (6 , PA BAM) C C SET UP TEMPERATURE, DENSITY, AEUNDANCES, EIUX FACTOB C DENE=DEN*FE DEOID=DENE IF(.NOT.FSER) GO TO 2005 SUM=0. C C CALCULATION OF ABUNDANCES C DO 101 IEL=1,NELMNT IF (IEL.GT.2) ABUND (IEL) =FAEUND*ABUND (IEL) I F ( ( I E L . GE. 3) . AND. ( I E l . L E . 5) ) ABUND (IEL)=CNOAB* $ ABUND (IEL) SDM=SUM+ABUND(IEL) 101 CONTINUE DO 102 IEL=1,NELMNT ABUND (IEL) =ABUND (IEL) /SUM 102 CONTINUE FABUND=1. CNOAB=1. IF(.NOT.TERSE.OR. . NOT. NGH AST) WRITE (6 ,1002) FABUND, $ RELHNT,ABUND 1002 FORMAT (• RELATIVE ABUNDANCES WITH FABUND= » , F6 .3 , $ 5X,'NELMNT=»,I3,/1X,9E13.3) C C BAIIATICN DILUTION APPLIED C 1  139  SFJ=WF*EJ WFTRAD=WFJ*TBAD I F (WFJ. EQ. WFJQLD) GO TO WFJ0LD=8FJ WMJ=»FJ*12.56637 C C C  ADJUSTMENT TO  200G1  FLUX MADE INCLUDING  A 4*PI MULTIPLICATION  EFTCT=FTOT*WF DO 103 IJ=1,76 PPHOT (IJ) = FHQT (IJ) *BMJ PPHEAT (IJ)=PHEAT ( I J ) *WMJ 103 CONTINUE FSEB=. FALSE. 20001 ICLOOP=ICLOOP+1 I F (ICLCOP.GT.NLOOP) GO TO 30000 2005 VERBO=VERBOS.AND. (CN VG. OB-. , NOT. EQUIM) WLIN=WLINE. AND. (CNVG.OB. . NOT. EQUIM) TK=T/11604.8 TKI=1./TK T4=T*1.E-4 ETHBEE=0.0 TM4 5=T**(-4.5) TSQBT=SQBT (T) I F ( . NOT . F I B ST. AND. EQUIM) NII-MINO <3,NITP) 122 I F (, NOT. CHABGX) GO TO 10004 CALL CHGEX(BUO,BDO, BUN, BDN,T) C C CHAEGE EXCHANGE CALCULATION FOB NITROGEN AND OXYGEN C U IS U EE ATE FOB I TO I I OF N AN E 0 C D I S DOWNBATE C I F ( ( X ( 1 , 1 ) .NE.1.E-07) .OB. (X (1 ,1) . NE. 0. ) ) G O T O 10004 X(1,1) = 1.E-07 X(2,1)=1. X (1 ,5)=0.5 X(2,5)=X(1,5) X (1,4) = 0.5 X(2,4)=X(1,4) 10004 DC 1 IT=1,NIT DO 2 1=1 NELMNT 11= IZED (I) ZED=II ENUCLD=1.E4 I F (WFTRAD.LE.O.) GO TO 45 C C THIS I S A CALCULATION OF LOWEST LEVEL I N EQUILIBRIUM WITH C CONTINUUM DUE TO RADIATION FIELD C RNUCLD=2.72 NNIT=0 44 ENCTNU=ZED*SQRT (3. *ALOG (RNUOLD) * 1 57802./WETfiAD) NNIT=NNIT+1 DIE N= R NOT NU -BNUOLD R N UOL D=R NO T N U-DIF N/(1.-.5/AIOG(RNOTNU)) I F (BNUOLD. LE. 1. ) BNUOLD=2. #  45  IF(NNIT.GT.4) GO TO 45 I F (ABS (DIFN/BNOTNU) . GT. TOL) GO TO 44  111=11+1  IZ=II DO 3 J = 1 , I I IJ=INDEX(J,I) CALL LEVEL (J ,1) BNOT=AMIN1(BNNOT,BNUGLD,2,)+.5 I F ( I . E Q . 1 ) BNNOTH=BNOT CALL CCLION <J,I) CALL B E C ( J , I ) IE (ULTEA, OB, ( ( I T . EQ. WIT.OB. QUIT) .AND. VEBBO) ) WRITE ( $ 6,1029)BNOT, $ ENNOT,ENUOLD, FUDJ 1029 FOBMAT {» RNOT,RNNCT,RNUOLD,FUDGE FACTOB•,2F20.1, $ 2E15.3) BNNOT=BNGT D B E C ( I J ) = D R E C ( I J ) * F DD J IF(.NOT.DIELEC) D B E C ( I J ) = 0,0 C C TH BEE ECDY BECGMBINATION FBOM SUMMERS AND BDBGESS C WITH A DIFFERENT Z DEPENDENCE C IF(,NOT.THREEB) GO TC 46 ETHBEE=1. 16E-08* (J**3) *TM45*DENE I F ( B B E C ( I J ) .EQ.O. 0) BTHBEE=0. 0 46 A B F C ( I J ) = R R £ C ( I J ) + E E E C ( I J ) + UB EC(IJ)+RTHREE C C B E C ( J , I ) IS RECOMBINATION BATE INTO J FROM J + 1 C C O L ( J , I ) I S COLLISION RATE OUT OF J TO J + 1 C I F ( ULTRA .OR . ( ( I T . EQ. NIT. OB. QUIT) • AND. VEBBO) ) WRITE ( $ 6,1001) $ I I , *3,CS (I J) , SLTE (IJ) ,PPHOT (IJ) 1001 FORMAT(« COLLISIONS, UPPER L E V E L S , PHOTO IONIZATION $ ,»BATE«, $ • ION (Z,N) ' ,2I3,3E15. 4) I E (ULTRA. OB, ( ( I T . EQ, NIT.OB wQDIT) . AND. VEBBO) ) WBITE{ $ 6,1000) $ I I , J , R R E C ( I J ) , DREC ( I J ) ,UREC (IJ) 1000 FOBMAT (• BAEIATIVE, EIELECTEONIC, UPPER L E V E L S , REC* $ »OMBINAIION $ *RATE* 213, 3E15.4) C C TOPOUT (J) IS RATE J TO J + 1 C TOPIN(J) I S RATE J+1 TO J C 3 CONTINUE IJJ=INDEX (1,1) TOPOUT(1)=PPHOT ( I J J ) + (CS (I J J ) + S L T E (I J J) ) *DE NE I F ( I . EQ. 1) TOPOUT(1)=TOPOUT <1)+-BDO*X <2,5)*ABUND (5) $ +BDN*X (2,4) *ABUND (4) I F ( I , EQ. 4) TOPOUT (1)=TOPOUT (1) +BUN*X (2,1)/AEUND (4) I F ( I . EQ.5) TOPOUT (1)=TOPOUT ( 1) +EUO*X (2, 1) /ABUND (5) TOPIN (1) = ABEC ( U J ) *BENE I F ( I . EQ. 1) TOPIN (1)=TOPIN (1) +EUO*X (1 ,5) * ABUND (5) f  $  24 25  +BDN*X(2,4) *ABUND (4) I F ( I . EQ.4) TOPIN(1)=T0PIN (1) +EDO*X (1,1) /ABUND (4) IF (I.EQ.5) TOPIN (1)=TCPIN (1) +BDN*X(1, 1) /AB UND (5) I F (AREC ( U J ) ,EQ. 0.0) GO TO 24 RATIO (1)=TOPOUT (1)/TOPIN { 1) GO TO 25 RATIO (1) = 1. 0 EEL (1) =RATIO (1) I F ( I . EQ, 1) GO TO 8 DO 4 J J = 2 , I Z IJ=INDEX ( J J , I ) TOPOUT (JJ) =PPHOT(IJ) + (CS ( I J ) +SLTE (IJ) ) *DENE TOPIN(JJ) = AEEC(IJ)*EEN E I F (TOPIN ( J J ) ,EQ. 0.0) GO TO 5 RATIO (JJ) =TOPOUT(JJ) /TOPIN (JJ) GC TO 4 RATIO (JJ) = 1. 0 CONTINUE  5 4 C C RATIO (J) I S POPULATION LEVEL J + 1 / LEVEL J C R E I ( J ) IS POPULATION RELATIVE TC LEVEL 1 C EEL (1) I S POP LEVEL 2 / POP LEVEL 1 C DC 6 J J = 2 , I I REL (JJ) =EATIO ( J J ) * B E I ( J J - 1 ) 6 CONTINUE 8 SUM-1.0 I F (AREC (INDEX (1,1) ) .EQ. 0.0) SUM=0.0 IF (I.EQ. 1) GO TO 31 C C I F RATE INTO LEVEL FROM TOP I S 0 SET POPUIATICN TO 0. C DO 7 J J = 2 , I I I F (AREC (INDEX ( J J , I) ) . EQ. 0.0) REL (JJ-1) =0. 0 S0M=SUM*REL ( J J - 1) 7 CONTINUE I F (AREC (INDEX ( I I , I ) ) .EQ.O.) R E L ( I I ) = 0. 31 SUM=SUM + BEL (II) FNOEM=1./SUM C C X ( J , I ) I S RELATIVE POP OF IONIZATION LEVEL J I N ATOM C SUM HITH I CONSTANT IS 1 C DO 9 J = 1 , I I I F ( A R E C ( I N D E X ( J , I ) ) . G T . 0 . 0 ) GO TO 16 X (J,I)=0.0 9 CONTINUE 16 NB=J NB1=NB+1 X (NE,I) = FNOEM DO 17 J=NB1,111 X ( J , I ) = REL (J-1) *FNOEM 17 CONTINUE 2 CONTINUE C C ELECTRON DENSITY CALCULATION  c  EENE=0. EO 21 1= 1,NELMNT II1=IZED <I)+1 DO 22 J = 2 , I I 1 DENE=DENE+X ( J , I) * (J-1) * JIB ON C (I) 22 CONTINUE 21 CCNTINUE DENE=DENE*DEN*WTNE+(1.-WTNE)*DEOLD EE= DENE/DEN WRITE (6,1010) DENE 1010 FOEMAT (• NEW ELECTRON DENSITY I S « , E 1 6 . 7 ) I F ( T E R S E ) GO TO 113 DO 20 1=1,NELMNT II=IZED(I) IZ 1 = 11 + 1 I F (ULTRA. OR. ( (IT, EQ, NIT.OR. QUIT) . AND, CN VG) ) WRITE ( $ 6, 1005) I I 1005 FORM A T ( ' RELATIVE AEUNEANCES FOR ELEMENT Z = » , I 3 ) I F (ULTRA. OR, (<IT. EQ.NIT.OR, QUIT) . AND. CN VG) ) WRITE(6 $ ,1004) $ ( J , X ( J , I ) ,J=1,IZ1) 20 CONTINUE 1004 FORMAT (1X,5 (15, £15.5) ) 113 IF(QUIT) GO TO 111 C C CHECKING FOR CONVERGENCE OF ELECTRON DENSITY C CONVERGENCE SEEMS TO EE SLOW WITH THIS METHOD C IF(AES(BEOLD-DENE)/DENE.LT.TOL) QUIT=.TBUE. DEOLD=DENE 1 CCNTINUE 111 CONTINUE C C NOTE THAT THERE I S NO LINE COOLING OF BARE IONS C I F (TOLD.EQ.T) GO TO 23 CALL LINCOL C C L I K E COOLING CALCULATED ONLY I F TEMPERATURE HAS CHANGED C I F (.NOT. EQUIM) TOLD=T 23 CONTINUE C C HYDROGEN LINE COOLING LOSSES C DONE AS ACCURATELY AS PGSSIELE SINCE COOLING IN 1E4 C 3E4 TEMPERATURE C RANGE IS CRUCIAL C ICOCL (1)=0. RNNOT=RNNOTH CALL HLINE (BLCOOL, 1) C THE 1 REFERS TO LOWER LEVEL FOR TRANSITIONS DO 30 N=1,9 LCOOL (1) =LCOOL (1) +HICOOL (N) HLCCOL (N) = HLCCOL(N) *X ( 1, 1) * ABUND( 1) *DENE/DEN  143  30 1011  C C C C C  CONTINUE I F (WLIN. AND. CNVG) WEITE(6, 10 11) HLCOOL FORMAT(' HYDROGEN LINE LOSSES A R E : » / 1 X , 9 E 1 4 . 4) IIKLCS=0. LRRAD=0. PHEET=0., DO 10 1=1,NELMNT II=IZED (I) DO 10 J = 1 , I I IJ=INDEX ( J , I ) ADJUSTMENT OF RECOMBINATION RATE TO ENERGY ON RATE USING FACTORS GIVEN BY SEATON FOR HYDROGEN  RECOMBINATION  UL=IF1 ( I J ) * T K I ULL2=.5*ALOG(UL) UL3=UL**(-.3333333) ABFACT= (-0. 0713+ULL2+0. 640*UL3) / ( G . 4288+ULL2+.469* $ UL3) BRCCGL=RBEC ( I J ) * ( I P 1 (IJ) +TK)*ABFACT C C C C C  DIELECTBONIC COOLING ASSUMES LOWEST ENERGY TRANSITION IS DOMINANT S T A B I L I Z I N G TRANSITION RDCCCL=DEEC(IJ)*(IP 1(IJ) +LOWLIN(IJ) ) LRRAD=LRRAD+ (RRCGOL + RDCOOL) *X (J+1 ,1) *ABUND(I) L C L X ( I J ) = X ( J , I ) *LCGOL ( I J ) * A FUND (I) * F E IIEICS=LINLCS+LCLX (IJ) CPHEAT(IJ) =PPHEAT(IJ) *X ( J , I ) * ABUND (I) PHE ET= PfiEET+CPHEAT (IJ) CONTINUE  10 C C ALL ENERGY RATES ARE IN ERG CM+3 S-1 C BEATING IS IN ERG S-1 C I-BBEMS=2.29E-27*SQBT (T) * ABUND (1) * F E LRRAD=LRRAD*FE*T.6 02192E-12 COCL= LBB EMS+LRR AD+LINLOS PHE'ETD=P BEET/DEN I F (.NOT.EQUIM) GO TO 38 I F ( .NOT. TERSE) WRITE (6,1021) PHEETD,CCOI,T 1021 FORM AT(* BEATING, COOLING RATES CM + 3 S - 1 « , 2 E 1 5 . 5 , $ 5X,»AT TEMPERATURE *,E15.5) DIFCLD=DIFF IF(PHEEID.GT.1.0E6*CCOL) DIFCLD=0. I F ( FfiEETD. LT. 1.0 E-6*COOL) D3FOLD=0. DIFF=COOL-PBEETD IF(CNVG) GO TO 20002 C C RADIATIVE EQUILIBRIUM TEMPERATURE CALCULATION C I F (.NOT. FIRST) GO TO 20000 TOID=T T=T*1.01  FIRST-. FALSE. GC TC 20001 20000 DERIV=(DIFF-DIFOLD)/(T-TOLD) IF(DERIV,EQ.O.) GO TO 20002 DELT=-DIFF/DERIV I F (. NOT. TERSE) MBIT I {6, 1022) T,DELT 1022 FORMAI(* T, DELTA T (DELT) * ,2E15. 5) I F (ABS(DELT/T)•GT.I'M AX) GO TO 20003 I F (ABS (DELT/T) .LT. TOL) CNVG=.TRUE. TOL D= T T=T+DELT GO TO 20001 20003 IOID=T T—I+DMAX*DELT/ABS(DELT)*T GO TO 20001 20002 CONTINUE 38 I F (.NOT. WLIN) GO TO 11 C C FEINTED OUTPUT OF DETAILS OF BEATING AND COOLING BATES C DG 12 1=1,NELMNT IZ=IZED (I) IJB=INDEX (1 ,1) IJE=IJB+IZ-1 WRITE (6,1009)IZ 1009 FORMAT (• LINE COOLING LOSSES FOB ATOM GF Z»,I3) WBITE(6,1003) ( L C I X ( U ) , IJ= I J E , I J E ) 1003 FORMAT(1X,8E15.3) 12 CONTINUE I F ( W F J . L E . 0.0. OB.. NOT. WLIN) GO TO 11 DO 13 1=1,NELMNT IZ=IZED(I) IJB=INDEX (1 ,1) IJE=IJB+IZ-1 WRITE (6,1 019) I Z 1019 FORMAT(» PHOTOIONIZATION SEATING BATES FOB ATOM Z=« $ ,13) WBITE(6,1003) ( C P H E A T ( I J ) , I J = I J B , I J E ) 13 CONTINUE C C INTERNAL ENERGY AND ENTHALPY IN UNITS OF EEG PER C CUBIC CM C 11 EINT=0. DO 14 1=1,NFLMNT IZ=IZED(I) DO 14 J=1,IZ IJ=INDEX ( J , I ) EINI=EINT + IE1 ( I J ) *X (J + 1,I) *ABUND(I) 14 CONTINUE EINT=(EINT*DEN+1.5*TK*(DEN+DENE)) * 1.602192E-12/DEN ENTHAL=EINT+(DEN+DENE)*TK*1.602192E-12/DEN CEINT=£INT/(TK*1.602192F-12) CENTHP=ENTHAL/ (TK* 1. 602192E- 12) C C OUTPUT CALCULATED QUANTITIES  c I F (OUTPUT) WBITE(7) DIN ,T, HF, F J , PHEET, COOL, L BE EM S , $ LRRAD,LINLOS, $ EI NT , ENTHAL,DENE, AEUND, X ,LCLX,CP HEAT, PFTOT IF{EQ DIM) WRITE (6,1051) ICLCOP 1051 FOBMAT (» NUMBER OF LOOPS TO CONV EBGENCE ,II) WRITE (6,1006) D E N , T , » F J , P H E E T D , C O O L , L B R E H S , L R B A D , $ IINLOS, $ EINT,ENTHAL,CEINT,CENTHE,DENE 1006 FOBMAT (»- PABAMETEBS WIRE: DENSITY, TEMPEBATOBE, • $,'DILUTION FACTOB•,3E15.5/ $ » TOTAL HEATING/DENSITY AND COOLING B A T E , 2 E 1 5 . 5 / $ » THE COOLING BAT IS FOB EE EMSSTRAHLUNG, », $ 'R ECO MBIN ATION RADIATION, AND LIKE LOSSES *,3E15.5/ $ ' I N T E R N A L EN ERG Y , EN TH ALP Y» ,2E1 5. 5 ,5X ,' AND COEFFI $ ,» CIENTS* ,2115.7/ $ • ELECTRON DENSITY*,E15.5) I F (. NOT.TERSE) WRITE (6, PABAM) NOWAST=.TRDE. GO TO 10000 3OCO0 WBITE (6,1049) NLOOP 1049 FOBMAT (« SOBBY EUT MAX NUMBIB OF TEMPEBATUBE LOOPS $ ^EXCEEDED',13) GG TO 10000 1000 1 STOP END 1  1  PROGRAM  SjgECHJAT  SUBROUTINE CHGEX(BBC,BDO,BUN,BDN,T) C C CHARGE EXCHANGE TAKEN FEOM: C 0 FIELD AND STEIGMAN C N STEIGMAN, HEFNER, AND GELDON C C EUO I S EETA FOR OI TO O i l THAT IS UP FI(X)=ERF(SQRT(X) ) -1. 1 2 83 8*EXP (-X) *SQBT {X) XAC=6.034/T XAD=732.8/T XC=0.812336/T XD=S8.64/T BDO=1. 97E-0 9* {, 3864 15**1 (XAC) ••0. 5* (PI (XAD) - F l (XAC)) $ + 0.529412* ( 1 . $ E I (XAD)) ) +2. 11E-09*( (0.115385*EXP (- XC) * $ FI(XAC-XC) + I 0.0294118*EXE (-XD) * F I (XAD-XD) ) ) £UO=EXP (-227. 45/T) * ( i . 97E-0 9-BDO) BUN=1. S7E~0 9*(EXP (-11031. 5/T) *.333333+EXP(- 11102. 3/ $ T)*.333333+ $ EXP (-11220. 7/1) *. 151515) EDN=1.97E-09-EXP(11031.5/T)*BUN BETUBN  END  ELOCK DATA COMMON /A/ INDEX,IZED,DEN,DENE,T,IK,TKI,T4,TSQRT, $ VEBECS,LAST INTEGER IZED (9) DATA IZED /1,2,6,7,8,10, 12,14,16/ DIMENSION INDEX (16,9) DATA INDEX /1,15*0,2,3,14*0,4,5,6,7,8,9,10*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,8*0, $ 25,26,27, 28, 29, 30, 3 1,3 2, 33,34,6*0, $ 35,36,37,38,39,40,41,42,43,44,45,46,4*0, $ 47,48,49,50,51,52,53,54,55,56,57,58,59,60,2*0, $ 61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76/ E  N  D  SUBROUTINE COLIGN(J,1) C C BOUTINE FOR CALCULATION OF COLLISIONAL C IONIZATION BATES FOB ALL ELEMENTS BUT C HXEEOGEN C DIMENSION INDEX (16,9) ,IZED (9) EEAI IP1 (76) ,IP2 (76) DIMENSION NUM1 (76),NUM2 (76) ,CS ( 7 6 ) , S L T E ( 7 6 ) BEAL CHIT (76) LOGICAL VERBOS ,SEMICO,ULTRA COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBOS,LAST COMMON /CION/ IP2,NUM1,NUM2,CS,SLTE COMMON /COLBEC/ IP 1,CHIT,BNNOT  3 101  COMMON /CONTRO/ SEMICO,ULTRA IJ=INDEX (J,I) IF(J.NE.IZED(I) ) GO TO 199 I E (.NOT. SEMICO) RNC=1.0EQ6 CS(IJ) =COLH (RNNOT,IE 1 (IJ) ,1) SLTE(IJ)=0. RETURN  C C CORRECTION FACTOR FROM P 205 MCWHIRTER ,R. W. P. , IN ATOM C IC AND C MOLECULAR PROCESSES IN ASTROPHXSICS, ED EY MCE HUBBARD C AND H C NDSSBAUMER, GENEVA OBSERVATORY, SAUVERNY, SWITZERLAND, C 1975. C 199 CS(IJ) = NUM1 (IJ) * EXP(-IF1 (IJ) *TKI) / (IP 1 (IJ)*IP1 (IJ) ) $ /(4.88+TK/IP1 (IJ) ) IF ( NUM2 (IJ) . LE. 0) GO TO 200 CS (IJ)=CS (IJ) +NUM2 (IJ) *EXP (-IP2 (IJ) *TKI) /(IP2 (IJ) * $ IP2(IJ)) / (4. 88+TK/IP2 (IJ)) 200 CS (IJ) =8.35E-08*TSQET*CS (IJ) C CHIT(IJ)=(2.8E-28*IP1(IJ)*DENE*DE NE*TKI)**. 142857143 CHIT (IJ) =IP1 (IJ) / (RNNOT*RNNOT) SITE (IJ)=4. 8E-06*CHIT (I J) / (IP 1 {IJ) *IP1 (IJ) *TSQRT) * $ EXP(-IP 1 (IJ) $ *TKI) IF(.NOT.SEMICO) SLTE(IJ)=0. BETUEN C C CHIT IS ESTIMATE OF IONIZATION- POTENTIAL OF LOWEST LEVEL C IN EQ'M WITH CONT * M C SITE FROM WILSON C END FUNCTION COLH(RN0,XIP,N) C C COLLISICNAL IONIZATION RATE FOR HYDROGEN C DIMENSION INDEX (16, 9) DIMENSION IZED (9) LOGICAL VERBOS COMMON /A/ INDEX,IZID,DEN,DENE,T,TK,TRI,T4,TSQRT, $ VERBOS,LAST X0=1.-1./(RN0*RN0) X02=1./(X0*X0) X03=X02/X0 RN=FLOAT (N) EN2=SN*RN EPKT-XIP*TKI/RN2 Y=X0*EPKT C FOR OPTICALLY THICK CALCULATIONS CAN USE N OTHER THAN 1 IF (N. EQ. 1) GO TO 2 A=1.9602805*RN*X03*(.3595-0.05798/X0+5.894E-03*X02) B=. 6666667*BN2/X0* (3. + 2./X0 •». 11 69*X02 ) Z=X0* (0.653 + EPKT) GO TO 4  2  A=1.9602805*X03*(.37767-0.1C 15/X0 + 0, 01<4028*X02) B=. 6666667/X0* (3.+2./XO-0.603*X02) Z=XO*(0.45+EPKT) 4 I F ( Z . G E . 170. ) GO TO 3 COLH= 1.093055E-10*R N2/X0*TSQRT*Y* v* $ (A* (EGNE (Y,IY) /Y-EONE (Z,INZ) /Z) • $ (B-A*ALOG (2.*RN2/X0) ) * (ZETA (Y) -ZETA (2) ) ) CCLH=CCLH*(13.598/XIP) **2 I F (IY.EQ.O.OR.INZ.EQ.O) G O T O 1 EETURN 1 WRITE(6, 1000) Y,Z,IY,INZ 1000 FORMAT (' * * * * * * ERROR I N EQNE*,4E15.4) RETURN 3 COLH=0. BETURN C C JOHNSON'S COILISICNAL IONIZATION FORMULA C CURRENTLY ONLY FOB IONIZATION FfiCM LEVELS 1 AND 2 C END FUNCTION ZETA (T) C C L I T T L E FUNCTION REQUIRED BY COLE C EE-EXP (-T) E0=EF/T E1 = EONE (T, IN D) IF(IND.EQ.O) WRITE (6,100) T 100 FORMAT(• * * * * * ZETA EGNE ERBOR*,E15.4) I2=EF-T*E1 ZETA=E0-2. *E1*E2 BETUBN END SUBROUTINE R E C ( J , I ) C C CALCULATION OF RECOMBINATION RA1ES FOB C ALL ELEMENTS BUT HYDROGEN C USES ALDROVANDI AND PEQUIGNOT TABLE C DIMENSION INDEX (16,9) DIMENSION IZED(9) REAL ARAD (76) , ETA (76) ,TMAX(76) ,TCRIT (76) ,ADI(76) , $ TO (76) , EDI (76) , $ T l (76) ,RREC(76) ,DBEC (76) REAL C H I T ( 7 6 ) , U R E C ( 7 6 ) , I P 1 ( 7 6 ) LOGICAL VEREOS ,ULTBA,SEMICO,FUDGE COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBOS,LAST COMMON /RECO/ RREC,DREC,U8EC,ARAD,ETA,TMAX,TCRIT,AD $ I,T0,BDI,T1 COMMON /COLREC/ I P 1,CHIT,RNNOT COMMON /CONTEO/ SEMICO, ULTRA COMMON /CFUDJ/ F UD J,R N OT,F UDG E IJ=INDEX ( J , I ) FUDJ=1. I F ( J . NE. IZED (I) ) GO TO 199  I F (IP1 (1*3) * T K I . GT. 170.) GO TO 200 DP.EC ( I J ) = 0 . RREC(IJ)=0.0 Z=FLGAT (J) NNOT=BNOT NTGP=MIN0 (9,NNGT) DO 111 N=1,NTOP C CAN CHANGE DO LOOP BANGE TO 2,9 FOB OPTICALLY THICK TO C LYMAN ALPHA BBEC (IJ) = BBEC ( I J ) •Z*BHII (IP 1 (IJ) , N) 111 CONTINUE I F (NTOP.EQ.9) GO TO 112 BBEC ( I J ) =RREC ( I J ) + (BNOT-FLOAT (NTOP) ) *Z*BHII (IP1 ( I J ) $ , NTCF+1) 112 CONTINUE UBEC ( I J ) = 0 . 0 BETUEN C C FACTOR OF 3 I S TO MAKE UP FOB TENDENCY OF TMAX QUOTED TO C EE MUCH TO LOW C 199 I F (T. GT. 3. *TMAX (IJ) ) GO TO 200 I F ( T . L T . T M A X ( I J ) / 2 0 0 0 . ) GG TO 200 E B E C ( I J ) =ABAD(IJ) * I 4 * * (-ETA (IJ) ) UREC ( I J ) = 1. 8E-14*IP1 (IJ) *TK** (- 1. 5) * C H I T ( I J ) I F (.NOT. SEMICO) U R E C ( I J ) = 0 . GO TO 299 s 200 BBEC ( I J) =0.0 DBEC(IJ)=0.0 UREC(IJ)=0.0 BETORN 299 I F ( T . L T . T C B I T { I J ) / 1 0 . ) GO TO 300 C C FACTOR OF 10 AN ATTEMPT TO MAKE TRANSITION SMOOTHES C DREC (IJ) =ADI ( I J ) *T** (-1.5) *EXP (-TO ( I J ) / T ) * (1. tBDI (IJ) $ *EXP (-T1 ( I J ) / $ T)) I F (» NOT. FUDGE) RETURN AEG=12. 55-7.*ALOG10 (BNOT) IF(ARG.LE.0.) ARG= 0. DELA=.01458333*ARG*ARG+0.09166667*ARG FUDJ=10.** (-DELA) BETORN 300 DREC (IJ) =0.0 C C ALL BUT HYDROGEN FROM FORMULAE OF ALDROVANI AND PEQUINO C T IN AA C H L I K E FROM JOHNSON C RETURN END BEAL FUNCTION RHII(XIP,N) C C RECOMBINATION TO HYDROGEN C 101  DIMENSION INDEX(16,9) DIMENSION IZED (9) LOGICAL VEBBOS BEAL IP1 (76) ,CHIT (76) COMMON /A/ INDEX,IZED,DEN ,DENE,T,TK,TKI ,T4 , TS QBT , $ VEBBOS,LAST COMMON /COLBEC/ IP 1,CHIT,BNEOT COMMON /CFUDJ/ FUDJ,RNOT,FUDGE NNOT-BNNOT X0=1,-1./ (BNOT*BNOT) X02=1./(X0*X0) EIN=1./N FIN2=FIN*FIN XIN=XIP*FIN2*TKI XTIN=XO*XIN X2=XTIN*XTIN C C ABECMOWITZ AND ST EG UN EXPRESSION FOB EXP (X) EONE (X) C NOTE THAT X=X0*IPN/KT, AND NEED TO MAKE COERECTION TO C EXTEBIOB EXP(IPN/KT) C I F (XTIN. I E . 10.0) GO TO 4 EXE1=(X2+4.03640*XTIN+1.15198)/(X2+5.03637*XTIN+ $ 4.19160)/XTIN GO TO 5 4 EXE1=EXP (XTIN) *ECNE (XTIN,INX) I F ( I N X . E Q . 0) WBITE (6, 1000) XTIN 1000 FOBMAT(* *******EONE EBBOB I N BHII******,E15.6) 5 EXE2=1. -XTIN*EXE1 EXE3=0.5*(1.-XTIN*EXE2) IF(N.GT.2) GO TO 3 I F (N. EQ, 2) GO TO 2 G0=1. 133 G1=-0.4059 G2=.07O14 GO TC 1 2 G0=1.0785 G1=-0.2319 G2=0.02947 GO TC 1 3 G0=0.9935+0.2328*FIN-G.2196*FIN2 G1=-EIN*(0.6282-0.5598*FIN+C.5299*FIN2) G2=FIN2* (0. 3887-1. 181*FIN+1. 470*FIN2) 1 BHII=5.197E-14*XIN**1,5*EXP(XIN/(BNOT*BNOT))* $ (GO* EXE1+G1*EXE2/X0 +G2*EX E 3*X 02) C MULTIPLY ANSWER BY Z OF ION BETUBN END BEAL FUNCTION G E N ( I E , I J , I Z , Y ) C C GAUNT FACTOB CALCULATION C USES MEWE AND KATO DATA C ANE MEWE APPBOX FOB EXP(Y)EGNE(Y) C DIMEKSICN A(49) ,B(49) ,C(49) ,D{49) DIMENSION INDEX (16,9)  1  DIMENSION IZED (9) LOGICAL VERBOS REAL IP1 (76) ,CHIT (76) COMMON /A/ INDEX,IZED,DEN,DENE,T,IK,T Kl ,T4 , TS QRT , $ VERBOS,LAST COMMON /GFACI/ A,B,C,D COMMON /COLREC/ IE 1,CHIT,RNNOT X=1./(IZ-3.0001) I E (ID. GT. 28) GO TO 1 I F (ID.EQ.20) GO TO 100 I F (ID. EQ. 22) GO TO 10 3 I F (ID.EQ.21) GO TO 104 I F (ID. EQ. 23) GO TO 101 I F (ID.EQ.28) GO TO 102 GFN= A (ID)+ (B(ID) *Y-C(ID) *Y*Y+B(ID) ) * ( ALOG ( (Y+1.) /Y) $ - 0 . 4 / ( ( Y + 1 . ) * ( Y + 1 . ) ) ) + C ( I D ) *Y RETORN  C  C ALL A LA MEWE C WITH ADDITIONS DUE TO KATO C 100 A ( I D ) = 0 . 7 * ( 1 . - . 5*X) B(ID) = 1.-0.8*X C <ID)=-0.5* (1.-X) GO TO 1 101 A (ID)=0. 11* (1.+3. *X) GO TO 1 102 A(IE)=0.35*(1.+2. 7*X) B (ID) =-0. 1 1* (1. *5. 4*X) GO TO 1 103 A (ID) = -0. 16* (1.+2. *X) B (ID) =0.8* (1.0-0. 7*X) GO TO 1 104 A(ID) =-0.32* (1.-0. 9*X) B(IB)=0. 88* (1. -1. 7*X) C (ID) = 0. 27* (1.-2. 1*X) GO TO 1 END SUBROUTINE LINCOL C C L I N E COOLING C RITH MODIFICATIONS FOR F I N I T E OPACITY C USES L I N E L I S T FROM MORTON C AND MORTON AND HAYDEN SMITH C REAL LCCOL (76) , ELINE (407) ,F (407) INTEGER IIND (407) ,JIND (407) ,IDENT (407) DIMENSION INDEX (16,9) DIMENSION IZED (9) INTEGER IB (16) / 1 , 2, 3*0, 3, 4, 5,0, 6, 0, 7, 0, 8, 0, 9/ REAL X(17,9) ,ABUND (9) COMMON /THICK/ X,ABUND,DV,TAUMAX COMMON / I I N E / L C O O L , E L I N E , F , I DENT,IIND,JIND,NLINE COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQBT, $ VEREOS,LAST TAU=1.  152  I F ( D V ) 11,11,12 COLUMN=DEN/DV CONTINUE DO 10 IJ=1,LAST 10 I C G O T ( I J ) = 0. DO 1 L=1,NLINE II=IIND(L) I F ( I R { I L ) .GT.LAST) GO TO 1 I F ( I L . EQ, 1) GO TO 1 C NOTE THAT THIS I S THE 2 OF THE ION JL=JIND(L) IV=IR(IL) IJ^INDEX <JL,IV) Y=ELINE(L)*TKI I F ( T A U MAX) 5,5,4 4 TAU=3.2905 E - 6 * F ( L ) * A B U N D ( I ? ) * X ( J L , 1 V ) *CGLUMN/ELINE(L) I F ( T A U - . O I ) 6,7,7 6 TAU=1. GO TO 5 7 TAU= (1.-EXP(-TAU))/TAU 5 G=GFN (IDENT(L) , I J , I L , Y ) 3 ICCCL (IJ) = L C C O L ( I J ) + F (L) *G* EXP {-Y) *TAU 1 CONTINUE DC 2 IJ=1,LAST LCOOL ( I J ) = 2 . 7 1 E - 1 5 / 1 S Q R T * L C C C L ( I J ) 2 CONTINUE BETUBN END SUBROUTINE HLINE(HLCCCI,NBOT) C C L I N E COOLING FOR HYDBOGEN C BEAL IP1 (76) ,CHIT (76) ,HLCCCI (9) INTEGEB INDEX (16,9) ,IZED(9) LOGICAL VERBOS COMMON /COLEEC/ IP 1,CHIT,RNNOT COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VEEEOS,LAST NNOT=RNNOT FB=FLOAT(N BOT) FB2=FB*FB EE4=FB2*FB2 FIN=1./FB FIN2=FIN*FIN IF(NBOT.GT.2) GO TO 3 I F (NEOT. EQ. 2) GO TO 2 G0=1. 133 Gl=-0.4059 G2=.07014 GO TO 11 2 G0=1.0785 G1=-0.2319 G2=0.02947 GO TO 11 3 G0=0.9935*0.232 8*FIN-0.2196*FIN2 G1=-FIN*(0.6282-0.5598*FIN+C.5299*FIN2) 12 11  153  G2=FIN2* (0. 3887-1. 1 81*FIN*-1 .470*FIN2) SE=-0.603 IF (NBOT.EQ.2) SB=.1169 RN=0. 45 IF (NBOT.GE.2) RN-1.94*FB** (-1.57) DG 9 N=1,9 HLCOOL(N)=0.0 9 CONTINUE NTOP=MIN0(9,NNOT) KE1=NBOT+1 DO 1 N=NB1,NTOP FN= FLCflT (N) FN 2=FN*FN FN3=FN*FN2 X=1.-(FB/FN)**2 ANN=3.920561 *(FB/FN)**3/X**4*(G0+G1/X+G2/(X*X)) BNN=4.*FB4/(FN3*X*X) * ( 1 . + 1. 233333/X + SE/(X*X) ) Y=13.59 8*TKI/FB2*X ENN=BN*X Z=RNN+Y E1Y=EGNE(Y,INY) E1Z=E0NE(Z,INZ) I F (INY. EQ.0.OB.INZ.EQ.0) WR I T E (6, 1000) Y,Z 1000 FOBMAT {• *******ECNE EEBOB IN HLINE',2E15.5) HLCOOL(N)=2.3814724E-21*TSQRT*FB2*Y*Y* (ANN* ( (1./Y+. $ 5)*E1Y $ - (1. /2 + . 5) *E1Z) • (BNN- ANN*ALCG (2.*FB2/X) ) * $ ( ( E X P ( - Y ) - Y * E 1 Y ) / Y - ( E X P ( - Z ) -Z*E1Z)/Z) ) *X C C T E E FINAL MOST X IS FOB T B E ENERGY OF THE TRANSITION C T E E CONSTANT HAS A B O U T I N 13.598EV AND AN EV TO EBG C CONVERSION C 11  1  CONTINUE BETDBN END  SUBBOUTINE LEVEL ( J , I ) C C LOWEST LEVEL IN EQUILIBRIUM WITH CONTINUUM C DIMENSION IP1 (76) ,CHIT (76) DIMENSION INDEX (16,9) , IZED (9) LOGICAL VERBOS,SEMICO,ULTRA  COMMON /CONTBO/ SEMICO,ULTBA  COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VEBBOS,LAST COMMON /COLBEC/ IP1,CHIT,RNKCT C C SEATGN'S ESTIMATE OF LOWEST LEVEL IN EQUIIIEBIUM WITH C C CNTINUUM C FOLLOWS WILSON, BUT WILSON'S FUMBERS USED. C RJ=J BNCE= (1. 4E15*RJ**6.*TSQRT/DENE) **. 1428571 RNNOT=RNCE I F ( J . NE. IZED (I) ) RETURN  154  2 3 1000  Y=6.337053E-06*T/J**2 CCNS= < ¥ / ( D E N E * D E N E ) ) * * . Q 5 8 8 2 3 f 3 * R J * * . 823594 DO 2 IT=1,5 CQNS2=. 23529i»/(RNCE**3*Y) BNC=126.*CONS*EXP(CGNS2) DIF=BNCE-RNC I F (TABS (DIE) . LT. 0.5) GO TO 3 fiNCE=BNCE-DIF/(1.+BNC*3.*CGNS2/BNCE) CONTINUE IF(.NOT.SEMICO) RNC=1.0E06 I F (VEBBOS) WBITE (6, 1000) BNC, IT FOBMAT(* CONTINUUM IEVEL*,F10.1,» ITEEATION #',I3) BNNCT=BNC BETUBN END  155  FBOGBAM COEF C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  A A PBOGBAM TO GFNEE ATE PHYSICAL PARAMETERS WHICH GO INTO THE COEFFICIENTS CALCULATED FOE THE DISPERSION RELATION N INPUT IS FROM UNIT 1 AND CONSISTS OF THE DERIVATIVE OUTPUT FEOM THE BEATING AND COOLING PROGRAM IN THE FORM (N,T),{N ,T+/-DELT), (N +/-DELN,T) . OUTPUT IS TG UNIT 7 AND CONSISTS OF THE 2EE0 ORDER QUANTITIES AND THEIR DENSITY AND TEMPERATURE DERIVATIVES. NAMELIST PARAMETERS WFUDGE A FUDGE FACTOR FOB ALT IRING FLUX GRAV THE GRAVITATION IN CM S-2 VO THE GAS VELOCITY DV VELOCITY DEBTVATIVE WHICH MAY BE CALCULATED INTERNALLY, BUT A STARTING VALUE IS NEEDED., NCNEQ CALCULATION OF ENERGY FQUILIBBIUM BY MAKING OP DIFFERENCE WITH CON DUCT ION, DNON AN DENSITY DIFFERENT THAN USED I N HEATING AND CCCLING TNGN A TEMPERATURE DIFFERENT THAN USED FOR HEATING AND COOLING DRHO DNON CONVERTED TO DENSITY (GM CM-3) DT TEMPERATURE DERIVATIVE D2T SECOND DERIVATIVE GF TEMPERATURE WITH RESPECT TO DISTANCE, CONKAP CONDUCTION CONSTANT. NIINE NUMBER OF LINES IN FORCE CALCULATION CEF THE INVERSE OF THE FACTOR TG CONVERT FROM ZERO TO FIRST MOMENT OF THE RADIATION FIELD VERBOS CONTROLS PRINTING I F ON SEE DETAILS OF X DERIV ATIVES,ETC. SKIP I F CN NG NEW HEATING COOLING DATA READ, JUST C ALCULATES FORCE SLAB FOB A STATIC ATMOSPHEBE THIS I S THE E F F E C T I V E COLUMN DEPTH CCNDUC I F OFF THE CONDUCTION QUANTITIES ABE SET TO ZE BO TEAD IS THE BRIGHT NESS TEMPERATURE OF THE RADIATION FTELD/WF USED IN THE CALCULATION OF THE WAVE DAMPI NG DUE TO RADIATION DTCL IS THE LEVEL GF T E E LOGARITHMIC DERIVATIVE BEL OW WHICH I T I S TO ZERO.  156  C C C C C C C C C C C C C C C C C C C C C C C  DYNEQ  I F ON THE MOMENTUM EQUATION IS EALANCED BY AEJ USTING EV DYNIT NUMBER OF ITERATIONS I K DYNEQ DYNTCL ACCURACY BEQUIRED OF DV IN MOMENTUM BALANCE WLINE I F ON THE FORCE CALCULATION FOB EACH LINE IS 0 UTPUT SLIM A CONVERGENCE AID IN DXNEQ, WHICH SHOULD BE SE T TC ABCUT 1.5 SPHERE I F ON THE DERIVATIVE OF THE DENSITY I S CAICULA TED FOR A SPHERICAL COORDINATE SYSTEM , ASSUMING SYMMETBY. STABMU I S THE COSINE OF THE ANGLE TO THE STAR. RSTAR I S THE DISTANCE FROM THE CENTRE CF THE STAR. 0 NLY USED BY THE DENSITY AND VELOCITY. DERIVATIVE. THE GRAVI TY MUST BE ADJUSTED. TEE OUTPUT QUANTITIES ABE MOSTLY SELF EXPLANATORY EN ENDINGS ARE DENSITY DERIVATIVES, DT ABE TEMPERATU RE REAL LEREMS , LRR AD, LINLCS,X (17,9) , LCLX{76) ,CPHEAT<76) ,FTOT INTEGER INDEX(16,9) / 1 , 15*0,2,3,14*0,4,5,6,7,8,9,1 $ 0*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,6*0, $ 2 5,26,27,28,29,30,31,32,33,34,6*0, $ 35,36,37,38,39,40, 4 1, 42, 43,44, 45, 46, 4*0 , $ 47,48,4 9,50,51,52,53,54,55,56,57,5 8,59,60,2*0, $ 61,62,63,64,65,66,67,68,6 9,70,71,72,73,74,75,76/ REAL AD (5) ,AT(5) ,AWFJ (5) , AGAIN (5) ,ACOOL (5) , AE (5) , AH S (5), $ ADE (5) ,AX(17,9,5) REAL KBCLTZ /1.38062E-16/ REAL RGAS /8.314E7/ REAL ABUND (9) REAL AMASS(9) /1.008,4.0026,12.0111,14.0067,15.9994 $ 20.179 $ ' 24.305,28.G86,32.06/ DIMENSION ELINE { 1000) ,F (1000) ,11 ( 1000) , J J ( 1 0 0 0 ) ,FLU $ X(100) , $ FPT(100) INTEGER I Z E D ( 9 ) / 1 , 2 , 6 , 7 , 8 , 1 0 , 12,14,16/ LOGICAL FREEZE,NONEQ,VEREOS,SKIP,CONDOC,DYNEQ,WLINE LOGICAL SPHERE BEAL*8 SLIM BEAL VTHEHM(9) EEAI*8 DLOG10,DVOLD,EGBDEV,ESQBT,DSIGN,DABS BEAL*8 N0,T0,V0,DN,DT,DV,RHOCV,SCUND, $ NEO,HO,EO,DNEDT,DNEEN,DHDT,DHDN,DEDT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GRAV,GRAD,GRADE,DGRDT, $ DGRDN, $ GO,L0,DGDT,DGDN,DLDT,DLDN $  157  REAL PHOT (76) , PHEAT (76) , SIGH A ( 100, 76) ,DELE (100) SEAL*8 GRIJ (16,9) ,DXDT(16,9) ,DXDN(16,9) EEAL*8 DIFF,DELV,DIE C L D , D F D E V , DP BE AL*8 V,T,DEN REAL*8 CK,FRAP,DGL,GL,TAU,DEXF INTEGEB NUO (76) ,NUF(76) INTEGEB DYNIT COMMON / ? / ELINE,F,II,JJ,FLUX,FPT,NLINE,NFLUX COMMON /A/ ABUND,X,VTHERM,SLAB COMMON /PHOBCE/ GRAD,DT,DGRDN,DXDT,DXDN,GRIJ,DGR $ DGRDDV COMMON /CONTBO/ WLINE,SPBERE,STARMU,RSTAR COMMON/DER/ DNU,DNL,DTU,DTL ,DEN,T NAM ELIST /PABAM/ F B FEZ E,W FUIG£,GR AV,V 0,DV,NONEQ,D NO $ N,TNON,DRHO, $ DT,D2T,NLINE,CHE,VERBOS,SKIP,SLAB,CONDUC^T $ RAD $ ,DTOL,DY NEQ,DYNIT,DYNTOL,HLINE,SLIM,SPHEBE,STARM $ U,RSTAR NAMELIST /PHYSIC/ NO,TO,VO,EN,DT,DV,RHOCV,SOUND, $ NE0,H0,E0,DNEDT,DNEDN,DHDT,DHDN,DEBT,DEDN, $ D2T,CONKAP,DK DT,DKDN,GRAV,GRAD,GRADE,DGRDT, $ DGRDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN NFLUX=97 FREEZE=. FALSE. HFDDGE=1. GBAV=1. E4 C C V>0 AWAY FROM STAR, SIMILAB1Y Z INCREASES UPWARDS C V0=0. DV-0. NCNEQ=. TRUE. DTOL=.05 EYNEQ=. FALSE. , DYNIT-10 DYNTOL=1.E-3 SLIM=.5D0 TRAD=50000. CONDUC=.TRUE. DNON=1. E11 TNON=0. T= 1. E5 DRHO=0. DT=0. D2T=0. NIINE=874 CHF=1. SLAE=0. VERBOS=.TBUE. WLINE=. FALSE. SKIP=.FALSE. SPHEBE=. FALSE. STARMU=1. EST AR= 1. E12  158  BEHIND 1 BEHIND 2 C C C  BEAD IN LINE AND  FLUX IAT A  (USUALLY  F I L E L FOB C ED AT)  BEAD (2) E L I N E , F , I I , J J , F L U X , F P T C C INCOMING FLUX HAS BEEN MULTIPLIED BY 4*PI/C C DG 49 18=1,96 DEL E (IN) =FPT (IN + 1) -FPT (IN) 49 CONTINUE BEHIND 3 BEAD(3) PHOT,PHEAT,SIGMA,FTOT1 BEHIND 4 BEAD (4,9774) (NDO (IJ) , NUF< IJ) , I J = 1, 76) 9774 FOBMAT (214) C C BEADS F I L E CONTAINING INTEGBATTON FBEQUENCIES WHICH HAS C USED BY PHOTION C 998 BEAD (5,PABAM, END=999) IF(.NOT.VEBBOS) WLINE=.FALSE. I F {. NOT. NONEQ. AND. VO. EQ.O.) V0=1. D7 WBITE(6,1014) C C BEAD IN SET OF 5 OUTPUT QUANTITIES FROM HEAT COOL PBGGBAM C 1014 FOBMAT(*1») WBITE(6,PABAM) I F (SKIP) GO TO 111 DC 14 1=1,5 BEAD(1) D,T A,W FJ,FJ,GAIN,COOL,LBB EMS,LBRAD,LINLOS, $ EINT,ENTH,DE,ABUND,X,LCLX,CPHEAT,FTOT AD(I) = D AT (I) =TA AHFJ (I) = WFJ AGAIN(I)=GAIN AC0CL(I)=CO0L AE(I) =EINT AH (I) = ENTH ADE (I) =DE DO 15 IZ=1,9 IQ1=IZED (IZ) + 1 DO 15 J=1,IQ1 AX(J,IZ,I)=X(J,IZ) 15 CONTINUE 14 CONTINUE FA=ABUND(3)/3.0 35918E-04 WBITE (6,1000) FJ,FA,FTOT 1000 FOBMAT('OFJ,IA,FTOT*,2F11.6,E15.7,/* DEN,TEMP,WFJ,H* $ ,*EAT,COOL,«, $ * EI NT, ENTHALPY, DEN E* ) DO 1 1=1,5 AGAIN (I) = AGAIN ( I ) / A D ( I ) 1 CONTINUE  $ $  WBITE (6,1001) (SD (ID) , AT (ID) , AW FJ (ID) ,AGAIN (ID) , ACOOL (ID) ,AE(ID) , AH (I D) , ADE (IE) , ID = 1 , 5 ) FOBMAT(1X,8E15.1)  1001 C C SET UP CENTRAL VALOES C DEN=AD (1) DENE=ADE(1) T= AT (1) N0=DEN N10=DENE TG=T DEINT=AE(1) DENTH=AH (1) INTH= AH (1) ST4=SQRT(SNGL(T)*1.E-04) C C THERMAL VELOCITY ALREADY DIVIDED BY C C DO 11 1=1,9 VTHERM (I) = 4. 28 33E-5/SQRT (AM ASS (I) ) *ST4 11 CONTINUE DEN 1= ADE (1) C C CALCULATE UPPER AND LOWER DERIVATIVES DIFFERENCES C DNU=AD (4)-AE (1) DNL=AD(1)-AD(5) DTU= AT (2) - AT (1) DTL= AT (1) - AT (3) C C DERIVATIVES OF THE IONIZATION FRACTIONS C DO 16 IZ=1,9 IQ=IZED(IZ) DO 17 J=1,IQ GRIJ (J,IZ)=0. DXDT(J,IZ)=0. EXDN ( J , I Z ) = 0 . T F ( A X ( J , I Z , 1) . L E . 1.E-10) GO TO 17 DXDT ( J , IZ) = • 5* ( (AX (J , IZ , 2) - AX (J , I Z , 1) )/DTU + $ (AX (J,IZ,1)-AX(«3,IZ,3) )/DTL) I F ( E AES ( EX DT ( J , IZ) *T/AX (J ,IZ,1) ). LE. DTOL) DXDT (J,IZ) $ =0.D0 DXDN(J,IZ) = . 5*( (AX (0,IZ,4) - AX (0 , I Z , 1) )/DNU + $ (AX (J, I Z , 1) - AX ( J , IZ , 5) )/DNL) I F (DABS(DXDN (J,IZ)*DEN/AX (J ,IZ,1)).LE.DTOL) DXDN (J, $ IZ)=0. DO 17 CONTINUE 16 CONTINUE I F ( .NOT.VERBOS) GO TO 20 DC 19 IZ=1,9 IQ=IZED(IZ) WRITE (6,1029) IQ 1029 FORMAT(* DXDT,DXDN FOB ATGM Z=*,I2)  1030 19 20  WBITE (6,1030) (DXDT(J,IZ) ,J=1,IQ) FOBMAT(1X,10E13.5) WBITE (6,1030) (0XDN(J,IZ) ,J=1,IQ) CONTINUE DO 18 IZ=1,9 IQ=IZED(IZ) DO 18 J=1,IQ X(J,IZ)=AX(J,IZ,1) CONTINUE  18 C C PHYSICAL DEBIV ATIV ES C DU=0. DL=0. CALL DERIV(ADE,DNEIN,DNEDT,DTOL) WBITE (6,1003) DNEDN,DNEDT 1003 FOBMAT {* DNEDN, DNEDT* , 2E1 5. 7) DN2=AD <1)*AD (1) CALX DEBIV(ACCOL,DLDN,DLDI,DTOL) DLDN= AD (1) *2.*ACOOL (1) +DL DN *DN2 DLDT=DLDT*DN2 BL=ACCCL (1) *DN2 WRITE(6,1005) DLDN,DLDT , EL 1005 FOBMAT ( * DLDN,DLDT,LOSS ES *,3E15.7) CALL DEBIV (AGAIN,DGDN,DGDT,DTOL) GAIN= AGAIN (1)*DN2 DGAINC=GAIN/2.9979E10 DGDN=AD (1) *2.*AGAIN (1) +DGDN*DN2 DGDT= DGDT* DN2 WRITE (6,1007) DGDN,DGDT,GAIN 1007 FORM AT ( * DGDN, DGDT , GAINS * , 4E15. 7) CALL DERIV(AE,DEDN,DEDT,DTGL) C= D EDT/KBOLTZ WRITE (6,1009) DEDN,DEDT,C 1009 FOB MAT (* DEDN, DEDT , CV » , 4E15. 7) CALL DERIV (AH,DHDN,DHDT ,DTOI) C=DHDT/KBOLTZ C C CCNDUCTION CALCULATION FBGM SPITZEB C WRITE (6,1011) DHDN, DHDT, C 1011 FORMAT(* D H D N , D H D T , C P » , 4 E 1 5 . 7 ) I E (CCNDUC) GO TO 110 DKDT=0. DKDN=0. CONKAP=0. GO TO 113 110 CONTINUE COULOG = 9.00 + 3. 45*ALOG10(SNGI (T) )-1. 15*ALOG 10 ( DENE) CGNKAP=1.8E-5*T**2.5/COULOG DKDT=2.5*CONKAP/T+CCNKAP/ (COHLOG*T) *3.45 DKD N = C C N K A P / ( C O U L O G * D E N £ ) * (-1 . 15) C C THE MEAN MASS OF AN ATOM COEFFICIENT C 113 WRITE (6, 1013) CCNKAP, DKDT, DKDN  161  1013  FORMAT(* C O N K A P , D K D T , D K D N » , 3 E 1 5 . 7 ) EHCCCN=0. BO 10 1=1,9 BHGCGN=RHOCGN*AMASS (I) *AEUND(I) CONTINUE  10 C C TEE FORCE DUE TO CONTINUUM RADIATION C RHOCV=RBOCON*1.660531E-2U 111 CONTINUE GRADC=0. DGCDN=0. DGCDT=0. DO 51 1=1,9 IZ=IZED(I) DO 52 J=1,IZ IJ=INDEX(J,I) XAC=ABUND(I) *X ( J , I ) I F ( X A C . L T . 1.E-10) GO TO 52 KUB=NU0(IJ) NUE=NUF (IJ)  50  52 51  IF(NUE. EQ« NUB) GO TO 52 GCL=0. DO 50 IN=NUB,NUE GCL=GCL+.5* (FLUX(IN + 1) * S I GM A (IN • 1, IJ) +FLUX ( I N ) * $ SIGMA ( I N , I J ) ) * $ DELE (IN) CONTINUE GR ADC=GR ADC+GCL*X AC DGCDN=DGCDN*GCL*DXDN<J,I) *ABUND(I) DGCDT=DGCDT + GCL*DXDT ( J , I) *AEUND (I) CONTINUE CONTINUE DGCDN=DGCDN*CHF/RHOCV DGCDT= DGCDT*CHF/EHOCV GRADC=GRADC/RHOCV*CHF TRKAPC=0. RHO=DEN *RHOCV DDEN=0. IFIT=0 DIFF=0.  C C I F DYNEQ EALANCE MOMENTUM EQUATION C I E (DYNEQ) WRITE (6, 1062) 1062 FORMAT(6X, DIFF ,11X,,«DDEN«,11X, DV«,13X,«BELV»,11X I * DGRDDV',9X,•GRAD*,11X,*DP«,13X,»IFIT*) 201 CONTINUE GRAD=0. ,, GRADE=0. DGBDT=0. DGRDN=0. I F (FTOT. EQ.O. ) GO TO 112 DGRDD V=0. C C CALCULATION OF LINE FORCE ,  J  ,  162  C CALL FORCE(V0,DV,T,DEN) EGRBDV= DGRDDV/RHOCV *CHF GRADL=GRAD/RHOCV*CHF GRADE=FTCT*2. 2 19E-35*DENE/ (DEN*RHOCV) *CHF DGEDT=GRADE*DNE DT/DENE DGR ET= DGRDT *CH F/RHO CV•DG E DT • DGCDT DGRDN=DGRDN*CHF/RHOCV+DGCDN C NOTE THAN GBIJ IS NOT MULTIPLIED BY CHF C NUMBER I S THOMSON CROSS SECTIONCVEB THE SPEED OF LIGHT GBAD=GBADL*GBADE+GRADC IF(.NOT.DYNEQ) GO TO 200 BN=~DV*N0/V0 IF(SPHERE) DN=DN~2.DG*N0/RSTAB DDEK=DN DIFOLD=DIFF DP=KBOLTZ/BHO* {DT* (NO+ NEO+DNEDT*T0/NO) + DN* (1.D0 + $ DNEDN) *I0) DIFF= DP $ *GBAV-GRAD+VO*DV I F ( I F I T . EQ. 0) GO TO 202 C C ADJUSTMENT OF DV/DZ FOR MOMENTUM EALANCE C DFDEV= (DIFF-DIFOLD) / (DV-BVOLD) DELV=-DIFF/DFDDV IF(DABS(DELV).GT.DAES(EV).AND.SLIM.NE.0.DO) DELV= $ DSIGN(1.D0,DELV) *DABS (DV) *SLIM EVOLD=DV DV=DV+DELV WRITE (6, 1061) DIFF,DDEN, DV, DELV, DGR DDV, GRAD, DP, I F I T 1061 FORMAT(1X,7E15.5,15)  202  200  1012 112  IFTT=IFIT + 1 IF(DABS(DELV/DV).LT.DYNTOL) GC TO 200 I F ( I F I T . L E . EYNIT) GO TO 201 GO TO 200 DVOLD=DV DV=DV*1.1D0 IFIT=1 GO TO 201 CONTINUE TBKAPC=GEAD/FTOT GB ATIO= GRAD/GBADE GAMMA=(GRAV-GRAD)/GRAV WRITE (6,1012) GRAD,GRADE,GBADL,GBADC,GBATIO,GAMMA, $ DGRDT,DGRDN FORMAT (* GRAD,GRADE,GRAEL,GRAEC,GRATIO,GAMMA,DGRDT, $ DGRDN */1X, $ 8E16.6) CONTINUE 10= (DEN + DENE) *T*KBOLTZ DP=RHO* (-V0*DV-GRA V+GRAD) I F ( NONEQ) GC TO 46 I F (VO.EQ.O.) GO TO 44 DRHO=-RHO*DV/V0  DDEN= DRHC/EHOCV 44 IF(NONEQ.AND.(VO.EQ.O.)) GO TO 45 C C ENERGY EQUILIBRIUM FORCED C EI USING CONDUCTIVE ENERGY TRANSPORT C RARELY USED C VC1=1.  C  IK FRAME OF STAR VC1=1 DT=(-KBOLTZ/BHOCV*(DDEN*(1.+DNEDN) ) *T/DEN $ -V0*DV+VC1*GBAD-GRAV)/ $ (KBOLTZ/RHOCV*(T/DEN*DNEDT+1. + DENE/DEN) ) GO TO 46 45 DDEN= ( (-GRAV+GR AD) *HHO- (DEN + DENE) *KBOLTZ*DT) / $ (KBOLTZ*T*(1.+DNEDN)) DBHG= DDEN*RBOCV 46 DH=DHDT*DI+DHDN*DRHO/RHOCV DKAF=DKDT*DT*DK EN*E EFN TN= (DEN+DENE)*T SOUND=SQRT (. 5*( ( (AD{4) +ADE(4) )*AT (4)-TN)/DNU + $ (TN- (AD (5) +ADE (5) ) *AT (5) ) /DNL) *RGAS/RHOCON) IF (NONEQ) GO TO 33 D2T=(-VC1*GAIN + RL + DDEN*V0*( 1.5*V0**2+ENTH)*RHOCV $ +DEN*RHOC¥*DV* (1. 5*V0**2+ENTH) $ •RHOCV*DEN*(DHBT*DT+DHEN*DEN))/{-CCNKAE) 33 IF (NGNEQ. AND. (TNON . GT. 0 .) ) T=TNON WRITE (6,1022) PO,RHO,DRHO,DP,SOUND,NONEQ,DT, $ EKAP,D2T 1022 FORMAT (» PHYSICAL PARAMETERS CALCULATED,P0,BHO,DRHO, $ »DP,SOUND' , 5E15.7,/' NONEQ, DT, DK AP , * , $ »D2T»,/1X,L8,3E15.7) CFLUX=CGNKAP*D2T*DKAP*DT WRITE (6,1024) CFLUX 1024 FORMAT( CONDUCTION FLUX=«, E15.7) IF (SKIP) GO TO 122 DN=DDEN H0= ENTH/RHOCV E0=EINT/RHOCV DHDT=DHDT/RHOC¥ DHDN=DHDN/RHOCV DEDT=DEDT/RHOCV DEDN=DEDN/RHOCV G0=GAIN L0=BL 122 CONTINUE C C OUTPUT OF PHYSICAL QUANTITIES C WBITE (7) N0,T0,V0,DN,DT,DV,BHOCV,SOUND, $ NE0,H0,E0,D NEDT,D NED N,DHDT,DH DN,DE DT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GRAV,GRAD,GRADE,DGRDT, $ DGRDN, $ GO , LO, DGDT, DGDN, DLDT, DLDN WRITE(6,PHYSIC) C C CALCULATION OF PHYSICAL LIMITING FREQUENCIES 1  c  1025  1033 519 999  C C C C C C C  WRAB=5.6S97E-03*TRAD**3*TRKAPC WCCCL=6.28319*10/(£0*RHO) 8REC=1.88E-10/DSQRT (T) *DEN EQC=C0ULOG*BENE*T** (-1.5) WEQPE-2.5E-02*EQC BEQEE=4.57E01*EQC WCGND=6.28*ABS (SN GL ( DK DT * DT + CO N K A P* D 2 T) ) /(DEDT*RHO) 9SITE (6,1025) WRAD,WCOQI,WREC ,WEQPE,WEQEE,WCOND FOR MAT(• WRAD,HCOOL,WREC,WEQPE,WEQEE,WCOND*,6E15.3) I F (DDEN. EQ.O. ) GO TO 519 HSCALE=DABS(DEN/DDEN) GAM=DHDT/DEDT WACS=GAM*GRAV/(2.*SCUND) WACH=SQBT (SNGL (GAM*GRAV/ (4. *HSCALE) ) ) WBVS=SQRT (GAM-1.)*GBAV/SOUNO WBVH=SQHT(SNGL( (GAM-1.) *GBAV/(GAM*HSCALE) ) ) WBITE (6,1033) HSCALE,GAM,WACS,WACH,WBVS,WBVH FORMAT(* H S C A L E , G A M , M A C S , W A C H , W B V S , W B V H , » , 6 E 1 4 . 3 ) CONTINUE GO TO 99 8 STOP END SUBROUTINE FOBCE(V,EV,T, DEN)  BOOTINE TO CALCULATE LINE FORCE WITH SIMPLE LUCY RADIATIVE TRANSFER FOE ONE SCATTERING LINE ALWAYS MOST BE SUPERSONIC FLOW MO OVERLAPPING LINES ALLOWED FOB LOGICAL VEBEOS INTEGEB INV(16) /1,2,0,0,0,3,4,5,0,6,0,7,0,8,0,9/ DIMENSION E L I N E ( 1 0 0 0 ) , F ( 1 0 0 C ) , I I ( 1 0 0 0 ) , J J ( 1 0 0 0 ) DIMENSION ABUND (9) ,2(17,9) , VTHEBM (9) ,FLUX(100) , $ FPT(100) SEA L*8 DXDT (16,9) ,DXDN(16,9) , GRAD,DGRDT,DGRDN , $ GRIJ(16,9) BEAL*8 V,DV,T,DEN,EVI REAL*8 CK,GL,DGL,TAO,TAUC,DGBDDV,FKAP,DGLDN B E A L* 8 DABS LOGICAL SPHEBE COMMON /PHORCE/ GBAD,DGBDT,DGBDN,DXDT,DXDN,GRIJ, $ DGRDDV COMMON /A/ ABUND,X,VTHEBM,SLAB COMMON / ? / E L I N E , F , I I , J J , F L U X , F P T , N L I R E , N F L U X COMMON /CONTEO/ VEREOS,SPHERE,STABMU,BSTAB IVI=DABS(DV) I F (SPHERE) EVI=£ABS (. 5*{1. •STABMU*STABM U) * $ (DV-V/RSIAR) + V/RSTAR) IF=1 FDF=1.-V/3.£10 COLUMN=DEN*SLAB I F (DVI. NE. 0.) COLUMN = 2. 9979 110/DVI*DEN IF(COLUMN.EQ.0.) GO TO 100 DO 10 L= 1,NLINE  J = J J (L) 1=11 (L) IV=INV(I) 3 I F (ELINE (L) . GT. FDF*FPT (II) ) GO TO 2 FNU= (FLUX (IF-1) +S* (ELINE (L) - F D F * F P T ( I F ) ) ) *FCF CK=1.0 976E-16*F (L) *ABUND (IV) C CONSTANT I S • P I * E * * 2 / ( H E * C ) / (EV TO HZ CONVEBSION) TAU=CK/ELINE(L)*COLUMN*X(J,1V) IF<X (J,IV) . I E . 1.E-10) G O T O 10 DGL=CK*FNU GI=DGL*X (J,IV) C C ELINE IS IN EV BUT NOTE THAN END IS EBG CM-2 S-1 EV-1 C TAUC=TAU*DVI I F ( I A U . L E . 1.E-3) GG TO 4 I F (TAU.GT.170.) GO TG 6 DGRDDV=DGBDDV+GL/TAUC*(1.-DEXP(-TAU)*(1.DO+TAU)) GL=GL*(1.-DEXP(-TAU) )/TAU DGLDN=-GL/DEN*DGL*DEXP (-TAU) * (DXDN ( J , IV) +X ( J , I V ) / $ DEN) 4 CGNTINUE DGBDDV=DGRDDV*GL/TAUC*TAU*TAU DGL= DGL * DEX P ( -T AU) GO TO 5 2 IF=IF*1 I F ( I F . GT.NFLUX) BETUEN S= (FLUX (IF) -FLUX ( I F - 1 ) ) / (FPT (IF) - FPT (IF-1) + 1. E-50) GO TC 3 6 CONTINUE DGB DDV=DGRBDV +GL/TAUC GL=GL/TAU DGL DN=-GL/DEN DGL=0. 5 GRAB=GRAD+GL G E I J ( J , I V ) = G R I J (J,IV) +GL DGB DT= DGB DT • DGL * DX DT (J , IV) DGRDN=DGRDN +DGLDN 1 IF<VEBBGS) 8 B I T E (6, 1000) L , ELINE(L) ,J,I,IV,FNU,TAU, $ GL,CK,FRAP,DGL 1000 FORMAT ( 1 X I 3 , F 1 0 . 5,3X5, 6E1S. 5) 10 CONTINUE C C ALSO, WHAT ABOUT THE CONTINUUM OPACITY C BETUEN C C OPTICALLY THIN CALC C 100 DO 101 L=1,NLINE #  J=JJ (I) 1=11 (L)  103  IV=INV (I) I F ( E L I N E ( L ) .GT.FDF*FPT ( I F ) ) GO TO 102 FN0= (FLUX ( I F - 1 ) +S* ( ELINE (L) -FDF*FPT (IF) ) )*FDF DGL=1.0976E-16*F (L) * ABUND (IV) *FNU  102  101  C C C C C  GL=LGL*1 (J,IV) GRAD=GRAD+GL GRIJ (J,IV) = GRIJ (J,IV) +GL D G R D1~ D G R DI + D G L * D X DT (J , I V ) DGFDN=DGRDN*DGL*DXDN ( J , IV) I F (VERBOS) WRITE (6, 1000) I-,EIINE(L) ,J,I,IV,ENU,GL,DGL GO 10 101 IE=IF+1 IF(IF.GT.NFLUX) RETURN S= (FLUX (IF) - F L U X ( I F - I ) ) / (FPT (IF) - FPT (IF-1) +1.E-50) GO TO 103 CONTINUE RETURN ENE SUBROUTINE DERIV (Q,DQDN,DQDT,DTOL)  A ROUTINE TO CALCULATE DERIVATIVES OF PHYSICAL QUANTITIES AND CHECK THAT THEIR LOG DERIVATIVES EXCEED SOKE MINIMUM, I F NOT THE ARE SET TO ZERO. REAL*4 Q(5) REAI*8 DQDN,DQDT REAL*8 DEN ,T COMMON /DER/ DNU,DNL,DTU,DTL,DEN,T I F ( Q ( 1 ) .EQ.O.) GO TO 1 DU* <Q(4)-Q(1))/DNU DL=(Q(1)-Q (5))/DNL DQDN=.5*(DU+DL) DLQ=DQDN*DEN/Q(1) I F (AES (DLQ).LT.DTOL) GO TO 2 « DU=(Q(2)-Q (1))/DTU DL= <Q(1)-Q(3))/DTL DQDT=.5*(DU+DL) DLQ=DQDT*T/Q (1) I F ( A B S ( D L Q ) . L T . D T O L ) GG TO 3 BETURN 1 DQDN=0.D0 DQDT=0. DO RETURN 2 WRITE (6,1001) DU,DL,DQDN,DLQ,DTOL 1001 FORMAT(* DU,DL,DQDN,DLQ,DTOL * ,5E15.7) EQDN=0. DO GO TO 4 3 WRITE (6,1002) DU,DL,DQDT,DLQ,DTOL 1002 FORMAT('DU,DL,DQDT,DLQ,DTOL * ,5E15.7) DQDT=0.DO RETURN END  PROGRAM C C C C C C C C C C  CGCALG  THIS PROGRAM CALCULATES THE COEFFICIENTS OF W AND K FOB THE DISPERSION RELATION POLYNOMIAL THE PHYSICAL QUANTITIES PBOEUCFD BY THE PROGRAM COEF ABI USED AS INPUT THE OUTPUT I S USFB BY T E E PBOGBAM DISPEB THIS I S A SUBROUTINE CALLED IN THE DISPEB* SUEBGUTINE CCCALC(*) BEAL*8 KBOLTZ,RMU,VC1,C,VG LOGICAL FREEZE LOGICAL MANY,RESTOR BEAI*8 CMASS,CMTM, CENE BEAL*8 C,VC1,MTMKN,MTMC N,MTHKT,MTMCT,DVG, $ EW N , EKN, ECN,EHT, FKT, i C T , EH V, EKV, ECV BEAL*8 N0 TO,VQ,DN,DT,DV,BHOCV,SOUND, $ NE0,H0,EO,DNE DT,DNECN,DHDT,DHDN,DEDT,DEDN, $ D2T ,CONKAP ,DKDT,DKD K ,GEAV ,GRAD0 ,GBADE,DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN BEAL* 8 CBD (5,4) , CI £(5,4) COMMON /COEFS/ CBD,CID COMMON /CNTB02/ MANY,BESTOB NAMELIST /PHYSIC/ NO,T0,VO,DN,DT,DV,BHOCV,S00ND, $ NEO,HO,EO,DNEDT,DNE DN,DHDT,DHDN,DEDT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GBAV,GBADO ,GBADE, DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,OLDN,FBEEZE NAMELIST /DISCO/ BMU,VC1,HTMKN,MTMCN,MTMKT,MTMCT,DVG, $ EWN,EKN,ECN,EWT,EKT,ECT,EHV,EKV,ECV I F (. NOT. BESTOB) GO TO 4 BACKSPACE 1 GG TO 3 CONTINUE I F (MANY) GO TO 2 BEAD(1,END=999) NO,TO,VO,DN,DT,DV,BHOCV,SOU ND, $ NEO,HO,EO,ENEDT,DNE DN,DHDT,DHDN,DE DT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GBAV,GBAEO,GBADE,DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN FBEEZ£=.FALSE. C=2.9979D10 KBGLTZ=1.380626D-16 BEAD(5,PHYSIC,END=999) BMU=KEGLTZ/BHOCV VC1=1.D0 #  4 3  2  C C THE NAMES OF THESE VARIABLES COMES FBGM C C THE LINEABIZATION OF THE EQDATICNS OF MOTION C  168  C EXCEPT NOTE THAT P THERE I S REPIACED EY HTM, C CMASS=DN*V0+DV*N0 CMTM=BMU*T0/N0* (DN + (INEEN*EN+ENEDT*DT) ) +DV* VO $ +RMU*DT*(1.D0+NE0/NO)-VC1*GRAD0+GRAV CENE=-VC1*GG+L0*CGNKAP*D2T+ (D KD N* D N* DKDT*DT)* DT + DN* $ V0*BHOCV*{1.5D0* $ V0**2+H0)+N0*RHOCV*DV*(1.5D0*V0**2+H0)+ $ RHOCV*N0*(BHDT*DT*DHDN*DN) WRITE (6,1001) CMASS,CMTM, CENE 1001 FORMAT(* CONSERVATION EQUATIONS CMASS,CMTM,CENE', $ 3D25. 12) VC1=1.-V0/C C CALCULATION DONE IN FRAME MOVING WITH GAS VG=G.D0 WRITE (6,PHYSIC) MTMKN=KBOLTZ/(N0*RHOCV)*(DNEDN*T0+T0) MTMCN=KBCLTZ/(N0**2*EHOCV)*{-BN*T0+N0*DNEDN*DT $ -NE0*DT- (DN£DN*DN*DNEDT*DT) *T0) -VC1*DGRDN MTMKT=KEOLTZ/(N0*RHOCV)*(NE0+DNEDT*T0+N0) MTMCT=KBCLTZ/(NO*BHGCV)*(DN+DNEDT*DT+DNEDT*DT+DNEDN $ *DN)-VC1*DGRDT DVG=DV+GRAD0/C EWN=(DEDN*N 0+(VG**2/2.+E0))*RHOCV EKN=VG*(.5D0*VG**2 + H0+DEBN*NC)*RHOCV + |- DKDN) *DT ECN=RHOCV* (DN*VG*DHDN+1.5*DV *VG**2+D¥*H0+ $ DV*NO*DHDN+VG*(DHDN*DN+DHDT*DT)) + D2T* (-DKDN $ )-VC1*DGDN+DLDN $ +RHOCV* { VO * (DEDN*DN+DEDI*DT) *• DEDN*VO*DN) EWT=DEDT*N0*RHOCV EKT=RHOCV*DHDT*VG*N0+(-DKDT)*DT+{ (-DKDT)*DT +(-DKDN) $ *DN) ECT-DLDT-VC 1*DGDT+ (DHDT* (DN*VG+DV*N0) ) *EHOCV+ (-DKDT $ )*D2T $ +RHOCV*DEDT*VO*DN E8V=VG*N0*RHOCV EKV=(1.5*VG**2+H0)*N0*RHOCV ECV= (DN*(1.5*VG**2 + H0)*3.*DV *VG*N0+ NO *{DHDT*DT+DHDN $ *DN) ) $ *RHOCV+G0/C $ +RHOCV*(N0*V0*DV) C C C CUTFUT LINEARIZED EQUATION QUANTITIES C THIS I S USEFUL FOR EXAMINING THE MAGNITUDES C OF THE PHYSICAL QUANTITIES WHICH C ARE DOMINATING THE SITUATION C WRITE (6,DISCO) CRD (1,1) = $ -ECN*MTMCT*DN-DVG*ECT*DV+ HTMCT*ECV*DV*MTMCN*DN*EC T CID(1,1)=0. DO CRD(2,1)=0.DO CID(2,1) =  169  $ VG* (-DVG*£CT+MTMCT*ECV-ECT*DV)-ECN*MTMCT*NO-ECN* $ MTMKT*DN-EKN* XMTMCT*DN-DVG*EKT*DV +MTMCT *EKV*DV+MTMKT*ECV*DV+MTMCN $ *DN*EKT>MTMCN* XNO*ECT+MTMKN*DN*ECT CRD (3,1) = $ VG**2*ECT+VG* (DVG*EKT-MTMCT*EKV-MTMKT*ECV+EKT*DV) $ +ECN*MTMKT* XN0 + EKN*H1MCT*N0+EKN*MBKT*DK + DVG* (-CONKAP) *DV-MTMKT $ *EKV*DV $ -BTMCN*DN* X(-CCNKAP) -MTMCH*N0*EKT-MTMKN*DN*EKT-HTMKN*NO*ECT CID (3, 1)=0. DO CRD (4, 1) =O.DO CID (4,1) = I VG**2*EKT+VG* (DVG* (-CONKAP) -MTMKT*EKV* (-CGNKAP) *DV  $  ) +  $ EKN*HTMKT*NOXHTMCN*N0*(-CONKAP) -MTMKN*DN* (-CGNKAP) -MTMKH*N0* EKT CED (5,1) = $ -VG**2*(-CONKAP)+ MTMO*N0* (-CCNKAP) CID (5,1)=0.DO CRD (1,2)=0.D0 CID(1,2) = $ EWN *MTMCT*DN+D VG*ECT*D VG*E KT*DV-MTMCT*ECV-MTMCT* $ EWV*DV-«IMCN* XDN*E8T*ECT*DV CRD (2,2) = $ VG* (-DVG*E8T+MTMCT*EWV-2*ECT-EHT*DV)-E8N*MTMCT*NO $ -EWN*MTMKT* XDN-DVG* EKT+MTMCT*EKV•MTMKT*£CV+MTMKT* EH V*D V+MTMCN* -$ N0*E8T + MTMO*DN X*EHT-EKT*DV CID (2,2)=0,D0 CRD (3,2) =0. DO CID (3,2) = $ -VG**2*EHT+VG* (MTMKT'*EWV-2*EKT) -EWN*MTMKT*NO-DVG* $ (-CGNKAP)+MTHKT X*EKV+MTMKN *NO*EBT-(-CONKAP)*DV CRD (4,2) = $ 2*VG*(-C0NKAP) CID (4,2) =0. DO CRD (5,2)=0.D0 CID <5,2)=0. DO CRD (1,3) = $ DVG*EwT-MTMCT*EwV4ECT+EHT*DV CID(1,3)=0.D0 CRD (2,3)=0. DO CID (2,3) = $ 2* VG* EHT- MT H KT * EH V • EKT CRD (3,3) = $ - (-COMKAP) C I D (3,3)=0.D0 CRD(4,3) =0.D0 CID(4,3)=0.DO CRD(5,3)=0.D0  170  CID (5,3) = 0. DO CBD(1,4)=0.D0 CID <1,4) = $ -EST CBD (2,4) =0. DO CID(2,4)=0.D0 CBD (3,4)=0. DO CID (3,4) =0.D0 CBD <4,4)=0. DO CID(4,4)=0.D0 CBD (5,4)=0. DO CID(5,4)=0.D0 WBITE (6,1000) ((CBD (J.I) ,J=1,5) , (CID ( J , I ) , J= 1,5) ,1= $ 1,4) 1000 FOB H AT (' 0 * , 5D25.10 , /, 1 X, 5 D2 5.10) EETUBN 999 EETOBN1 END  PROGRAM C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  DISPER  A EEOGBAM TO FIND THE BOOTS OF THE CUBIC DISPERSION RELATION FCUKD FOE THE CASE OF ONE DIMENSIONAL PLANE HAVES SUBROUTINE COCALC USES THE OUTPUT FBOM COEF TO ACTUAL FIND THE POLYNOMIAL THE PARAMETERS CONTROLLING THE EEOGBAM ARE: KLGG I F TBUE A LOGARITHMIC SERIES OF K ARE GENERATED KMIN MINIMUM K VALUE ( I F KLOG THEN THIS IS A LOG) KM AX MAXIMUM K VALUE PLREAL PLOT REAL FREQUENCIES ELIMAG PLOT IMAGINARY IBEQUFNCIIS KINC INCREMENT BETHEEN K VALUES TGI I F TWO BOOTS L I E CLOSES THAN THIS THEY ABE ASSUMED TO BE IDENTICAL EBB ACCUBACY OF NDINVT SOL'N TO B=0, DD=0 MAXIT # OF ITERATIONS IN NDINVT EQTCL ACCURACY OF NEWTON ITERATION ^BOOT IMPROVER NFILE F I L E LINENUMBEB WHERE COEFFICIENTS START USUAL 1+A MULTIPLE OF 5 LABEL TRUE TO LABEL PLOTS SUEDIV # OF SUBDIV USED EY NEXT BOOT ESTIMATOB PEMIN FOB DIFFEREN BEAL PLOT Y AXIS MINIMUM PBINC SAME EUT INCREMENT FROM MIN FOR 10" PLOT PIMIN SAME AS PRMIN BUT FOR IMAGINARY PART PIINC SAME AS PBINC EUT IMAG NPRINT EVERY NPRINT'TH K VALUE AND ROOT OUTPUT TO PRINTER ER SEMIV I F TRUE OUTPUT W i l l ALLOW NPRINT TO TAKE EFFECT REAL*8 DKR (210) REAL* 8 BD( 11), ID ( 1 1 ) , BOOTS (4) ,ROOTI(4) ,PKR(5) ,PKI(5 $ ) EEAL*8 EDIS(12) ,IDIS (12) SEAL*8 DELTAK REAL*8 DIST,DISTOL ,RSAVE,DERR INTEGER*4 KEEG (3) , KEND ( 3) REAL*8 X(4) ,F(4),ACCEST (4) ,EEB EEAL*8 BBC (4) ,IDC(4) SEAL*8 DONE /1.D0/ EEAL*8 WIM (3,210) ,WB(3,210) BEAL*8 KMIN,KMAX,KINC,TOL BEAI*8 DSIGN, DEE AL, DIM AG, DLOG10, DABS, DMIN1 ,DMAX1 COMPLEX*16 DCMPLX,CECOT,CDLCG REAL*8 CDABS,BLOG LOGICAL FREEZE,NONEQ,KLOG,PLREAL,PLIMAG,SOLVEQ,KNEG $ ,MANY,RESTOR,FIRST,DUPLIC,LABEL LOGICAL*1 I N S T A B ( 3 , 2 1 0 ) , A L A B E L ( 8 0 ) I NT EG EE* 4 SYM(3) / 1 2 , 2 , 5 / REAL*8 GEEL,EQTOL,COMPAE,LDEI,DPS,DPI REAI*4 VPHASE(3, 210) ,VG(3,210) , AX (210) ,AY(210)  C0MPLEX*16 CK,DC (4),DDC (4) ,8C,»C2 ,WC3,NDC(3) ,DDIS, DDDK C0MPLEX*16 DwDK,PRED (3) ,SLOPE (3) LOGICAL V EE EOS, ZEE IMG, FANCY, SEMIV,LPRINT INTEGER*4 SUBDIV INTEGER*4 INSTBI (3, 40) , NMAXL (3) REAL*8 BMAXL (3) ,DAWIM CGHHCN /NEWT/ DDIS,MAXIT COMMON /CCCALC/ CDC,EC,NEC COMMON /CPOL/ RDC,IDC COMMON /CONTEO/ SOLVEQ COMMON /CNTB02/ MANY,BESTOR COMMON / D I S / RDIS,IDIS NAMELIST /PARAM/ KLGG,KMIN,KMAX,PIREAL,PLIMAG,KlNC, $ TOL $ ,ERR,MAXIT,VERBOS,FANCY,PINC,PKMIN,MANY $ , RES TOR, EQTOL, NFILE,LABEL,S OBDIV $ ,PRMIN,PBINC,PIMIN,PIINC,NPRINT,SEMIV EXTERNAL FCN  $  C C SET UP DEFAULT VALUES C SUEBIV=3 NFILE=1 TCL=1. D-6 EQTOL=1.D-14 FLIMAG=. TBUE. PLBEAL=.TBUE. , KLGG=. TRUE. , KMIN=-12.D0 KMAX=-2. DO KINC=.1D0 VEBBCS=. FALSE. , NPBINT=5 SEMIV=.TEUE. FANCY=.FALSE. ZEBIMG= .TBUE. , MANY=.FALSE. RESTOB=. FALSE. MAXIT-100 EEB=1,D-15 FIBST-.TBUE. LABFL=. FALSE. 99S9 IF(MANY.AND..NOT.FIBST) GO TO 9990 NFIIE=1 READ (5,PARAM,END=9S98) C C READ IN PARAMETER L I S T OF WHAT TO DO C N F I L E 1=NFILE- 1 I F ( N F I L E I . L E . O ) GO TO 5 DO 6 ISKIP=1,NFILE1 READ (1) 6 CONTINUE 5 CCNTINUE IF(MANY.AND..NOT.FIRST) GO TO 9990  NK= (KMAX-KMIN)/KINC+1. 5 C C CALCULATE ABBAY OF K VALUES C I F (KLOG) GO TO 2 DO 1 I=1,NK DKB (I) = KHIN*(I-1) *KINC 1 CONTINUE GO TO 3 2 NKB= 1 SKE=NK DO U I=NKB,NKE DKB (I) = 10. DO** (KMIN • (I-NK B) *KINC) <4 CONTINUE 3 CONTINUE 99S0 WRITE (6,1000) 1000 FORMAT(* 1') I F (.NOT.LABEL) GO TO 9 BEAD(5,1066,END=9998) ALABEL WBITE (6,1067) ALABEL 1067 FORMAT(1X,80A1) 1066 FOBMAT(80 A1) 9 CONTINUE C C BLOT SCALING QUANTITIES C PINC=0. FKMIN=0, PRMIN=0. PBINC=0. PIMIN=0. EIINC=0. WRITE (6,PARAM) EIRSI=. FALSE. RRMN=9.E70 C C SEE I F ROOTS ARE PROPERLY SEQUENCED C RRMX=-9.E70 EIMN=9.E70 RIMX=-9.E70 DO 222 IN=1,3 VPHASE (IN, 1)=0. WMAXL (IN) = -9. D70 NHAXL(IN)=0 VG (IN, NK) =0. 222 CONTINUE CALL COCALC (69998) C C CALCULATE POLYNOMIAL COEFFICIENTS C GREL=0. DO 1=1 SOLVEQ=. FALSE. DDPLIC=.FALSE. C  17a  C FIND FIRST BOOT OF POLYNOMIAL C CALL D I S P C O ( D K B ( I ) ) CALL CPOLY1 (BBC, IDC,3,BOOTB,BOOTI,6999) 185 SOL VEQ=. TRUE. 99 DO 97 IN=1,3 CALL NEWTON (ROOTR (IN) ,ROOTI (IN) ,EQTOL) HR (IN,I) = BGGTR(IN) WIM(IN,I)=ROOTI (IN) 97 CONTINUE 181 DO 183 IN=1,2 IR=IN+1 184 I F ( I R . G T . 3) GO TO 183 WC=DCMPLX (WR (IN, I) ,HIM (IN, I) ) HC2=DCMPLX(HR ( I R , I ) ,HIM ( I f i , I ) ) C0MEAB=DMAX1 (CDABS (HC) , CDABS (HC2) ) IF(CDABS(HC-HC2)/COMPAR.LT.TCI) GO TO 170 IR=Ifi+1 GO TO 184 183 CONTINUE BUPLIC=.FALSE. 189 1=1+1 C C ESTIMATE THE VALUES OF THE NEXT SET OF ROOTS C I F (I.GT.NK) GO TO 98 DELTAK=DKR(I)-DKR(1-1) DO 96 IN=1,3 CALL ADVANC (ROOTR (IN) ,ROOTT (IN) , DELTAK , DHDK ,SUBDIV) VG (IN,I-1)=SNGL (DBEAL (DHDK) ) PEED (IN)=DCMPLX (ROOTR (IN) ,ROOTI (IN) ) 96 CONTINUE CALL DISPCO (DKR (I) ) C C CAICULATE COEFFICIENTS FOB NEXT K C GO TO 99 170 IF(DUPLIC) GO TO 188 I F ( I . EQ. 1) GO TO 188 SOLVEQ=.FALSE. C A L I DISPCO (DKB (I) ) C C I F DUPLICATE ROOTS ABE FOUND C GO BACK TO POLYNOMIAL ROOT FINDER C TO SEE IF ANOTHER ROOT CAN BE FOUND C I F SO TRY TO PROPERLY ORDER ROOTS C HBITE(6,1099) I , I N , I B 1099 FOBMAT(» SS6&S6SfiS8fiS6DUPLICATE ROOTS I,IN,IR«,3I5 $ ) CALL CPOLY1 (RDC,IDC,3, R O O T R , R O O T I , £ 9 9 9 ) NFIL=1 198 DISTOL=9.D70 DEBB=1.D70 DO 174 IN=1,3 I F (BOOTB (IN) . EQ.O. DO) G O T O 173  175  173 174  194  195 196  192 188 98  DEBB=DMIN1 (DABS (ROOTS (IN) ) , DEER) I F (EOOTI (IN) . EQ.O. DO) G O T O 174 DERH=DMIN1 (DABS(BOOT! (IN)) ,DERR) CONTINUE BEEE=DERR * 1.D-3 EPR=DLOG (CAES (DREAL (PRED(NFIL) ) ) +DERR) DPI=DLOG(DABS(DIMAG(PEED(NEIL))) + DERR) DO 195 IN=NFIL,3 DIST-DABS ( (DREAL (PEED (NFIL) ) - BOOTB (IN) ) * $ (DPH-DLOG (DABS (ROOTB (IN) ) +DERR) ) ) $ +DABS ( (DIHAG (PEED (NFIL) )-ROOTI (IN) ) * $ (DPI-DLOG (DABS (ROOTI (IN ) ) +DER B) ) ) I F (DIST.GT,DISTCL) GO TO 1S5 DISTOL=DIST INFIL=IN CONTINUE WR (NFIL,I) = BOOTR (INFIL) WIM(NFIL,I)=ROOTI(INFIL) I F (INFI1.EQ.NFIL) GO TO 192 ROOTB(INFIL)=ROOTR (NFIL) BOOTR(NFIL)=WR(NFIL,I) ROOTI(INFIL)=ROOTI(NFIL) ROOTI(NFIL)=WIM(NFIL,I) NFIL=NFIL+1 I F (NFIL. LT. 3) GO TO 198 GO TO 185 DUPLIC=. FALSE. GO TO 189 SOIVEQ=. FALSE. DO 100 1=1,NK LPRINT=. FALSE. I F (VERBOS) LP RINT=,TR U E. I F ( (• NOT. VERBOS. AN E. SEMI?) .AND. $ (MOD(I-1,NPRINT).EQ.0)) LPEINT=.TRUE. DG 100 IN=1,3  166 C C CHECK FOB I N S T A B I L I T I E S C IF(WR (IN,I) . EQ. 0. DO) G O T O 102 GR EL= WIM(IN,I)/WR (IB , I ) I F (WIM (IN,I) . LT.O. DO) G O T O 102 INSTAB(IN,I)=.TBUE. I F ( I . EQ. 1) GO TO 105 DAWIM=WIM (IN,I) I F (DAWIM.LT.O. DO) WMAXL (IN) = 0. DO IF(WMAXL(IN) .GT.DAWIM) GO TO 101 I F (INSTBI (IN, NMAXL (IN) ) .EQ.I-1) GO TO 105 NMAXL (IN) =NMAXL (IN) +1 104 I N S T B I (IN, NMAXL (IN) ) = I WM AXL (IN) =DAWIM GO TC 101 102 INSTAB(IN,I)=.FALSE. 101 CCNTINUE IF(DKB(I).EQ.O.DO) GO TO 288 C C FIND PHASE AND GBOUP V E L O C I T I E S  104  176  C CHECK FOB ICCAL MAXIMA IF ONSIAELE C VPHSSE (IN,I)=SNGL (WE (IN , I)/IKE (I) ) 288 IF (LP.BINT) WBITE (6,1033) I , DKR (I) , WR (I N,I) ,WIM (IN, $ I) $ ,VPHASE (IN,I) ,GREI,VG(IN,I) 1033 FORMAT(1X,13,3D26.14,3D 15.5) C C SAVE PLOT SCALING MAX AND MIN C IF(KLOG) GO TO 131 BIMX=AMAX1 (BIMX,SNGL(WIM(IN,I)) ) EIMN=AMIN1(RIMN,SNGI(WIM(IN,1) ) ) EBMX=AMAX1 (RRMX ,SNGI (WB (IN, I) ) ) RBMN-=AMIN1 (RRMN,SNGL (WR (IN,I) ) ) GO TO 100 131 R=SNGL(DABS (WR(IN,I))) I F (B.EQ.O. ) GO TO 132 RRMX=AMAX1(RRMX,R) BRMN= AMIN1 (FRMN, R) 132 R= SNGL (DABS (HIM (IN ,1) ) ) IF (B.EQ.O. ) GO TO 100 RIMX=AMAX1(RIMX,R) RIMN= AMIN1 (BIMN, B) 100 CONTINUE SOLVEQ=. FALSE. IF(.NOT.PLREAL) GO TO 300 WBITE(6,1011) RRMX,EFMN,RIMX,RIMN 1011 FORMAT(' RBMX,RRMN,RIMX,RIMN',4E15.7) IF (. NOT. KLOG) GO TO 201 RIMX=ALOG10 (RIMX) RIMN=ALOG10 (RIMN) £BMX=ALOG10 ( B R M X ) BRMN=ALOG10 (BBM 8) 201 I E (. NOT. FANCY) GO TO 260 RMM=AINT (ALOG10 (ABS (BRMN) ) ) IM=RRMN/10.**RMM HMIN=IM*10.**RMM BMM= AlNT(ALGG10 (BRMX-RBMN)) IM= (RRMX-RRMN) /10. **RMM WDX=IM*10. ?*RMM C C PLOTTING C DO SCALING C PLOT AXES C IAE EL C PLOT ROOT LINES C APEIY SPECIAL SYMBOLS IE C REAL(8)<0, OF IMAG(W)>0. C GO 10 261 260 WMIN= ERMN WDX=(RRMX-REMN)/10. 261 IF (PINC. EQ.O.) PINC= (KMAX-KMIN)/1 0. IF(PKMIN.EQ.O.) PKMIN=K MIN INC=(NK-1)/20 3  177  262 263  206  203 202 250  241 240  214 213  249 248 212  IF(PRMIN.NE.O.) WMIN=Ffi MIN IF(PBINC.NE. 0. ) WDX=PBINC CALL AXIS(0.,0.,'HAVE NUMBEE* ,-11,10.,0.,PKMIN,PINC $ ) CALL A X I S { 0 . , 0 . , « B E A L ANG FBEQ•,13,10.,90.,WMIN,WDX $ ) I F (.NOT. KLOG) GO TO 262 CALL SYMBOL (—0.3,9.0,.14,* LOG-LOG * ,90.,7) I F (.MOT. LABEL) GO TO 263 CALL SYMBOL(i.,9.75,.14,ALABEI,0. ,80) CONTINUE S=10./(NK-1.) DO 206 1=1,NK AX(I) =(1-1.) *S AY (I)=0. CONTINUE I F (KLOG) GO TO 250 DO 202 IN=1,3 DO 203 1=1,KK AY (I) = (SNGL (WB (IN,I) ) -WMI N) /WDX CONTINUE CALL LINE(AX,AY,NK,1) CONTINUE CALL PLOT(12.,0. ,-3) GO TO 300 DO 212 IN=1,3 NIN=0 DO 213 1=1,NK I F (SB (IN,I) . EQ. 0. DO) G O T O 214 AY (I) =(SNGL(DLOG10 (DABS (WE (IN,I) ) ) ) -WMIN) /WDX I F ( H B ( I N , I ) . G T . 0 . D O ) GO TO 213 IF ( I . EQ. 1) GO TO 240 IF(WR (IN,1-1) .LT.O.DO) GO TG 241 NIN=NIN+1 KBEG(NIN)=1 KEND (NIN) = I GO TO 213 KEND (NIN) — I GO TO 213 NIN=1 KBEG (NIN)=1 KEND (NIN) = I GO TO 213 AY(I)=0. CONTINUE CAIL LINE (AX, AY, NK, 1) IF(NIN.EQ.O) GO TO 212 DG 248 K=1,NIN KB=KBEG(K) KE= KE ND (K) DO 249 I=KB,KE,INC CALL SYMBOL (AX (I) , AY (I) ,. 14,SYM (IN) ,0.0,-1) CONTINUE CONTINUE CONTINUE CALL PLOT (12. ,0. ,-3)  178  300 369  360 361  IF(.NOT.PLIMAG) GO TC 399 DO 369 1=1,NK AY (I) =0. CONTINUE IF{ .NOT. FANCY) GO TO 360 E M M= AINT(AL CG10 (ABS (BIMN) ) ) IM=EIMN/1Q.**8MM WMIN=IM*10. **.BJ1M BMM=AINT (ALOG10 (BIMX-EIMN) ) IM=(BIHX-BIMN)/10.**BMM WDX=IM*10.**BMM GO TO 361 WMIN= BIMN WDX=(BIMX-RIMN)/10. IF (1DX. EO. 0.) GO TO 399 CONTINUE IF (PIMIN. NE. 0. ) WMIN=PIMIN IF(PIINC.NE.O.) WDX=PIINC CALL AXIS(0.,0.,»WAVE NUMEES•,-11,10.,0.,PKMIN,PINC $ ) CALL AXIS(0.,0.,•IMAG ANG FEEQ»,13,10.,90.,WMIN,SDX $  251 252 350  341 340  353 352  )  IF(KICG) GO TO 350 DO 252 IN=1,3 DO 251 1=1,NK AY (I)= (SNGL<aiM (IN, I) ) -HMIN) /WDX CONTINUE CALL LINE (AX, AY,NK, 1) CONTINUE G O TO 390 CALL SYMBOL(-0.3,9.0,.14,* LOG-LOG',90.,7) DO 351 111=1,3 NIN=0 DO 352 1=1,NK IF(WIM(IN,I).EQ.O.) GO TO 353 AY (I)= (SNGL (DLOG10 (DABS (WIM (IN, I) ) ) ) -WMIN) /WDX IF (WIM (IN,I) .LT.O.DO) GO TO 352 I F ( I . I Q . 1) GO TO 340 IF (WIM (IN, 1-1) . GT. 0. DO) GO TO 341 NIN=NIN+1 KBEG(NIN) = I KEND(NIN)=I GO TO 352 KEND(NIN)=1 GO TC 352 NIN=1 KBEG (NIN) = I KEND(NIN)=1 GO TC 352 AY(I)=0. CCNTINUE CALL LINE(AX,AY,NK,1) IF (NIN. EQ. 0) GO TO 351 DO 348 K=1,NIN KB=KBEG (K) KE=KEND (K)  DO 349 I=KB,KE,INC CALI S¥MBOL(AX(I) ,A¥(I) ,.14,SYM(IN) ,0.0,-1) 349 CONTINUE 348 CONTINUE 351 CONTINUE 390 CALL PLOT(12. ,0. ,-3) 399 DO 501 IN=1,3 WRITE (6,1013) (INSTAB (IN ,1) ,1=1, NK) 1013 FOBMAT (1X, 10 (3X , 10L1) ) NM=NMAXL (IN) I F (KH.EQ.0) GO TO 501 WfiITE(6, 1012) IN,NM, (INSTBI (IN,I) ,1=1 ,NM) 1012 FOBMAT(« INSTABILITY MAXIMA FOB GBOUP',12,110,/lX, $ 6(2X,5I4)) 501 CONTINUE C C USE THE LOCAL MAXIMA AS STARTING POINTS C FOB SOIUTICN TO D=0, D£/DK= 0 C DO 401 IN=1,3 NNX=NMAXL(IN) IF(NNX.LT.1) GO TO 401 DO 400 1=1,NNX I I = I N S T B I (IN,I) X(1)=WR(IN,II) X (2) = WIM (IN, II) X(3)=DKR ( I I ) X(4)=0.DO F ( 1) = 0.D0 F (2) = 0. DO C C DS E NEWTON PROCEDURE (FEOM BBC COMPUTER CENTBE) C TO SOLVE EQUATIONS C C MAXIT 200 USUALLY USED C C C NOTE ABOUT COEFFICIENT STRUCTURE: C I F K GOES TO -K*, THEN W GOES TO -W* C WHICH MEANS THAT THE SAME PHYSICAL ROOT RETURNS C CALL DISPCO (X (3) ) WC=ECMPLX(X(1) ,X(2) ) WC2=HC*WC WC3=WC*WC2 DDDK=DDC (1) *WC3 + DDC (2) *WC2+CDC (3) *WC+DDC (4) F(3)=DREAL (DDDK) F (4) = DIM AG (DDDK) WRITE (6, 1054) I,IN,X,F 1054 FORMAT(* OSTART",2I3,8D15.7) CALL NDINVT (4,X,F,ACCEST,MAXIT,ERR,FCN,S996) WRITE (6, 1015) ( X ( I C ) ,ACCEST (IC) ,IC=1,4) 1015 FOBMAT (* X ACCEST* , 4 (D18. 7, D10. 2) ) I F (X (2) .LT. 0.D0) GO TO 402 CALL DISPCO (X(3)) CALL CPOLY1 (BDC,IDC,3,BOOTB,ROOTI,&999)  1055 505 996 1056 402 400 401 9998 997 998 999 990 1020  DO 505 IIN=1,3 CALL NEWTON (BOOTH ( U N ) , BOOTI ( U N ) , EQTOL) WC=DCMPLX(ROOTR (IIN) ,BOOTI ( U N ) ) WC2=WC*WC WC3=8C2*WC DDDK= DEC (1) *WC3+DDC(2) *WC2+IDC{3) *WC + DDC(4) DDDK=-DDDK/ (NDC (1) *WC2+NDC(2) *8C+NDC(3) ) WBITE (6, 10 55) DDDK, BOOTB ( U N ) , BOOTI ( U N ) FOBMAT (» ######## ABSOLUTE INSTABILITY, GBOUP VELCC* $ ,'ITY',2D16.8, $ 5X,»W=»,2D15.5) CONTINUE GO TO 402 WBITE(6,1056)ERfi,X,ACCEST FOBMAT (» *******NDINVT FAILED**** EBR,X,ACCEST*/,1X, $ 9D13.5) CONTINUE CONTINUE CONTINUE GO TO 9999 CALL PLGTND STOP NP=997 GO TO 990 KP=998 GO TO 990 NP=999 WRITE(6,1020) NP FOB MAT ( * **************CPOLY1 TROUBLES****,14) GO TO 9999 END SUBROUTINE DISPCO(K)  C C CALCULATES COEFFICIENTS OF DISPEESICN RELATION FOR REAL C K C LOGICAL SOLVEQ REAL*8 DREAL,DIMAG,K,K2,K3,K4 COMELEX*16 EDC(4) , DC (4) ,NDC(3) , DCMPLX REAL*8 CRD (5,4) ,CID (5,4) ,RDC(4) ,IDC(4) COMMON /CCCALC/ D D C , £ C , N D C COMMON /CPOL/ BDC,IDC COMMON /COEES/ CRD, CID COMMON /CONTRO/ SOLVEQ K2=K*K K3=K2*K K4=K3*K IF(SOLVEQ) GO TO 1 DG 100 1=1,4 IB=5-I BDC (I) = CBD (1,IB) +CBD(2, IB) *K+CRD ( 3, IB) *K2 $ +CRD(4,IB) *K3+CRD(5,IB) *K4 IDC (I) =CID(1, IB) +CID <2,IB)*K+CID(3,IB)*K2 $ +CID(4,IE) *K3*CID (5, IB) *K4 100 CONTINUE 1 CONTINUE  181  102  DO 1C2 1=1,4 IB=5~I DC (I) =DCMPLX (CRD (1 ,IB) ,CID (1 ,IB) ) $ +DCMPLX (CBD (2, IB) ,CID(2, IB) ) *K $ +DCMPLX(CBD (3,IB) ,CID (3 ,IB) ) *K2 $ +DCMPLX(CRD(4,IE),CID(4,IB) )*K3 $ +DCMPLX (CRD (5,IB) ,CID (5,IB) ) *K4 DDC (I) =DCMPLX(CRD (2 ,IB) ,CID (2 ,IB) ) $ +DCMFLX (CRD(3,IB) , CID(3, IB)) *2.D0*K $ +DCMPIX(CBD (4,IB) ,CID (4 ,IB)) *3 . D0*K2 $ +DCMFLX (CBD (5, IB) ,CIE (5, IB) )*4.D0*K3 I F (IB.EQ. 1) GO TO 102 NDC(I)=DC(I) *DFLOAI(IB-1) CONTINUE RETURN END SUBROUTINE NEWTON(RE,SI,TCI)  C C DOES NE8TON METHOD IMPBOVEMENT OF ROOTS C VAIUES FROM ESTIMATE OB BOOT FIN DEB C ABE SUBSTITUTED BACK INTO THE FULL EQUATION C BEAL*8 BB,BI,TOL,BDC (4) ,IDC (4) ,BATIO BEAI*8 DREAL,DIMAG COMPLEX*16 DDC(4) ,DC (4) ,NDC(3),DIS,DDIS,DELW,WC,WC2 $ ,WC3 COMPLEX*16 DCMPLX REAI*8 CDABS,WAES COMMON /NEWT/ DDIS,MAXIT COMMON /CCCALC/ DDC,EC,NDC ILOGP=0 HC=DCMPLX(RR,RI)  2  wc2=wc*wc  WC3=WC2*WC DIS=DC (1) *WC3+DC (2) +WC2+DC (3)*WC+DC (4) DDIS=NDC(1) *HC2+NDC (2) *WC *NEC (3) I F (DREAL (DDIS) . EQ. 0. DO. AND. DIMAG (DDIS) . EQ. 0.D0) GO $ TO 3 DEIW=-DIS/DEIS WABS=CDABS(WC) I F (WABS. EQ.O. DO) G O T O 1 BATIO=CDABS(DELW)/SABS WC=WC+DELW IF(RATIO.LE.TOL) GO TO 1 ILOCP=ILOOP+1 I F (ILOOP.LT.MAXIT) GO TO 2 WRITE (6, 1000) WC, RATIO 1000 FOBMAT(* MAXIMUM NUMBEB OF ITERATIONS EXCEEDED.,W R « , f 'OCT IS NOW*, $ 2D25.15,« EBBOB= *,D 15.5) GO TC 1 3 WBITE (6,100 1) WC,DIS,BATIO 1001 FOEM AT (' DEBIVATIVE GOES TO ZEBO. WC, EIS,EEROB. • , $ 4D15.5) 1 CONTINUE BB=DBEAL (WC)  182  RI=DIMAG(WC) FETURN END SUBROUTINE ABVANC (WB, HI, DELTAK, DWBK , SUBDIV)  C  C ESTIMATES NEXT BOOT IN K SEQUENCE FROM PRESENT C ROOT AND DERIVATIVE C REAL*8 WR,HI,DREAL,DIMAG,DEITAK,DK,DFLOAT INTEGER*4 SUBDIV COMPLEX*16 DDC (4),DC (4) ,NDC(3),WC,DDIS,DHDK COMELEX*16 DCMPLX,WC2,WC3,WC4 COMMON /NEWT/ DDIS,MAXIT COMMON /CCCALC/ DDC,DC,NDC WC=DCMPLX(WR,WI) DK=DELTAK/DFLOA T(S U B DIV) DO 1 1=1,SUBDIV WC2=WC*WC WC3=WC2*WC DDIS=NDC(1)*WC2+NDC(2)*HC+NDC(3) £ H D K = - ( D D C (1) *WC3*ECC (2) *HC2+DDC (3)*HC+DDC (4) J/DDIS WC=WC+DWDK*DK 1 CONTINUE WR=DREAL(HC) WI=DIMAG(WC) RETURN END SUBROUTINE FCN(X,F) C C SUBROUTINE CALLED BY NDINVT C EVALUATES D AND DD/DK FCE COMPLEX H AND K C BEAL*8 X (4) ,F(4) COMPLEX*16 C K , C « , C W 2 , C W 3 , D D , D D D K COMPLEX* 16 DDC (4) , EC (4) ,NDC(3) COMPLEX*16 DCMPLX EEAL*8 DBEAL,DIMAG COMMON /CCCALC/ DDC,DC,NDC CK= DCMPLX (X (3) ,X (4) ) CALL DISCO (CK) CW= DCMPLX (X (1) ,X (2) ) CW2=CW*CW CW3=CW*CW2 DD=DC( 1) *CW3+DC (2) *CW2 + DC (3) *CH«-DC (4) F(1) = DBEAL (CD) F (2) = DIMAG (DD) BDDK=DDC (1) *CW3 + DDC (2) *CW2+DDC (3) *CW+DDC (4 ) F(3)=DBEAL(DDDK) F(4)=DIMAG(DDDK) BETUBN END SUEBOUTINE DISCO(CK) C C CALCULATES DISPEBSION POLYNOMIAL FOB COMPLEX K C BEAL*8 DBEAL,DIMAG  C0MPLEX*16 DDC (4) ,DC(4) ,NDC(3) CCKBLEX*16 CK,CK2,CK3,CK4 C0MPLEX*16 DCMPLX BEAL*8 CBD (5,4) ,CID<5, 4) COMMON /CCCALC/ DDC,DC,NDC COMMON /COEIS/ CBD, CID CK2=CK*CK CK3=CK2*CK CK4=CK3*CK DO 100 1=1,4 IB=5-I DC (I) = DCMPLX (CBD (1, IE) , CID ( 1, IB) ) $ +DCMPLX(CBD(2,IB) ,CID (2,IB) ) *CK $ + DCMPLX (CBD(3, IB) ,CID (3, IB) )*CK2 $ +DCMPLX(CBD(4,IB) ,CID(4,IB) ) *CK3 $ +DCMPLX(CBD(5,IB),CID(5,IB))*CK4 DDC(I) = DCMPIX (CBD (2, IE) , CID (2, IS) ) $ •DCMPLX (CBD (3,IB) ,CID (3 , I B ) ) *2 . DO *CK $ +DCMPLX (CBD (4, IE) , CI D (4, IB)) *3.D0*CK2 $ + DCMPLX (CBD (5,IB) ,CID (5,IB) ) *4.D0*CK3 CCNTINUE BETUEN END  

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.831.1-0085756/manifest

Comment

Related Items