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UBC Theses and Dissertations

A model of a magnetic star Carlberg, Raymond G. 1975

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A MODEL CF a MAGNETIC STAB by RAIHGND G. CARLEERG A THESIS SUBMITTED IH PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Astronomy and Geophysics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March,1975 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT A method of computing the e f f e c t s of a magnetic f i e l d cn the s t r u c t u r e cf a s t a r i s d i s c u s s e d . The f i e l d i s r e s t r i c t e d so that the magnetic f o r c e i s d e r i v a b l e from a p o t e n t i a l , which a l l o w s the development of a simple e x p r e s s i o n f o r the - f i e l d and d i s t o r t i o n to the s t a r . The technique i s used t o d e r i v e the perturbed e quations of s t e l l a r s t r u c t u r e , the o n l y d i f f e r e n c e from the unperturbed - equations being an a l t e r a t i o n to the e f f e c t i v e g r a v i t y , as a f u n c t i o n of r a d i u s . T h i s method i s a p p l i e d to the computation of the s t r u c t u r e of an upper main ..sequence s t a r c o n t a i n i n g a d i p o l a r magnetic f i e l d . , The . cases, o f the f l u x p e n e t r a t i n g the c o n v e c t i v e c o r e , and the f l u x : excluded from the core are c o n s i d e r e d . . The changes i n the s t r u c t u r e of the s t a r , the d i s t o r t i o n of the s u r f a c e and the expected changes i n the observable q u a n t i t i e s are c a l c u l a t e d . i i TJ £11 Of COm 1HTS Abstract- ...,.. . • •. .,••>.• • •, i Figures ....:. • • • •.... . i i i Tables . . . . . . , . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . • . . . . . . . . . . . . . . . i v •1. Introduction and Review .. ..1 1.1 Observations ....................•..... • •. ....... • • • 1 1.2 Emperical Ucdels ^ .............. 3 1.3 The Theoretical Problem .................... .3 1.3.1 The Surface E f f e c t s .. .....,........ ......... 4 1.3.2 The Interior F i e l d Structure .......... ....,. 5 1.3.3 The Dynamics .... ................ .7 1.3.4 Convection .... f............................ .10 1.3.5 S t a t i c F i e l d Models 11 1.i» The Purpose of This Investigation 16 2. The Model of a Magnetic Star 17 2.1 The Equations of S t e l l a r Structure ................. 18 2.2 The Magnetic F i e l d ...•,;•,.,>,,•.,.•.,...,......,..,... 20 2.3 The Magnetic F i e i a Force Potential 22 3. The Equations Of S t e l l a r Structure 28 3.,1 Expansion of Pot e n t i a l And Variables ............... 28 3.2 The Perturbed S t e l l a r Structure Equations ..........29 3.3 The Distortion ...................................,.31 4. .Results and Discussion ...................*.••.....«.....34 4.1 The F i e l d Structure .•.............................. 34 4.2 The Flux Free Convective Core ......................-44 4.3 The Rotating Magnetic Star 45 4.4 Ef f e c t s of the F i e l d on the Star ...................47 4.5 The Perturbation t c the Structure 51 4.6 The V a l i d i t y of the Approximation ..................59 5. •_, Conclusions •....".«••«. ..v.. • • •.... .. ••• • • '• •. v. ••.. •:«'•....... 60 6. -Acknowledgments ...... •...... ........• • • •.• •..... • •••••••61 Bibliography • •.... y.. • • .. • •« •••••• • •......•... • •• *.... 62 Appendix I: S t e l l a r Structure Equations ........ • ,,,........ ,69 Appendix I I : The D i s t o r t i o n Terms ...... , y.v•••«•••• ••••.,.. 80 Appendix I I I : The Larson-Demargue Equations ....82 Ap pen d ix IV: The Grey At mosphere .,•..........•. •,.......•..92 Appendix V: Computer Programs ..............................97 i i i FIGURES 1. The Ap S t a r s .........2 2. The Stream F u n c t i o n b(r) ..... ................ 35 3. Stream F u n c t i o n D e r i v a t i v e b 1 ( r ) ....................... 36 4. S t r e a m l i n e s f o r the Flux P e n e t r a t i n g t h e Core .......... 37 5. S t r e a m l i n e s f o r the F l u x Excluded from the Core ., 38 6. Components of the.Magnetic F i e l d v . 3 9 7. R a t i o of Magnetic to G r a v i t a t i o n a l Force 40 8. C e n t r a l R a d i a l F i e l d v s . P o l a r F i e l d 41 9. C e n t r a l Density Change v s . Po l a r F i e l d 48 10. C e n t r a l Temperature Change v s . P o l a r F i e l d 49 11. S t e l l a r Radius Change vs. P o l a r F i e l d ...50 12. The Magnetic Star i n the HR Diagram .................... 52 13. E q u a t o r i a l And P o l a r R a d i i D i f f e r e n c e , v . . . • » « 5 3 14. P o l a r And E q u a t o r i a l Temperature D i f f e r e n c e 54 15. The D i s t o r t i o n ................................•........55 16. Changes i n B and V I n d i c e s ............................. 56 17. Geometry of Star»s S u r f a c e ..91 iv TABLE'S -I : Comparison o f I n i t i a l Models . . . . . i . . . . . . . . . . . . . . . . . . . . . 31 I I : Comparison of Magnetic Stream Functions 12 I I I : Comparison of Changes to S t r u c t u r e .............51 1 1•• INTRODUCTION AND REVIEW 1. 1 Observations of S t e l l a r Magnetic F i e l d s S t e l l a r magnetic f i e l d s were f i r s t observed in sunspots by Hale in 1908, and general high in tens i ty s t e l l a r f i e l d s were discovered by Babcock in 1946. Since then a large amount of observat ional data has been accumulated on magnetic s t a r s . The measured quantity i s the e f f e c t i v e long i tud ina l magnetic f i e l d , i . e . the component of the f i e l d i n the observer 's l ine cf s igh t , averaged over the v i s i b l e sur face. The e f f e c t i v e f i e l d i s i n -ferred from the Zeeman s h i f t between r igh t and l e f t c i r c u l a r l y po lar ized components of a spect ra l l i n e (4) . The extremely small displacements produced by theZeeman e f fec t a r e . c n l y mea-surable because the magnetic s tars have very sharp l i n e s , which i s taken to mean that they.are slow rota tors (29). The quoted f i e l d s have to be viewed with some c a u t i o n . . Borra (10,11) has shown that there may be s i g n i f i c a n t errors in the measurement of the e f f e c t i v e f i e l d i f one only measures the cent ro id of the l i n e , while d isregarding the shape of the l i n e . In a d d i t i o n , the f i e l d s are found from an average over severa l spec t ra l l i n e s which r e f l e c t s i g n i f i c a n t i n t r i n s i c va r ia t ion in the f i e l d s t r e n t h . The magnetic f i e l d s found i n t h i s way vary between the l i m i t of observat ion, about 100 Gauss, and 34kG (4,5,50) which i s the largest f i e l d measured fo r a main sequence s t a r . Magnetic stars are confined to the narrow range cf spec t ra l 2 t y p e s f r o m B8 t o A O . Due t o t h e i r a n o m o l o u s a b u n d a n c e s r e l a t i v e t o o t h e r s t a r s i n t h i s t e m p e r a t u r e r a n g e , t h e y a r e c l a s s i f e d a s Ap s t a r s ( 2 9 , 50) . • F i g u r e 1 : The Ap S t a r s U n d e r l i n i n g i n d i c a t e s a magnetic f i e l d c (from Deutsch (11)) D e p e n d i n g o n t h e s u r f a c e t e m p e r a t u r e d i f f e r e n t e l e m e n t s a r e o v e r o r u n d e r a b u n d a n t (44) s o m e t i m e s a s much a s o n e t h o u s a n d t i m e s w i t h r e s p e c t t o a n o r m a l s t a r . Ap s t a r s w e r e l c n g known t o e x h i b i t p h o t o m e t r i c v a r i a t i o n s i n t h e i r b r i g h t n e s s a n d c o -l o u r , u s u a l l y w i t h an e x t r e m e l y r e g u l a r p e r i o d . I n a d d i t i o n t h e y e x h i b i t s p e c t r a l v a r i a t i o n s , p a r t i c u l a r y o f t h e p e c u l i a r a l e m a n t s . • T h e d i s c o v e r y c f a m a g n e t i c f i e l d w h i c h i n m o s t Ap s t a r s v a r i e d r e g u l a r l y w i t h e s s e n t i a l l y t h e same p e r i o d a s t h e s p e c -t r a l a n d p h o t o m e t r i c v a r i a t i o n s was h a i l e d a s t h e u n i f y i n g c h a r a c t e r i s t i c . O f c o u r s e t h e r e a r e many e x c e p t i o n s , some A p s t a r s show no m a g n e t i c f i e l d , a n d some m a g n e t i c s t a r s show no r e g u l a r i t y o f f i e l d v a r i a t i o n . A number o f e x c e l l e n t r e v i e w s o f t h e m a g n e t i c s t a r s c a n be f o u n d i n t h e r e f e r e n c e s (4 , 5 , 2 1 ,29 , 3 7 , 4 9 ) . Main sequence spectra/types 67 BS AO AZ A* A7 AS F2 _ _ i i ! : I I '• (P-v)e — Fig. 6. The color-absoluterr.acnitu ' le-spwtral type d i a -g r a m in the region of the peculiar A S t a n (Apt a n d me-tallic-line stars (.UL). .'Adjptrd irotn u. E&cxx: Astromm. J . tt. 4 ; (I9:7)-J See text. 3 1. 2 Emperical Models of Ma <jnetic Stars The empirical model which has been best elucidated, and which provides an extremely neat and e f f i c i e n t framework for the observations i s the oblique rotator model. It was f i r s t pro-posed by Stibbs (102) and later elaborated by Deutsch (29) . One of the best recent uses of the model i s Pyper*s analysis of o< 2 Canum Venaticorum, which includes detailed maps of the magnetic f i e l d and element d i s t r i b u t i o n over the the surface of the star. A simple d i p o l e . f i e l d i n the oblique rotator model gives only sinusoidal variations of the f i e l d and hence cannot explain the assymmetrical shapes of the magnetic f i e l d v a r i a t i o n . Con-sequently Landstreet (48) proposed a decentred dipole (decentred as much as 2/3 of the s t e l l a r radius for 53 Cam) which provides a much nicer f i t to the observations. However t h i s may be mostly due to the introduction of an additional free parameter. Chiam and Monaghan (16) provided an equally close f i t by mixing a quadropole component with the dipole. 1.3 The Theoretical Problem The t h e o r e t i c a l problem posed by magnetic stars i s more or less separated into two areas;.the explanation of the observed c h a r a c t e r i s t i c s of the spectrum, which means consideration of surface e f f e c t s of the magnetic f i e l d ; and the investigation of 4 t h e u n o b s e r v a b l e i n t e r i o r f i e l d s t r u c t u r e . 1.3.1 T h e S u r f a c e E f e c t s P r e s t o n (84) h a s w r i t t e n a n e x c e l l e n t summary o f t h e s u r -f a c e e f f e c t s o f m a g n e t i c f i e l d s . O n l y t h e m a j o r p o i n t s w i l l be d i s c u s s e d h e r e . M i c h a u d (60) p r o p o s e d t h a t t h e m a g n e t i c f i e l d w o u l d s t a b i -l i z e p a t c h e s . a g a i n s t a n y s m a l l t e n d e n c i e s t o mix t h e . a t m o s p h e r e , t h u s a l l o w i n g t h e r a d i a t i o n p r e s s u r e t o d r i v e a d i f f u s i o n p r o -c e s s . , T h e r e i s a l s o t h e p o s s s i b l i t y o f s u r f a c e , n u c l e a r r e a c -t i o n s (35) . P a r t i c l e s a c c e l e r a t e d t o v e r y h i g h e n e r g i e s by t h e m a g n e t i c f i e l d s u r r o u n d i n g t h e s t a r w o u l d come c r a s h i n g , i n a t t h e p o l e s t o r e a c t w i t h t h e s u r f a c e e l e m e n t s . I t i s i n t e r e s t i n g t o n o t e t h a t a l l e r a n d C o w l e y (4) t e n t a t i v e l y i d e n t i f i e d Pm i n a m a g n e t i c s t a r , t h e most s t a b l e i s o t o p e o f Pm h a v i n g a h a l f l i f e o f a b o u t 18 y e a r s . O t h e r c u r r e n t t h e o r i e s i n v o l v e b i n a r y s u p e r -n o v a e a n d a c c r e t i o n . T h e p h o t o m e t r i c c h a r a c t e r i s t i c s a r e now g e n e r a l l y b e l i e v e d t o be a c o n s e q u e n c e . o f t h e n o n - u n i f o r m d i s t r i b u t i o n o f e l e m e n t s , w h i c h r e d i s t r i b u t e s t h e f l u x t h r o u g h b a c k w a r m i n g a n d l i n e b l a n -k e t i n g . S p e c i f i c c o m p u t a t i o n s were made b y P e t e r s o n . (81) f o r a r a n g e o f S i a b u n d a n c e s a n d W o l f f . a n d W o l f f (114) f o r t h e d i f -f e r e n t i o n i z a t i o n s t a t e s o f the r a r e e a r t h e l e m e n t s . B o t h o f t h e s e were a b l e t o p r o d u c e v a r i a t i o n s o f t h e p r o p e r m a g n i t u d e , a l t h o u g h t h e r e a r e p r o b l e m s w i t h r e s p e c t t o s p e c i f i c t y p e s o f Ap 5 s t a r s . Trasco (107) has magnetic f l u x tubes emerging over the s u r f a c e , and the magnetic pressure c r e a t e s l o c a l hct s p o t s . A l s o the d i s t o r t i o n of the s t a r by.the magnetic f i e l d would pro-duce l i g h t and c o l o u r f l u c t u a t i o n s . S t r i t t m a t t e r and N o r r i s (103) have t r i e d to u n i f y the s u r -f a c e e f f e c t s of magnetic f i e l d s t o present a p l a u s i b l e o u t l i n e of the e v o l u t i o n a r y h i s t o r y of a magnetic Ap s t a r . They c o n s i -der how the magnetic f i e l d i n t e r a c t s with r o t a t i o n , c i r c u l a t i o n , c o n v e c t i o n , a c c r e t i o n , mass l o s s , and d i f f u s i o n . . In b r i e f , they f i n d t h a t i f the magnetic f i e l d exceeds an i n i t i a l c r i t i c a l value the e x t e r n a l f i e l d w i l l not vanish beneath the s u r f a c e , thus a l l o w i n g magnetic braking t o slow the s t a r . As the- s t a r slows the d r i v i n g f o r c e . f o r c i r c u l a t i o n l e s s e n s . u n t i l the magne-t i c f i e l d i s a b l e to e n t i r e l y suppress i t , and a d i f f u s i o n pro-cess can e s t a b l i s h the abundance inhomogeneities. 1.3.2 The I n t e r i o r F i e l d S t r u c t u r e The i n t e r i o r f i e l d s o f magnetic s t a r s are s p e c u l a t i v e quan-t i t e s s i n c e there i s no d i r e c t o b s e r v a t i o n a l evidence to guide one through an extremely complex and i n t r a c t a b l e problem. There are s e v e r a l immediate q u e s t i o n s ; how the f i e l d got t h e r e i n the f i r s t p l a c e , what t h e . d e t a i l e d s t r u c t u r e of the f i e l d i s , and how the i n t e r n a l f i e l d i s r e l a t e d to the o b s ervable s u r f a c e f i e l d . . The only g e n e r a l t h e o r e c t i c a l equation which covers r e a l 6 stars with magnetic f i e l d s i s the v i r i a l theorem. Invoking the global s t a b i l i t y condition.that the t o t a l energy must be nega-tive Chandrasekhar and Fermi (22) showed that the maximum RMS magnetic f i e l d i s limited by: What i s the o r i g i n of the f i e l d within the star? For strong magnetic f i e l d s the f o s s i l theory seems to be the most l i k e l y answer. During the collapse of the protostar from the i n t e r s t e l l a r clouds the gas becomes ionized and the magnetic f i e l d l i n e s are "frozen" into the collapsing cloud. The i n t e r -s t e l l a r magnetic f i e l d of 10~ 6 G i s more than s u f f i c i e n t to pro-vide the maximum, f i e l d allowed by the v i r i a l theorem. With simple arguments for a p o l o i d a l f i e l d Cowling (21) shewed that the decay time i s of the order of 10 1 0 years. Since a f o s s i l f i e l d would provide magnetic braking, t h i s theory provides a mechanism for the explanation of the generally slow rotation of the magnetic stars. I t i s possible that f o s s i l f i e l d s may be present i n many stars, but the i n t e r i o r and exterior f i e l d s have been separated and the exterior f i e l d l a t e r l o s t . The dynamo theory provides a.magnetic f i e l d by transforming the k i n e t i c energy of large scale mass motions into magnetic energy, but the dynamo requires a "seed" f i e l d . f o r i t s i n i t i a l operation. This mechanism i s not p a r t i c u l a r l y simple, since Cowling's celebrated anti-dynamo theorem precludes the main-tenence of an.axisymmetric f i e l d by symmetric mass motions. The battery e f f e c t (9 1) converts thermal energy into magne-t i c energy, through the g r a v i t a t i o n a l and r o t a t i o n a l forces pro-7 ducing a s l i g h t , charge separation between the electrons and ions. This i s the only mechanism envisioned so far which re-quires no i n i t i a l , f i e l d for i t s operation. Unfortunately the mechanism w i l l not work.at a l l i f any poloidal magnetic f i e l d component i s present,. and i n any case i t i s only capapable of generating very weak f i e l d s . 1.3.3 The Dynamics The discussion of the o r i g i n of the magnetic f i e l d serves to introduce the c r u c i a l interaction between the magnetic f i e l d and the mass motions within the sta r . The problem i s e s s e n t i a l -ly dynamic and s t a t i c solutions may impose too great a r e s t r i c -tion on the f i e l d to be representative of the true f i e l d . In sp i t e of a desire to provide truly general solutions the.problem rapidl y becomes so hopelessly complex that l i t t l e can be dene. The s t a b i l i t y of a star with a magnetic f i e l d has been d i s -cussed i n a number of recent papers. Wright (119) and Markey and Tayler (53) both used Bernstein's energy p r i n c i p l e (6) to f i n d that a purely polo i d a l f i e l d i s unstable tc "kink" and "sausage" type i n s t a b i l i t i e s . A purely t o r o i d a l f i e l d with a non-zero current density on the axis of the star i s found to be unstable to interchanges of the flux tubes (105, 109). This would occur near the centre of the star where convective motions are important, the interaction being unclear. These pure f i e l d s are unstable, but i f a polo i d a l and t o r o i d a l f i e l d cf roughly 8 squal magnitudes are.mixed, the r e s u l t i n g f i e l d i s l i k l y to be s t a b l e . It may be noted that the i r analyses made a number of s i m p l i f y i n g , assumptions which may lessen the impact of these i n s t a b i l i t i e s . The .equations were incapable of saying anything about the s i ze of the o s c i l l a t i o n s , merely.whether or not the state was s t a b l e . , Consequently the o s c i l l a t i o n s , although pre-sent , may be q u i t e . s m a l l . In add i t ion the geometry was s i m p l i -f i e d so that any s i g n i f i c a n t departure from equi l ibr ium would severely s t r a i n the assumed f i e l d s t r u c t u r e . . It i s i n t e r e s t i n g to note that the predicted per iod of o s c i l l a t i o n i s of the order of the s i z e of the reg ion , d iv ided by the Alfven wave v e l o c i t y . This period i s of the order of an hour, which i s roughly the per iod of the high frequency o s c i l l a t i o n s observed i n the . l i g h t var ia t ions ( 4 9 ) . . Only, the most rap id ly o c c u r i n g , . l a r g e s t scale i n s t a b i l i t i e s were inves t iga ted . In addi t ion there, are r e s i s -t ive i n s t a b i l i t i e s a n d . m i c r o - i n s t a b i l i t i e s which-may have an an important r o l e . These i n s t a b i l i t i e s put added r e s t r i c t i o n s on the i n t e r i o r f i e l d s t ructure on.a r e a l s t a r , and reguire more complex models to f u l f i l the s t a b i l i t y c r i t e r i o n . . I n . p a r t i c u l a r they suggest that probably a t o r o i d a l and po lo ida l f i e l d coex-i s t . The most common t h e o r e t i c a l models of the magnetic s tars fol low from the oblique rotator model, i . e . a magnetic f i e l d f ixed in a ro ta t ing s t a r . Mestel and Takhar (59) have i n v e s t i -gated the i n t e r n a l dynamics of the oblique ro ta tor . Ey c o n s i -dering energy d i s s i p a t i o n . i t i s found that the angle between the magnetic axis and the ro ta t ion ax is decreases for an oblate star and increases for a pro la te s ta r . The time scale cf the process 9 i s e a s i l y capable of changing the angle of o b l i q u i t y within the time s c a l e of a s t a r . . The i n t e r n a l motions a l s o lead t c mixing o f m a t e r i a l between the e v o l u t i o n a r y core and the s u r f a c e l a y e r s . R o t a t i n g s t a r s . a r e well known to e x h i b i t meridian c i r c u l a -t i o n , and i n g e n e r a l one would expect that even a non - r o t a t i n g magnetic s t a r would.have some c i r c u l a t i o n . I t i s p o s s i b l e t h a t the magnetic f i e l d i s s u f f i c i e n t l y w e l l f r o z e n i n t c the m a t e r i a l t h a t no c i r c u l a t i o n - a c r o s s f i e l d l i n e s i s allowed. For a r o t a -t i n g magnetic s t a r the two f i e l d s may i n t e r a c t i n such.a way the c i r c u l a t i o n i s e n t i r e l y suppressed, i n which case F e r r a r o ' s law of i s o r o t a t i o n r e q u i r e s t h a t t h e angular v e l o c i t y be constant.on stream l i n e s (34) . Maheswaran (57) has c a l c u l a t e d the e v o l u t i o n of a p r e s c r i b e d magnetic f i e l d i n a p r e s c r i b e d c i r c u l a t i o n f i e l d and f i n d s t h a t the f l u x i s e x p e l l e d from the the middle of the c i r c u l a t i o n zone, l e a d i n g to a c o n c e n t r a t i o n of the f l u x towards the p o l e s of the s t a r . The s p e c i f i c e f f e c t s of r o t a t i o n have d i f f e r e n t q u a n t i t a -t i v e f e a t u r e s dependent on the r a t i o of the r o t a t i o n a l to magne-t i c e n e r g i e s , and on.the geometry of the f i e l d . Of p a r t i c u l a r i n t e r e s t i s Mestel's .remark (57) t h a t a weak_ magnetic f i e l d , which c o n s t r a i n s a . s t a r t o uniform r o t a t i o n , s e t s up a c i r c u l a -t i o n which des t r o y s the i n i t i a l l y c o n s e r v a t i v e , i . e . c u r l f r e e , c e n t r i f u g a l f o r c e . 10 1.3.4 Convection The problem of s t e l l a r c o n v e c t i o n i s q u i t e d i f f i c u l t , even without the c o m p l i c a t i n g e f f e c t s of a magnetic f i e l d , r o t a t i o n , or a combination.of the two. I f . i n i t i a l l y t h e r e i s a magnetic f i e l d i n a zone which i s unstable to c o n v e c t i o n then the work of Weiss (111) s u g g e s t s . t h a t the eddying motions of the c o n v e c t i o n would expel the f i e l d . T h i s process r e s u l t s i n the d e s t r u c t i o n of some f l u x at the c e n t r e . o f the eddy through r e s i s t i v e d i s s i -p a t i o n , and a c o n c e n t r a t i o n of the e x p e l l e d f l u x at the borders of the eddy. T h i s process occurs on a time s c a l e of a few times the eddy tu r n o v e r . t i m e , which f o r a . c o n v e c t i v e c o r e , would be c o n s i d e r a b l y s h o r t e r than the main seguence l i f e t i m e c f the s t a r . T a y l e r , h i s co-workers (74,75, 105), and Kovetz (16) have d i s c u s s e d c r i t e r i a f o r c o n v e c t i v e s t a b i l i t y . Gough and Ta y l e r obtained a r e l a t i o n .for s t a b i l i t y i n an i n f i n i t e l y conductive plane p a r a l l e l compressible f l u i d . The s i m p l i f i e d s t a b i l i t y aquation i s : 6 »c-+'•<•«• I > V - S7aeL . I t i s r e a d i l y seen t h a t the f i e l d has a s t a b i l i z i n g i n f l u e n c e on c o n v e c t i o n , but f o r t y p i c a l c e n t r a l s t e l l a r p ressures of 1 0 1 7 dynes/cm 2, very l a r g e magnetic f i e l d s are r e q u i r e d t c supress 11 c o n v e c t i o n i n the core. T h i s c r i t e r i o n may be u s e f u l i n the core where the g r a d i e n t can be s e t egual to be a d i a b a t i c g r a -d i e n t to very high accuracy, but p r o v i d e s no equation f o r s t e l -l a r s t r u c t u r e i n the envelope, where some dynamical theory, e.g. mixing l e n g t h theory i s r e q u i r e d . S t o t h e r s and C h i n (104) and Boss and T a y l e r (75) have com-puted models of upper main sequence s t a r s with c o n v e c t i o n com-p l e t e l y suppressed i n the c o r e s , although i t i s p o s s i b l e t h a t the magnetic f i e l d may i n t e f e r e with the c o n v e c t i o n only to the extent of making i t a l e s s e f f i c i e n t energy t r a n s p o r t mechanism. The models found have e v o l u t i o n a r y h i s t o r i e s i ncompatible with o b s e r v a t i o n s o f c l u s t e r s . Hence r a d i a t i v e c o r e s do not seem to be v i a b l e . , 1.3.5, S t a t i c F i e l d S t r u c t u r e Models To make any progress i n the c o n s t r u c t i o n of models of mag-n e t i c s t a r s a gr e a t number o f s i m p l i f y i n g approximations must be made.. The e a r l i e s t models were very simple i n d e e d , but se r v e d t o o u t l i n e many of the q u a l i t a t i v e f e a t u r e s of more complex mo-d e l s , and a l s o p r o v i d e d a path of a n a l y s i s which c o u l d be ex-tended to more r e a l i s t i c cases. S e v e r a l e x c e l l e n t reviews are a v a i l a b l e (21, 55) . B r i e f l y the f i r s t models assumed t h a t the s t a r was b a r y t r o -p i c . „ T h i s i m p l i e s t h a t the magnetic f o r c e / u n i t mass must be d e r i v a b l e from a p o t e n t i a l , i . e . 12 (V x H ) * H 1 ~ i r r f J " U ' Ferraro applied t h i s condition to a l i q u i d star model (34). Later Wentzel added a t o r o i d a l f i e l d component (112) to a l i q u i d star with a po l o i d a l f i e l d . These two components of the f i e l d i n concert with the barytrcpic condition greatly r e s t r i c t the possible f i e l d structures. Wentzel was confronted by the prob-lem of surface boundary - conditions f o r the f i e l d . Either force free f i e l d s give r i s e to large surface stresses, which must be balanced by an external f i e l d ; or the surface f i e l d must vanish. An exact solution.with the surface f i e l d vanishing was found by Prendergast . (87), which has the interesting property of being spherical. Woltjer (115,116,117) and Wentzel (113) extended these r e s u l t s to include more.realistic density d i s t r i b u t i o n s . Attempts were made to establish the s t a b i l i t y of the f i e l d structure, but since thermal equilibrium i s e n t i r e l y ignored these analyses have . l i t t l e . a p p l i c a b i l i t y to r e a l i s t i c . s t a r s . Any non-spherical perturbing force i s capable of driving c i r c u l a t i o n , which i n turn i s capable of d i s t o r t i n g the o r i g i n a l perturbing force. Roxburgh looked for solutions where the c i r -c u lation and magnetic f i e l d had come into equilibrium. Solu-tions were obtained only for two cases: a dominant.rotational force with a weak .magnetic f i e l d maintaining the star i n nearly uniform rotation,.and a dominant t o r o i d a l f i e l d . Roxburgh (90).presents f i e l d s f o r r e a l i s t i c , non-polytropic stars which are made self - c o n s i s t e n t by r e s t r i c t i n g the pertur-bing magnetic f i e l d to be independent of the c i r c u l a t i o n which V x 13 i t drives. This i s done for a weak.toroidal f i e l d dominanted by rotation and . for a t o r o i d a l f i e l d which dominates the rotation and the poloidal f i e l d . Roxburgh also.examines a mixed toroidal and poloidal f i e l d i n a polytrope (92). This r e s u l t s i n an eigenvalue eguation f o r the magnetic stream function. The f i e l d i s forced to vanish at the surface to meet the condition of being derivable from a stream function. -Van der Borght has extended Roxburgh's re-s u l t s , i n p a r t i c u l a r c a l c u l a t i n g the shape of the star, Monaghan.has extensively investigated of polytropes with large dipole magnetic f i e l d s . He sets out the perturbation te-chnique for the calculation-of the change i n the star's struc-ture (61). any variable Q(r,0) i s expanded as where Q 0(r) i s the unperturbed zero order value, Ay a parameter of order of the r a t i o of the magnetic f i e l d enrgy to the gravi-t a t i o n a l energy, Ql0is the f i r s t order spherical perturbation, and Q the nonspherical term. The resulting polytropes are distorted into oblate spheroids, contracted with respect to the o r i g i n a l non-magnetic model. Later Monaghan (64) repeated the analysis with poly-tropes with a s i m p l i f i e d and improved perturbation expansion which incorporates.some of the magnetic effects in the spherical model (not the. zero order . model) . . The basic type of perturbation expansion has been extended by Monaghan to .more r e a l i s t i c s t e l l a r models (63), i . e . they include thermal equilibrium. In turn t h i s has been extended 14 again to r o t a t i n g magnetic s t a r s by Davies (28),. Wright (118), and Monaghan and Robson (70). None.of these models i n c l u d e c i r -c u l a t i o n , The f i r s t order.terms comprise a coupled s e t of non-l i n e a r d i f f e r e n t i a l . e q u a t i o n s , i n v o l v i n g the n o n - s p h e r i c a l per-t u r b a t i o n s t o - t h e . p r e s s u r e , d e n s i t y , temperature, and g r a v i t a -t i o n a l f i e l d . The magnetic f i e l d stream f u n c t i o n i s found by using an i t e r a t i v e . t e c h n i q u e on the s e t of n o n - l i n e a r . e q u a t i o n s . The s o - c a l l e d p s e u d o - p o l y t r o p i c magnetic f i e l d o b t a i n e d from 77*. -7. ~ A fa r * , i s used as a s t a r t i n g approximation. Note t h a t i m p l i e s t h a t the f i e l d i s only dependent on the s p h e r i c a l model. .The f i n a l f i e l d found e x h i b i t s a. q u a n t i t a t i v e change of about 25% from the s t a r t i n g approximation, but r e t a i n s the b a s i c q u a l i t a t i v e f e a -t u r e s . . . . . ... For the case of r o t a t i n g s t a r , i f the s u r f a c e p c l a r f i e l d s t r e n g t h i s held constant as the r o t a t i o n v e l o c i t y i s i n c r e a s e d , then the r a t i o of c e n t r a l f i e l d to p o l a r f i e l d s t r e n t h i n c r e a s e s from about 30 f o r a non-rota t i n g s t a r , to about 1200 f o r a s t a r with a r a t i o of r o t a t i o n a l to magnetic f o r c e of about 106-. Sim-i l a r l y i f the i n t e r i o r magnetic f l u x i s h e l d constant as the r o t a t i o n v e l o c i t y i n c r e a s e s , the emergent f l u x decreases t o zero at some f i n i t e l i m i t l e s s than the break up v e l o c i t y . These r e s u l t s are dependent only on the r a t i o of r o t a t i o n a l to magne-t i c f o r c e and not on t h e i r . a b s o l u t e v a l u e s r e l a t i v e to the t o t a l energy of the s t a r .(for weak f i e l d s anyway). The shape o f the s t a r as determined by Monaghan.and fiobson i s dependent on what one takes the s u r f a c e t o be, a s u r f a c e of c o n s t a n t pressure or 15 constant temperature, which are not c o i n c i d e n t s u r f a c e s i n these models. In f a c t the constant pressure s u r f a c e s are c o n s i d e r a b l y more o b l a t e . T h i s type of .model has been extended to more complex f i e l d s t r u c t u r e s . Monaghan _ (67) has p l a c e d the d i p o l a r f i e l d a t an angle to the r o t a t i o n a x i s and found that f o r a given s u r f a c e f i e l d the maximum . c e n t r a l f i e l d o c c u r s when the angle cf o b l i -q u i t y i s z e r o . U n f o r t u n a t e l y these r e s u l t s are not very com-p l e t e because„of a problem of convergence. Chiam and Monaghan (16) i n c o r p o r a t e d a m u l t i p o l e magnetic f i e l d i n a p o l y t r o p e , using the same b a s i c a n a l y s i s as Monaghan's e a r l i e r work, but with a s t r e a m . f u n c t i o n extended to h i g h e r m u l t i p o l e s . These types of c a l c u l a t i o n s were f u r t h e r extended to simple r e a l i s t i c s t a r s by Monaghan (68) and Moss (72) .. „ ..The set of e-quations which r e s u l t from the p e r t u r b a t i o n expansions are an e x t e n s i v e s e t of n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s , which are s o l v e d by recourse to the p s e u d o - p o l y t r o p i c approximation f o r a s t a r t i n g s o l u t i o n . The r e s u l t i n g f i e l d s t r u c t u r e i s s i m i l a r to the r o t a t i n g d i p o l a r magnetic s t a r s ; the i n t e r i o r c o n c e n t r a t i o n of the f i e l d i n c r e a s e s with r o t a t i o n , f o r a g i v e n f l u x t h e r e i s a maximum r o t a t i o n r a t e f o r the e x i s t e n c e of a s o l u t i o n , and t h e r e i s a minimum value t h a t the r a t i o of magnetic f i e l d energy to r o t a t i o n a l , e n e r g y c a n have as the r o t a t i o n i n c r e a s e s . . In c o n t r a s t , the s u r f a c e f i e l d need not vanish above a c e r t a i n r o -t a t i o n v e l o c i t y , s i n c e the quadropole component dees not vanish with the d i p o l e component. Recently attempts have been made to move on. from, these s t a t i c models to more r e a l i s t i c dynamic models. Monaghan (69) 16 has c a l c u l a t e d the decay of the f i e l d and Moss (73) t r i e d t o i n c l u d e some c i r c u l a t i o n . A d i f f e r e n t type o f model has been put f o r t h by Trasco (106) . He assumes that the s t a r c o n t a i n s a random magnetic f i e l d which i n t r o d u c e s an i s o t r o p i c magnetic pressure. The f i e l d i s obtained from a f l u x - f r e e z i n g c o n s i d e r a t i o n , where H*-f3 Trasco's model i s s p h e r i c a l and t h e r e f o r e amenable t c standard s t e l l a r model computer programs modified o n l y by i n c l u d i n g a magnetic pressure term t o the gas p r e s s u r e . There are no " f i r s t o r d e r " or " n o n - s p h e r i c a l " terms to be e x p l i c i t l y c a l c u l a t e d . 1.H The Purpose of T h i s I n v e s t i g a t i o n T h i s t h e s i s c a l c u l a t e s the magnetic f i e l d and the s t e l l a r s t r u c t u r e by a method i n the s p i r i t of Trasco, a n d . s i m i l a r to the one used f o r r o t a t i n g s t a r s . That i s , a r e a l i s t i c pre-s c r i b e d d i p o l e f i e l d , based upon the s t a r ' s s p h e r i c a l s t r u c t u r e i s used t o provide.a magnetic f o r c e .which i n c o r p o r a t e s . t h e b a s i c s t r u c t u r a l changes i n . t h e s p h e r i c a l model, thereby a v o i d i n g the problem of e x p l i c i t l y c a l c u l a t i n g the n o n-spherical-terms. The Henye technique f o r c a l c u l a t i n g s t e l l a r models i s n e a r l y t r i -v i a l l y adapted to t h i s . , In a d d i t i o n i t can be e a s i l y expanded to i n c l u d e dynamical and e v o l u t i o n a r y changes. A11 these p o s s i -b i l i t i e s are of great p r a c t i c a l value f o r a model of a magnetic s t a r . . 17 2. THE MODEL OF A MAGNETIC STAR 2 . 1 The Equat ions of S te 11ar S truet_ure The f u l l set. of- equations which must be s a t i s f i e d for a complete dynamical.description of a .magnetic star are- described i n Roxburgh . (90) . and elsewhere. Only the equations which are l e f t after making the various simplifying assumptions w i l l be repeated here. F i r s t , the greatest s i m p l i f i c a t i o n . This model.has no dynamical features, which immediately.means a l l quantities are independent of time,..and hence a l l time derivatives are zero. Next, there are no.large scale, ordered f l u i d motions.within the star, such.as c i r c u l a t i o n currents. This asssumption of a s t a t i c model greatly a l l e v i a t e s the computational.burden by a l -lowing us to ignore such diverse e f f e c t s as viscous energy dis-s i p a t i o n , energy transport by c i r c u l a t i o n , balance of the t o r o i -dal forces, .angular ..momentum transfer, and d i f f e r e n t i a l r o t a -tion. These e f f e c t s although important i n . themselves,.can be investigated -separately and are not necessarily c r u c i a l to un-derstanding how the magnetic f i e l d would e f f e c t the star. To simplify the e l e c t r o s t a t i c equations we assume the.star i s a perfect conductor, which considering the high densities and tem-peratures i s not. unreasonable. These assumptions leave us with the following equations. Hydrostatic equilibrium. 18 VP f 1 (2. 1) P i s t h e p r e s s u r e , .-^ i s t h e d e n s i t i y , - * V i s t h e r a t e c f a n g u l a r r o t a t i o n , u3 i s . t h e . d i s t a n c e f r o m t h e a x i s o f r o t a t i o n , 3" i s t h e c u r r e n t d e n s i t y , H i s - t h e m a g n e t i c f i e l d v e c t o r , a n d $ i s t h e p o t e n t i a l f i e l d a c t i n g on t h e s t a r , w h i c h i s u s u a l l y o n l y s e l f g r a v i t a t i o n . , . . T h e g r a v i t a t i o n p o t e n t i a l i s g i v e n b y P o i s s o n ' s e g u a t i o n , F o r e n e r g y t r a n s p o r t by r a d i a t i o n t h e f l u x i s 3 ' <fif T h e e n e r g y c a n be t r a n s f e r e d b y c o n v e c t i o n i f . (2.2) (2.3) (2.4) T h e r a t i o o f g a s p r e s s u r e t o t o t a l p r e s s u r e i s g i v e n by (3 T h e e q u a t i o n o f s t a t e f o r a p e r f e c t gas g i v e s t h e g a s p r e s s u r e 19 P1 - £ f T (2 .5) The r a d i a t i o n p r e s s u r e i s (2 .6) Energy c o n s e r v a t i o n , with the above assumptions i s simply (2.7) where ^ i s the l o c a l energy g e n e r a t i i o n r a t e per gram o f ma-t e r i a l . . P e r f e c t c o n d u c t i v i t y i m p l i e s E=0 everywhere i n the s t a r so Maxwell's equations reduce t o (2 .8) (2 .9) 20 2.2 T h e M a g n e t i c F i e l d T h e L o r e n t z f o r c e p r o d u c e d b y t h e m a g n e t i c f i e l d . r _ T y H rL - , (2.10) c a n be d e c o m p o s e d , by u s i n g (2.8) a n d s p l i t t i n g t h e f i e l d i n t o t o r o i d a l (Hj) a n d p o l o i d a l (Hp) c o m p o n e n t s , t h e n Hn fc = C * X ) * & ttv*"tj*v* f (y*£t)* Mj* ' (2.11) T h e l a s t t e r m i n (2.11) i s t h e o n l y t o r o i d a l f o r c e a n d can o n l y be b a l a n c e d by t h e c o n v e c t i o n o f a n g u l a r momentum by c i r c u l a -t i o n . S i n c e c i r c u l a t i o n h a s b e e n e x p l i c i t l y a s s u m e d n o t t o e x i s t , t h e t o r o i d a l f o r c e must v a n i s h . T h e r e f o r e , Cvx Tu ) x ~tp =q, (2.12) w h i c h i m p l i e s (2.13) W i t h (2.9) t h i s means t h a t t h e f i e l d i s d e r i v a b l e f r o m a s t r e a m f u n c t i o n . Now (2.8) when d e c o m p o s e d s a y s 21 \7x ^ -*fj;. (2 .14) We r e q u i r e t h a t J p g o e s t o z e r o a t t h e s u r f a c e . C o n s e q u e n -t l y ( 2 . 1 3 ) c o m b i n e d w i t h (2 .14) i m p l i e s t h a t f o r a m i x e d p o l i d a l - t o r o i d a l f i e l d , t h e p o l o i d a l f i e l d must go t o z e r o at t h e s u r f a c e . Hence i t c a n n o t m a t c h o n t o an e x t e r n a l , f i e l d . T h i s c o n d i t i o n on t h e p o l o i d a l c o m p o n e n t o f t h e c u r r e n t c a n a l s o be s t a t i s f i e d i f H f o r H ^ , a r e z e r o t h r o u g h o u t t h e s t a r . T h i s b o u n d a r y c o n d i t i o n c a u s e s t h e m a g n e t i c f i e l d s t o be p a r t i t i o n e d i n t o t h r e e d i s t i n c t s e t s ; 1) a p u r e l y t o r o i d a l f i e l d , 2) a m i x e d t o r o i d a l - p o l o i d a l f i e l d w h i c h i s c o n f i n e d t o t h e s t a r , . 3) a p u r e l y p o l o i d a l f i e l d w h i c h m a t c h e s o n t o a n e x t e r -n a l f i e l d . T o o b t a i n an o b s e r v a b l e f i e l d , t h e t h i r d c h o i c e was t a k e n . A t t h e o u t s e t i t s h o u l d b e p o i n t e d o u t t h a t W r i g h t and M a r k e y a n d T a y l e r h a v e shown t h a t a p u r e l y p o l o i d a l f i e l d i s u n s t a b l e t o MHD i n s t a b i l i t i e s . i n t h e s t a r . T h e i n s t a b i l i t i e s o c c u r at t h e 0 t y p e n e t r a l p o i n t , w h i c h i s l o c a t e d a t a b o u t 1 /3 t h e s t e l -l a r r a d i u s f r o m t h e c e n t r e . T h u s t h e o s c i l l a t i o n s may be g r e a t -l y damped i n t h e o v e r r i d i n g . l a y e r s o f t h e s t a r . I n s p i t e , o f t h e s e i n s t a b i l i t i e s i t was f e l t t h a t p u r e l y p o l o i d a l f i e l d s were a r e a l i s t i c a p p r o x i m a t i o n w o r t h d o i n g . O t h e r r e s e a r c h h a s b e e n done on m a g n e t i c f i e l d s o f t h i s t y p e s o 22 t h a t a t l e a s t t h e m e t h o d o f c o m p u t i n g t h e c h a n g e s t o t h e s t e l l a r s t r u c t u r e c o u l d b e . c o m p a r e d . A d d i t i o n a l l y t h e t e c h n i q u e c a n be a a s i l y e x t e n d e d t o t o r o i d a l a n d m i x e d p o l o i d a l a n d t o r o i d a l f i e l d s . 2 . 3 T h e M a g n e t i c F i e l d F c r c e P o t e n t i a l F o r s i m p l i c i t y o f c a l c u l a t i o n o f t h e s t e l l a r s t r u c t u r e , w h e r e m e t h o d s e x i s t t o h a n d l e p e r t u r b a t i o n s due t c r o t a t i o n a l f o r c e s d e r i v a b l e f r o m p o t e n t i a l f u n c t i o n s , we r e s t r i c t t h e f i e l d t o t h a t s u b s e t f o r w h i c h t h e f o r c e p e r u n i t mass i s d e r i v a b l e f r o m a p o t e n t i a l f u n c t i o n , t h a t i s (2. 15) T h e n t h e e q u a t i o n o f h y d r o s t a t i c e q u l i b r i u m r e d u c e s t o (2. 16) w h e r e $ c a n . be d e c o m p o s e d i n t o a g r a v i t a t i o n a l p o t e n t i a l , j> p l u s a p o t e n t i a l f o r t h e p e r t u r b i n g f i e l d , ^ 5 (2. 17) 23 Consequently a l l p h y s i c a l s t r u c t u r e q u a n t i t i e s P,T,andf are c o n s t a n t on s u r f a c e s o f constant jEj, and hence the o p a c i t i e s and energy g e n e r a t i o n r a t e s are constant a l s o . T h i s c o n s i d e r -ably s i m p l i f i e s the c a l c u l a t i o n of s t e l l a r s t r u c t u r e . HOw r e a l i s t i c can a magnetic f i e l d t h i s r e s t r i c t e d be? T h i s f i e l d i s not intended t c g i v e a r e p r e s e n t a t i o n of the de-t a i l e d i n t e r n a l s t r u c t u r e of the f i e l d , r a t h e r i t i s proposed as a simple and c o m p u t a t i o n a l l y guick way to c a l c u l a t e the gross a f f e c t s of a magnetic f i e l d on a s t a r , and the gross f e a t u r e s of that f i e l d . T h i s r e s t r i c t i o n of p o t e n t i a l d e r i v a b l e d i s a l l o w s o n l y non-conservative magnetic f o r c e f i e l d s , and s t i l l l e a ves a vast s e l e c t i o n which should adequately perform the f u n c t i o n s we r e q u i r e of them. How as noted from (2.13) the magnetic f i e l d i s d e r i v a b l e from a stream f u n c t i o n , j^p", such t h a t (2.18) Taken with (2.9) t h i s s p e c i f i e s f ( r , G ) ana t h e r e f o r e i n s p h e r i c a l c o - o r d i n a t e s , (2.19) The major c o n t r i b u t i o n to the observed f i e l d s i s assumed to be a d i p o l e f i e l d . T h e r e f o r e we choose 'f' to r e p r e s e n t a f i e l d which w i l l match onto an e x t e r n a l d i p o l e . . A d i p o l e has a stream f u n c t i o n ^ f l * r - 1 s i n 2 6 so f o r the i n t e r n a l f i e l d we chose 24 Y - (6 b ( r ) c ^ e , (2 .20) where (B i s c h o s e n s o t h a t b(R^.)=1. S u b s t i t u t i n g (2 .20) i n t o (2 .19 ) g i v e s f ( 2 ,21 ) N o t i n g t h a t t h e v e c t o r t e r m i n b r a c k e t s i n (2 .21) i s t h e g r a d i e n t o f b ( r ) s i n 2 ( 6 ) , a l l we r e q u i r e f o r t h e p o t e n t i a l i s t h a t (2 .22 ) He r e q u i r e a g e n e r a l b (r) w h i c h s a t i s f i e s t h i s n o n l i n e a r e q u a t i o n (and n o t e . t h a t ^ i s c o u p l e d t o b i n a f i r s t o r d e r c o r r e c t i o n ) . M o n a g h a n * s w o r k w i t h p o l y t r o p e s shewed t h a t a n e x a c t s o l u t i o n f o r b ( r ) f r o m t h e f i r s t o r d e r p e r t u r b a t i o n e x p a n -s i o n , i s g i v e n by I / ) , / , 2 i> ) - C o ^ s T H l « " f poly pe, 1 1 (2 .23) T h i s h a s b e e n s u c c e s f u l l y u s e d a s a s t a r t i n g p o i n t f o r the s o l u t i o n o f t h e n o n l i n e a r e q u a t i o n s r e s u l t i n g f r o m t h e same e x -p a n s i o n made t o f i r s t o r d e r f o r r e a l s t a r s . T h e s e s o l u t i o n s 25 were f o u n d t o d i f f e r l i t t l e f r o m t h e i n i t i a l a p p r o x i m a t i o n t o b ( r ) , t h e s o c a l l e d p s e u d o - p o l y t r o p i c s t r e a m f u n c t i o n , g i v e n b y ~f° ^ ^ (2 .24 ) T h i s i s what i s u s e d h e r e i n t o o b t a i n t h e m a g n e t i c f i e l d , w i t h t h e a d d e d d i s t i n c t i o n t h a t ^ » ( r ) i s no l o n g e r t h e z e r o o r d e r d e n s i t y d i s t r i b u t i o n , r a t h e r i t i s t h e s p h e r i c a l p a r t o f t h e d e n s i t y , now p e r t u r b e d b y t h e m a g n e t i c f i e l d . . . . . . . S u b s t i t u t i n g (2 .24) i n t o ( 2 . 2 1 ) a n d i n t e g r a t i n g we o b t a i n f o r t h e m a g n e t i c f o r c e f o t e n t i a l . ^ JL? h (,) ^ > * Hrrk (2. 25) I t s h o u l d be n o t e d t h a t . (2 .24) d o e s n o t g i v e a c o m p l e t e l y c o n s i s t e n t e g u a t i o n f o r t h e f i e l d , s i n c e t h e f o r c i n g f u n c t i o n ^ r 2 a n d t h e b o u n d a r y c o n d i t i o n s a r e e v a l u a t e d o n t h e s p h e r i c a l p a r t o f t h e m o d e l , n o t on t h e o b l a t e s p h e r o i d w h i c h t h e m a g n e t i c f i e l d c a u s e s t h e s t a r t o a s s u m e . T h e b o u n d a r y , c o n d i t i o n s f o r t h e m a g n e t i c f i e l d a r e q u i t e s i m p l e . T h e f i e l d must m a t c h o n t o an e x t e r n a l d i p c l e , s o a t t h e s u r f a c e we h a v e (2. 26) A t t h e c e n t r e we r e q u i r e t h a t t h e m a g n e t i c f o r c e v a n i s h , so 26 t ( 6 ) = 0 , {>'(<>)= V. (2 .27) E q u a t i o n (2 .24 ) i s l i n e a r a n d t h e s o l u t i o n c a n be e x p r e s s e d a s a p a r t i c u l a r s o l u t i o n p l u s a c o n s t a n t t i m e s t h e s o l u t i o n t o t h e h o m o g e n o u s e q u a t i o n . The s o l u t i o n o f t h e h o m o g e n o u s e q u a -t i o n i s C o n s e q u e n t l y i f t h e f i e l d e x t e n d s t o t h e c e n t r e o f t h e s t a r t h e b o u n d a r y c o n d i t i o n (2 .28) . r e q u i r e s t h a t a^=0. On t h e o t h e r hand t h e f i e l d may n o t e x t e n d t o t h e c e n t r e a n d a n o t h e r f i e l d s t r u c t u r e a r i s e s . A l t h o u g h t h e i n f l u e n c e o f c o n v e c t i o n o n t h e m a g n e t i c f i e l d i s . n o t w e l l u n d e r s t o o d , t h e r e a r e s e v e r a l e q u i l i -b r i u m p o s s i b i l i t i e s . The f i e l d may s u p p r e s s c o n v e c t i o n , t h e f i e l d may be e x p e l l e d . b y t h e c o n v e c t i v e . m o t i o n s , o r t h e r e may be some s o r t o f c o e x i s t e n c e . A s m e n t i o n e d a b o v e i t seems t h a t a s t a r w i t h t o t a l l y s u p p r e s s e d c o n v e c t i o n i n t h e c e r e i s . n o t v e r y p l a u s i b l e ( 1 0 4 ) . W e i s s (111) c o n d u c t e d i n v e s t i g a t i o n s i n t o t h e e x p u l s i o n o f f l u x by e d d i e s , w h i c h i n d i c a t e d t h a t t h e f i e l d may be e x p e l l e d f r o m t h e c o r e . To r e p r e s e n t a m a g n e t i c f i e l d w h i c h h a s b e e n e x p e l l e d we s e t r (2 .28) (2 .29) 27 w h e r e r c i s t h e r a d i u s o f t h e c o n v e c t i v e c o r e . F o r t h e s o l u t i o n o f t h e h o m o g e n o u s , e q u a t i o n we m a t c h (2 .28) o n t o (2 .29) w h i c h s l i m i n a t e s one o f t h e c o n s t a n t s , a n d (2 .30) U n f o r t u n a t e l y (2 .30 ) i m p l i e s t h a t a t t h e b o u n d a r y , (2 .31) w h i c h r e s u l t s i n a d i s c o n t i n u i t y i n r a d i a l f o r c e a c r o s s t h e b o u n d a r y o f t h e c o n v e c t i v e c o r e . B u t a s t h e r e s u l t s o f W e i s s show t h e r e i s an e x t r e m e l y h i g h c o n c e n t r a t i o n o f f l u x a t t h e edge o f t h e e d d i e s . . T h u s w h i l e t h e r e a r e . n o d i s c o n t i n u i t i e s we m i g h t e x p e c t an e x t r e m e l y r a p i d r i s e i n . b * ( r ) a c r o s s t h e b o u n -d a r y . , A d i s c o n t i n u i t y i s n o t v e r y p a l a t a b l e , b u t . i t was f e l t t h a t t h e a l t e r n a t i v e o f f i t t i n g a p o l y n o m i a l t o s m o o t h i t o u t , o r a n y o t h e r a r t i f i c i a l d e v i c e w o u l d be e v e n l e s s a c c e p t a b l e . A l s o t h e d i s c o n t i n u i t y i n t o t a l f o r c e , m a g n e t i c p l u s g r a v i t a t i o n i s r e l a t i v e l y s m a l l , , s o . a s . l o n g as t h e s t e l l a r s t r u c t u r e . p r o g r a m was a b l e t o c o n v e r g e , t h e a p p r o x i m a t i o n was deemed a d e q u a t e . 28 3 . T H E EQUATIONS OF S T E L L A R S T R U C T U R E 3 . 1 E x p a n s i o n Of P o t e n t i a l a n d V a r i a b l e s T h e m a g n e t i c f o r c e p o t e n t i a l o f (2 .25) c o m b i n e d w i t h t h e g r a v i t a t i o n a l p o t e n t i a l c a n now be w r i t t e n a s , J * t'(*,B)+ * fc>u; f | _ p x ( c o » e ; ] . (3.1) T o s p e c i f y t h i s p o t e n t i a l c o m p l e t e l y , we n e e d . t o know f> w h i c h i s c o m p u t e d f r o m P o i s s c n ' s e q u a t i o n . N o t e t h e p e r t u r b a -t i o n i s c o m p o s e d o f a s p h e r i c a l l y s y m m e t r i c t e r m p l u s a P*. (/*) t e r m , w h e r e y U = c o s 9 . Any v a r i a b l e Q c a n be e x p a n d e d i n a L e g e n d r e s e r i e s , ~ = i (3 .2 ) We w o u l d e x p e c t t h a t t h e d o m i n a n t p e r t u r b a t i o n t o be t h a t o n e f o r c e d b y t h e P J ^ , ) t e r m o f t h e m a q n e t i c p o t e n t i a l . We w i l l e x p l i c i t l y make t h i s a s s u m p t i o n . P , T , a n d ^ w i l l be e x p a n d e d as (3 .3 ) I f Q b i s o f o r d e r 1 t h e n i s o f o r d e r 29 a J8_2 IS (3 .4 ) W i t h t h e e x p a n s i o n f o r a s a b o v e t h e P o i s s o n ' s e q u a t i o n f o r t h e g r a v i t a t i o n a l p o t e n t i a l b e c o m e s (3 .5 ) O n l y t h e fia a n d (f>± t e r m s a r e l e f t w i t h <f e x p a n d e d a s a b o v e . 3 . 2 T h e P e r t u r b e d S t e l l a r S t r u c t u r e E q u a t i o n s T h e p e r t u r b a t i o n i s h a n d l e d i n a manner s i m i l a r t o t h a t u s e d b y F a u l k n e r , R o x b u r g h a n d S t r i t t m a t t e r (32 ) . a n d t h e J 2 method o f P a p a l o i z o u a n d W h e l a n ( 7 7 ) . T h e s p h e r i c a l t e r m s a r e f o u n d by e v a l u a t i n g , t h e e q u a t i o n s a t t h e p o i n t o n a s u r f a c e o f c o n s t a n t , p o t e n t i a l . T h i s i s an e x a c t t r e a t m e n t o f t h e p e r t u r b a t i o n , . e x c e p t f o r a p p r o x i m a t i o n s l a t e r made when e v a -l u a t i n g t h e i n t e g r a l s o v e r t h e n o n s p h e r i c a l s u r f a c e . The f u l l d e v e l o p m e n t o f . t h e . e q u a t i o n s i s g i v e n i n A p p e n d i x I . T h e r e s u l -t i n g e q u a t i o n s a r e , t o f i r s t o r d e r , f to " V } (3 .6) 30 (3 .7 ) (3 .8 ) (3 .9 ) --5 f - r , >U/3 1 ' ( 3 .10 ) (3. 11) a £ 2 3 4rrK ^ r (3 .12) T h e e s s e n c e o f t h i s m e t h o d . o f c a l c u l a t i o n o f t h e s t e l l a r s t r u c t u r e i s s i m p l i c i t y . By t a k i n g t h e d i s t a n c e t o a p o t e n t i a l s u r f a c e a l o n g t h e l i n e P ( ) - 0 a s t h e r a d i a l c o o r d i n a t e , t h e s t e l l a r s t r u c t u r e e q u a t i o n s r e t a i n t h e i r o r g i n a l f c r m . T h e o n l y 31 change i s a f i r s t o rder c o r r e c t i o n t c the mass. S i m i l a r l y the v a r i o u s p e r t u r b a t i o n s to the s t a r s s t r u c t u r e are e a s i l y c a l c u -l a t e d . The d e t a i l s of the c a l c u l a t i o n are o u t l i n e d i n Appendix The energy g e n e r a t i o n r a t e was taken from Larsen and Demargue (47). The o p a c i t i e s were found from the a n a l y t i c f i t to the K e l l e r - H e y e r o t t o p a c i t y t a b l e s used by Sackmann and Arand (96).^ The equations were i n t e g r a t e d using the r e l a x a t i o n t e c h -nique o u t l i n e d by Larson and Eemarque although the equations were changed t c n e g l e c t degeneracy pressure and to i n c l u d e r a -d i a t i o n p r e s s u r e . The composition chosen was X=.80 and Z=.Q2, corr e s p o n d i n g t o a po p u l a t i o n I s t a r . The s p h e r i c a l , non-magne-t i c s t a r t i n g models c a l c u l a t e d agree w e l l with the r e s u l t s cf other c a l c u l a t i o n s . Table I 2 Comparison of I n i t i a l Models r B/B© ~T i o g ( i / i o ) j i o g ( T e ) T ~ i o g ( B / E 0 ) I | Iben's Models X=.708 Z=.02 I ± -I - h — — — I- - H | 9 | 3.65 | 4.41 | .54 | I 5 I 2.80 | 4.29 | .35 | | 3 | 1.97 | 4.14 | .24 | H — ' i _ J . . H | Unperturbed Models Used Here X=.80 Z=.02 I | 10~ {"* 3.61 ] 4.36 J .61 "1 I 5 | 2.26 | 4.19 | .42 I | 3 | 1.69 | 4.05 | .28 | l : l — . L _ , ; I 32 3.3 The D i s t o r t i o n S u b s t i t u t i n g the expanded p h y s i c a l v a r i a b l e s i n t o the s t e l -l a r s t r u c t u r e e g u a t i c n s and equating the n o n s p h e r i c a l terms, i . e . those with a c o e f f i c i e n t of P ? f/0 # c r i t s d e r i v a t i v e , g i v e s a s e t of equations i n -ft# Pi# , and 7^. , He p o i n t cut th a t the n o n s p h e r i c a l p a r t of the magnetic p e r t u r b i n g p o t e n t i a l i s determined from the s p h e r i c a l model alone. / S i n c e a l l q u a n t i t i e s are constant on s u r f a c e s of c o n s t a n t ^ , we . only need to use t h e s e : g u a n t i t i e s to determine the shape.of the s u r f a c e c f con-s t a n t • These s u r f a c e s are e a s i l y found by making a simple T a y l o r ' s expansion about a s p h e r i c a l s u r f a c e . T i e r a c i a l d i s -tance from the ce n t r e to any p o i n t i s g i v e n , t c f i r s t o rder, by (3. 13) The expansion i s made on the p o t e n t i a l , the o n l y p a r t of which needs to be s p e c i a l l y determined i s the n o n s p h e r i c a l p a r t o f the g r a v i t a t i o n a l p o t e n t i a l . The e f f e c t i v e tempertature of the s t a r i s e a s i l y determined from (3.14) To f i n d the v a r i a t i o n of the temperature over the s u r f a c e of the s t a r we use Von Z e i p e l ' s theorem which s t a t e s t h a t the r a d i a t e d f l u x i s d i r e c t l y p r o p o r t i o n a l t c the e f f e c t i v e g r a v i t y a t th a t p o i n t . Hence 33 V Ju,f>-dt~~ (3.15) Using t h i s temperature d i s t r i b u t i o n a simple plane p a r a l l e l grey atmosphere was f i t t t e d to the s u r f a c e , from which the V and B i n d i c e s c o u l d be. o b t a i n e d . , . The 0 index was net attempted s i n c e the grey approximation, is-, net v a l i d f o r s t a r s of T^>10*6K. The c a l c u l a t i o n i s o u t l i n e d i n Appendix I I I . 14. R E S U L T S AND D I S C U S S I O N 4 . 1 T h e F i e l d S t r u c t u r e The s t r e a m f u n c t i o n f o r t h e m a g n e t i c f i e l d was c a l c u l a t e d f r o m t h e . p s e u d o - p o l y t r o p i c a p p r o x i m a t i o n ( 2 . 2 4 ) . T h e . d e n s i t y was t a k e n f r o m t h e s p h e r i c a l p a r t o f t h e m o d e l . , . T h i s s t r e a m f u n c t i o n was t h e n u s e d i n t h e m a g n e t i c f o r c e p o t e n t i a l t o o b t a i n a new s p h e r i c a l m o d e l , f r o m w h i c h a new b ( r ) was c a l c u l a t e d , a n d s o on u n t i l t h e m a g n e t i c f i e l d c o n v e r g e d . F o r a r e a s o n a b l e s t e p s i z e f r o m t h e p r e v i o u s f i e l d , f i e l d s o f o r d e r 1 0 7 G c o n v e r g e d t o 0.1% i n a b o u t .5 i t e r a t i o n s . E g u a t i o n (2 .25) - o b v i o u s l y i m p l i e s t h a t t h e a c c u r a c y o f t h e s t r e a m f u n c t i o n i s . d e p e n d e n t on t h e a c c u r a c y o f t h e r u n o f t h e d e n s i t y f r o m t h e s p h e r i c a l m o d e l , w h i c h i n t u r n i s d e p e n d e n t on t h e r e a l i t y o f t h e m o d e l u s e d . T h e j u d g e m e n t . o f t h e . .mcdel c a n be s e p a r a t e d i n t o two p a r t s ; f i r s t , t h e method o f c a l c u l a t i o n o f t h e p e r t u r b e d s t r u c t u r e ; a n d s e c o n d , t h e . i n p u t p h y s i c s f o r t h e q u a n t i t i e s r e q u i r e d by t h e s t r u c t u r e c a l c u l a t i o n . , . T h e p h y s i c s f o r t h e o p a c i t y was a c u r v e f i t t o d e t a i l e d t a b l e s . The e n e r g y g e n e r a t i o n r a t e was f o u n d . f r o m f o r m u l a s b a s e d upon s t e p , b y . s t e p r e a c t i o n s , u s i n g e x p e r i m e n t a l l y d e t e r m i n e d c r o s s - s e c t i o n s . I t i n t e r e s t i n g , t o c o m p a r e . t h e s t r e a m f u n c t i o n b (r) f o u n d h e r e t o t h e . s t r e a m f u n c t i o n f o u n d . b y M o n a g h a n . a n d R c b s o n (MR r e f . 7 0 ) a n d D a v i e s (28) f o r a d i p o l a r . f i e l d . T h e s e a u t h o r s u s e d a n o n l i n e a r c a l c u l a t i o n a n d b (r) was n o t r e s t r i c t e d t c be a p o -F i g u r e 3 : D e r i v a t i v e o f S t r e a m F u n c t i o n o F i g u r e 5 : Streamlines f o r the Flux Excluded from the Core CO oo TO EQUATOR -60 Figure 7 : Rat io of Magnetic to G r a v i t a t i o n a l Force 41 60 Figure 8 : Central Radial f i e l d vs. Polar Field H(r=0) <J> 0 55 © 5Q(^ © 3 M 0 rt © 45 40 35 30 pole 0 0 © 5 M G (J) © 0 0 10 M, 0 0 H pole 0 © 0 0 0 O 2 4 6 8 10 12 14 6  . I I 1 I I 1 x 1 0 ° G. 42 t e n t i a l f u n c t i o n . MR u s e d t h e p s e u d o - p o l y t r o p i c s t r e a m f u n c t i o n as an i n t i a l a p p r o x i m a t i o n . . T h i s i s g i v e n i n t h e c o l u m n l a -b e l l e d PP i n T a b l e I I . H o t e t h e r e i s c o n s i d e r a b l e . . d i f f e r e n c e b e t w e e n t h e s t r e a m f u n c t i o n s o f t h i s p a p e r a n d MR, t h e c n l y d i f -f e r e n c e i n c a l c u l a t i o n b e i n g t h e s p h e r i c a l m o d e l u s e d . B o t h MR a n d D a v i e s u s e d . a v e r y s i m p l e e n e r g y g e n e r a t i o n f o r m u l a ( Q^^T*1) a n d e l e c t r o n s c a t t e r i n g o p a c i t i e s . S i n c e t h e c h a n g e t o t h e n o n l i n e a r s o l u t i o n i s o n l y a s m a l l f r a c t i o n ( 25%) f r o m t h e i n i t i a l p s e u d o - p o l y t r o p i c s o l u t i o n , i t c a n be s e e n t h a t o n e d o e s n o t n e e d t o . g o t o n o n l i n e a r c a l c u l a t i o n s f o r a v a l i d m o d e l o f a n o n - r o t a t i n g m a g n e t i c s t a r . . -T a b l e III C o m p a r i s o n o f M a g n e t i c S t r e a m F u n c t i o n s | ~ | T h i s P a p e r | MS (70) I D a v i e s (28) | | x=r /R | i... ,,. ...r i |b (r) | I PP I n o n - | l i n e a r | n o n -l i n e a r F r 1 0 | 0 l 0 | 0 1 0 1 0.1 | .537 I .206 | . 258 | .22 | 0.2 | 1.51 | .706 | .864 | .78 I 0.3 | 2. 02 | 1.24 | 1.47 | 1.34 | 0.4 | 2.07 I 1.57 | 1.78 | 1.66 | 0.5 | 1. 87 | 1.65 | 1.77 | 1.69 | 0.6 | 1.63 I 1.56 | 1.61 | 1.56 I 0.7 | 1.42 | 1.40 | 1.42 | 1.38 | 0.8 | 1.23 I 1.25 | 1.25 | 1.22 I 0.9 | 1. 13 I 1.11 | 1.11 | 1.09 I 1.0 | 1.0 | 1.0 | 1.0 | 1.0 L J I L_ i _ j T h e s o l u t i o n s . a r e r e m a r k a b l y d i f f e r e n t i n t h e c e n t r a l r e -g i o n s , where "X, t h e r a t i o o f m a g n e t i c t o g r a v i t a t i o n a l f o r c e , i s t h e h i g h e s t , a n d t h e r e f o r e t h e r e g i o n w h e r e m a g n e t i c f o r c e s a r e e s p e c i a l l y i m p o r t a n t ( see f i g . 7 ) . The s t r e a m f u n c t i o n s f o u n d h e r e i n t h i s r e g i o n a r e . a t l e a s t t w i c e t h e v a l u e f o u n d by t h e o t h e r s . T h i s d i f f e r e n c e i s a t t r i b u t e d t o t h e g r e a t e r c e n t r a l 43 d e n s i t y c o n c e n t r a t i o n o f t h e more d e t a i l e d m o d e l s u s e d . N o t e t h e l o c a t i o n o f t h e 0 t y p e n e u t r a l p o i n t i s s l i g h t l y c l o s e r t o t h e o r i g i n i n t h e m o d e l s f o u n d h e r e , w h i c h may be o f i m p o r t a n c e when c o n s i d e r i n g t h e m a g n i t u d e o f t h e i n s t a b i l i t y c f t h e p o l o i -d a l f i e l d . , F i g u r e 8 shows t h a t as t h e s u r f a c e m a g n e t i c f i e l d g r o w s i n s t r e n g t h , t h e c e n t r a l c o n c e n t r a t i o n of t h e f i e l d , i . e . fl d i a l ( r = 0 ) / H ro.4,'<d ( r = . P 0 l e ) , i s r e d u c e d . A l s o t h e n e u t r a l p o i n t moves o u t w a r d s l i g h t l y w i t h r e s p e c t t o t h e r a d i u s o f t h e s t a r , w i t h i n c r e a s i n g f i e l d . , T h e s e r e s u l t s c a n be u n d e r s t o o d on an i n t u i t i v e p h y s i c a l b a s i s . A s t h e s u r f a c e f i e l d i n c r e a s e s , the s t a r c o n t r a c t s a n d t h e s u r f a c e i s t h e n s l i g h t l y c l o s e r t o t h e m a j o r c u r r e n t s o u r c e . N e a r t h e s u r f a c e t h e m a g n e t i c f i e l d v a r i e s w i t h r - 3 so f o r a g i v e n i n c r e a s e i n s u r f a c e f i e l d , o n l y p a r t o f t h e i n c r e a s e n e e d come f r o m a d i r e c t l y p r o p o r t i o n a l i n c r e a s e i n t h e i n t e r n a l c u r -r e n t , t h e r e s t c o m i n g f r o m t h e c o n t r a c t i o n o f t h e s t a r . T h u s t h e c e n t r a l f i e l d , w h i c h v a r i e s d i r e c t l y w i t h t h e i n t e r i o r c u r -r e n t , r i s e s l e s s r a p i d l y t h a n t h e s u r f a c e f i e l d a n d t h e r e f o r e t h e r a t i o o f t h e two f a l l s . S i m i l a r l y t h e o u t w a r d move o f t h e n e u t r a l p o i n t c a n b e v i e w e d as a r e s u l t o f t h e i n c r e a s e d d e n s i t y o f f i e l d l i n e s i n t h e c o r e . The f i e l d l i n e s r e p e l e a c h o t h e r a n d t r y t o e x p a n d away i n o r d e r t o m i n i m i z e t h e f i e l d e n e r g y ; c o n s e q u e n t l y t h e n e u t r a l p o i n t moves o u t s l i g h t l y . 44 4 . 2 T h e F l u x F r e e C o n v e c t i v e C o r e F i g u r e 2 shows t h e . s t r e a m f u n c t i o n f o r t h e c a s e o f t h e mag-n e t i c f i e l d e x c l u d e d f r o m t h e c o n v e c t i v e c o r e a n d f i g u r e 4 shows t h e s t r e a m l i n e s o f - t h e f i e l d . F o r r>1 /2 R t h e f i e l d r a p i d l y c o n v e r g e s t o t h e c o r e p e n e t r a t i n g f i e l d . F o r i d e n t i c a l , v a l u e s o f t h e s u r f a c e f i e l d t h e t o t a l f l u x c o n t a i n e d b e t w e e n t h e c e n t r e a n d t h e n e u t r a l p o i n t i n t h e e q u a t o r i a l p l a n e i s g i v e n by o T h i s f l u x i s a b o u t 10$ l o w e r f o r t h e f l u x e x c l u d e d c a s e . T h e m a g n e t i c f i e l d e n e r g y i s t a k e n a s (4 .2 ) where t h e m a g n e t i c f i e l d i s g i v e n by 45 2 r (4 .3 ) T h e r a t i o o f t h e m a g n e t i c f i e l d e n e r g y t o t h e g r a v i t a t i o n a l e n e r g y i s a l w a y s l o w e r f o r t h e f l u x e x c l u d e d f i e l d , g i v e n e q u a l s u r f a c e f i e l d . 4 . 3 T h e R o t a t i n g M a g n e t i c S t a r F i e l d s were c a l c u l a t e d f o r a r o t a t i n g s t a r w i t h a p o l o i d a l f i e l d w h i c h p e n e t r a t e s t h e c o r e , w i t h t h e a x i s o f r o t a t i o n p a r a l l e l t o t h e m a g n e t i c a x i s . T h e t e c h n i q u e o f c a l c u l a t i o n u s e d r e q u i r e s o n l y , t h a t t h e perfuming p o t e n t i a l . b e e x p r e s s i b l e i n t e r m s o f a s p h e r i c a l l y s y m m e t r i c t e r m p l u s a t e r m w i t h a n g u -l a r d e p e n d e n c e P ^ ( c o s Q ) . I f p i s t h e a n g l e o f o b l i q u i t y , t h e n t h e p e r t u r b i n g p o t e n t i a l i s (4 .4 ) w h i c h r e d u c e s t o s i n 2 © a n d c o s 2 8 t e r m s o n l y f o r y?=0 a n d {? - ^/2, O n l y t h e p=0 c a s e was d o n e a t t h i s t i m e . R o t a t i o n was n o t a d d e d t o t h e f l u x e x c l u d e d f r o m t h e c o r e 46 c a s e b e c a u s e i t was f e l t t h a t t h i s t r e a t m e n t o f t h e i n t e r a c t i o n o f r o t a t i o n w i t h t h e d i s c o n t i n u i t y i n b ' ( r ) a t t h e c e r e b o u n d a r y w o u l d be c o m p l e t e l y l a c k i n g i n a n y p h y s i c a l m e a n i n g . A v e r y s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h i s c a l c u l a t i o n c f a r o t a t i n g m a g n e t i c , s t a r a n d o t h e r s . (MR, W r i g h t , a n d . D a v i e s ) i s t h a t t h e o n l y - c o u p l i n g b e t w e e n t h e r o t a t i o n a n d t h e m a g n e t i c f i e l d i s t h r o u g h t h e s p h e r i c a l m o d e l . C o n s e q u e n t l y t h i s c o u p -l i n g i s a l w a y s . f a i r l y - s m a l l . On t h e o t h e r h a n d , . M R ' s a n d D a v i e s ' m o d e l s a r e c o u p l e d . . t h r o u g h t h e n o n s p h e r i c a l , d i s t o r t i o n t e r m s . T h e s e t e r m s a r e . o f o r d e r . . . o f . t h e s t r o n g e s t p e r t u r b i n g f o r c e a c t i n g , s o . i f r o t a t i o n i s d o m i n a n t , t h e m a g n e t i c f i e l d w i l l be g r o s s l y a l t e r e d f r o m t h e f i e l d o f a n o n - r o t a t i n g m o d e l . As a r e s u l t o f t h e i r method MR f i n d t h a t t h e i r s t a r i s c o o l e r a t t h e p o l e s , w h e r e a s t h e s t a r f o u n d h e r e i s h o t t e r a t t h e p o l e s , as a r e most o b l a t e . s t a r s . The m o d e l s f o u n d h e r e do n o t e x h i b i t t h e v a s t i n c r e a s e i n i n t e r i o r f i e l d c o n c e n t r a t i o n , n o r - t h e v a n i -s h i n g o f t h e s u r f a c e f l u x a t s o m e . f i n i t e r o t a t i o n a l v e l o c i t y . T h e c e n t r a l f i e l d . c o n c e n t r a t i o n i s e x p e c t e d t c r i s e , due t o t h e s a m e , e f f e c t , o f t h e . c h a n g e i n r a d i u s a s b e f o r e w i t h a n o n -r o t a t i n g s t a r , e x c e p t t h a t . h e r e t h e r e w i l l be an e x p a n s i o n due t o r o t a t i o n , r a t h e r t h a n a c o n t r a c t i o n , w h i c h w i l l c a u s e t h e r a t i o t o r i s e r a t h e r t h a n f a l l . F o r a f a i r l y weak f i e l d o f 1 0 s G i t i s f o u n d t h a t H r ( r = 0 ) /tty (r=R^) a p p r o x i m a t e l y d o u b l e s as g o e s f r o m z e r o t o a . v a l u e s u c h t h a t t h e r a t i o o f t h e . r c t a t i c n a l f o r c e t o g r a v i t a t i o n a l f o r c e a t t h e e g u a t o r i s a b o u t 75%. M c n a -g h a n a n d R o b s o n , a n d W r i g h t f i n d t h a t t h i s r a t i o i n c r e a s e s a b o u t 20 t i m e s f o r a s i m i l a r c h a n g e i n J l . 47 4 » 4 E f f e c t s o f t h e F i e l d on t h e S t a r A l l t h e - w e l l known r e s u l t s f o r a p o l o i d a l f i e l d a r e c o n -f i r m e d f o r b o t h t h e m a g n e t i c f i e l d s t r u c t u r e s , m a g n e t i c . f l u x p e r m e a t i n g t h e c o r e a n d f l u x e x c l u d e d f r o m t h e c o r e . T h e d i f -f e r e n c e s a r e l a r g e l y q u a n t i t a t i v e . . T h e c h a n g e s i n t h e s p h e r i c a l m o d e l as a f u n c t i o n o f f i e l d s t r e n g t h a r e shown i n t h e . . a c c o m -p a n y i n g f i g u r e s . , . . ( 9 , 1 0 R - 1 1 ) . T h e l u m i n o s i t y o f t h e s t a r i s r e -d u c e d w i t h i n c r e a s i n g m a g n e t i c f i e l d . T h e a d d e d m a g n e t i c f o r c e i n t h e r a d i a l d i r e c t i o n a t t h e c e n t r e a l l o w s t h e e q u i l i b r i u m t o e x i s t w i t h a . l o w e r - P J V I 4 + P r ^ , so t h e c e n t r a l t e m p e r a t u r e - d r o p s , w i t h a c o n s e q u e n t d r o p i n t h e t h e r m o n u c l e a r r e a c t i o n r a t e . F r o m t h e n e u t r a l p o i n t o u t w a r d s t h e f i e l d r e i n f o r c e s t h e . g r a v i t a -t i o n a l f o r c e , . w h i c h b r i n g s a b o u t a n e t c o m p r e s s i o n o f t h e s t a r , a n d a r i s e m . t h e c e n t r a l d e n s i t y . S i n c e C«?l . . , t h e s l i g h t r i s e i n ^ i s i n s u f f i c i e n t t o o f f s e t t h e f a l l i n T . F o r t h e _ e f f e c t i v e t e m p e r a t u r e , T c °< -J r^i j , t h e c h a n g e i n t h e l u m i n o s i t y must be o f f s e t a g a i n s t t h e c h a n g e . i n r a d i u s . F o r t h e f l u x e x c l u d e d mag-n e t i c f i e l d , t h e h i g h - f o r c e s a t t h e c o r e - b o u n d a r y a r e p r o p o g a t e d t h r o u g h t h e s t r u c t u r e s u f f i c i e n t l y t h a t t h e l u m i n o s i t y i s r e -d u c e d t o a g r e a t e r . . e x t e n t . t h a n t h e f o r t h e f l u x p e n e t r a t i n g c a s e . , On t h e o t h e r h a n d , s i n c e t h e two f i e l d b f u n c t i o n s c o n -v e r g e s t r o n g l y , beyond. , t h e n e u t r a l p o i n t , t h e n e t i n w a r d f o r c e when a v e r a g e d o v e r t h e s t a r i s g r e a t e r f o r t h e f l u x p e n e t r a t i n g c a s e , t h a n . t h e f l u x e x c l u d e d c a s e . C o n s e q u e n t l y , t h e s t a r c o n -t r a c t s l e s s f o r t h e f l u x e x c l u d e d f i e l d . T h e c o n t r a c t i o n i n b o t h c a s e s i s more t h a n e n o u g h t o o f f s e t t h e . f a l l i n t h e l u m i n o -s i t y , s o T r i s e s , t h e e f f e c t b e i n g l e s s f o r . t h e f l u x e x c l u d e d c a s e . T h e s e c h a n g e s i n t h e l u m i n o s i t y a n d e f f e c t i v e t e m p e r a t u r e F i g u r e . 9 . : C e n t r a l Density Change v s . P o l a r F i e l d 3 M„ A A A A o A o o A 6 8 Polar Magnetic F i e l d A f l u x excluded from core Q f l u x penetra t ing core I I _ 10 12 x 10 co Figure 10 : Central Temperature Change vs. Polar Field Polar Magnetic Field A 6 8 10 12 x 10 6 -.01 -.02 A T> -.03 o -.OA A flux excluded from core O flux penetrating core - . O A Figure 11 : Stellar Radius Change vs. Polar Field Polar Magnetic Field 8 10 O A O A 12 x 10' -.08 R -.12 -.16 O A o A A flux excluded from core O flux penetrating core Ln o 51 mean t h a t t h e s t a r moves t o t h e l e f t and down i n t h e HR d i a g r a m , i . e . b e l o w t h e main s e q u e n c e . T h i s i s i n c o n t r a s t t o T r a s c o ' s m o d e l s . H i s s t a r s h a v e a . r a n d o m f i e l d w h i c h a d d s a n . e v e r y w h e r e p o s i t i v e m a g n e t i c p r e s s u r e , - w h i c h c a u s e s t h e . s t a r s t o e x p a n d . T r a s c o ' s s t a r s . h a v e . a r e d u c e d l u m i n o s i t y , b u t . a n i n c r e a s e d r a -d i u s , s o t h a t t h e y move t o t h e r i g h t a n d d o w n , w h i c h i s a b o v e t h e m a i n s e g u e n c e . F o r a c o m p a r i s o n . o f r e s u l t s , . M o n a g h a n a n d R o b s o n ' s p e r t u r -b a t i o n s a g a i n s t t h o s e f o u n d h e r e . a r e p r e s e n t e d b e l o w . . T a b l e I I I I C o m p a r i s o n o f C h a n g e s t o S t r u c t u r e f --• -1 T • T ! 1 I Kfole, I A l ° g l - I 4 l o g R | J l c g T e | Monaghan and R o b s o n r • H 1- h 6x106 j - . 0 2 9 | - . 0 4 2 | .008 11x106 I - . 0 8 2 | - . 1 1 0 | .045 14x10* | - . 1 2 7 | - . 1 6 2 | .068 j J i T h i s P a p e r 5x106 | - . 0 1 3 | - . 0 4 9 | . 0 0 5 3 10x10* | - . 0 3 7 | - . 1 2 1 | . 0 1 4 15x10* | - . 0 6 1 | - . 1 8 5 | . 0 2 2 M R ' s r e s u l t s i n d i c a t e l a r g e r c h a n g e s t o t h e s p h e r i c a l m o d e l t h a n t h o s e p r e s e n t e d h e r e . T h i s . i s p e r h a p s due t o t h e u s e o f s i m p l e r i n p u t p h y s i c s f o r t h e o p a c i t y a n d e n e r g y g e n e r a t i o n r a t e . 4 . 5 T h e P e r t u r b a t i o n s t o t h e S t r u c t u r e Figure 12 : The Magnetic Star in the HR diagram -.05 A L o g ^ L © -.10 - 1 « I , . . . „ r > i • A 3 A 5 A 3 8 A 10 _ A o * 12 . 5 — -o 10 - -A flux excluded from core O f l u x penetrating core 1 1 1 Fields in 10 Gauss 1 1 .025 .020 C .015 .010 .005 A log T e % 10 Figure 13 : Oblateness of Star v s . P o l a r F i e l d 5 M„ A o A O 2, A O A o H pole A f l u x excluded from core O f l u x penetra t ing core 10 121' x l O 6 G-. J I Lo 300 Figure 14 : Polar minus Equatorial Temperature Difference K 250 A o 200 A o 150 100 A o A flux excluded from core O flux penetrating core '50 6 4 H ju xlO G J L_ . . pole lei RADIUS 56 Figure 16 : Changes in B and V indices -.025 -.020 -.015 -.010 -.005 57 B o t h p o l o i d a l m a g n e t i c f i e l d s p r o d u c e f o r c e s w h i c h p e r t u r b t h e s t a r i n t o an o b l a t e s p h e r o i d . T h e d i f f e r e n c e s b e t w e e n t h e e q u a t o r i a l a n d p o l a r r a d i i i s p l o t t e d i n F i g u r e 13 . F o r w e a k e r f i e l d s t h e d i f f e r e n c e i s c o n s i d e r a b l y g r e a t e r f c r t h e f l u x e x -c l u d e d f i e l d . W i t h l a r g e r f i e l d s , t h e f i e l d i s s u f f i c i e n t l y s t r o n g t o p r o d u c e p e r t u r b a t i o n s i n t h e s p h e r i c a l s t r u c t u r e w h i c h r e a c t b a c k on t h e f i e l d s t r u c t u r e . T h i s p r o d u c e s s u f f i c i e n t " s p r e a d i n g " o f t h e f i e l d , i . e . b ^ ^ i s r e d u c e d , a n d t h e p o s i t i o n o f b ^ a y moves o u t w a r d s o m e w h a t , s u c h t h a t t h e d i f f e r e n c e b e t w e e n t h e e q u a t o r i a l - a n d p o l a r r a d i i a n d t h e t e m p e r a t u r e c h a n g e c o n -v e r g e t o w a r d s . a common v a l u e f o r t h e two f i e l d s t r u c t u r e s . T h e g r a p h . o f T ^ b e t w e e n t h e p o l e a n d t h e e q u a t o r , c l e a r l y i n d i c a t e s t h a t t h e o b s e r v e d m a g n e t i c f i e l d s (maximum H o f a b o u t 1 0 s G) d o e s n o t p r o d u c e s u f f i c i e n t p e r t u r b a t i o n t o t h e s t r u c t u r e t o be t h e e x p l a i n i n g f a c t o r f o r t h e l i g h t . v a r i a t i o n o f m a g n e t i c s t a r s . F o r a 5 KQ s t a r w i t h t h e f l u x p e n e t r a t i n g t h e c o r e t h e d i f f e r e n c e i n t e m p e r a t u r e i s a b o u t 0 . 0 5 ° K , w h e r e a s t h e r e q u i r e d t e m p e r a t u r e d i f f e r e n c e i s . o f t h e o r d e r o f 1 0 3 . P K ( 1 0 7 ) . I t i s p o s s i b l e t h a t v e r y l a r g e i n t e r n a l f i e l d s do e x i s t , . b u t t h a t t h e r e i s some s u r f a c e m e c h a n i s m w h i c h t r a p s most c f t h e f l u x w i t h i n t h e s t a r . A l l o f t h e s e s t a r s h a v e r a d i a t i v e e n v e l o p e s , a n d t h e o b s e r v a t i o n s i n d i c a t e l o n g p e r i o d s t a b i l i t y c f t h e n o n -h o m o g e n o u s d i t r i b u t i o n o f e l e m e n t s o v e r t h e s u r f a c e s o any f l u x c o n t a i n i n g m e c h a n i s m must b e . a r e l a t i v e l y q u i e t p r o c e s s . I t i s p o s s i b l e t h a t . . s i g n i c a n t c i r c u l a t i o n . . m i g h t d e v e l o p , somewhere b e l o w t h e s u r f a c e w h i c h w o u l d t e n d t o s h e a r t h e f i e l d l i n e s a n d s u p p r e s s t h e . f l u x t h r o u g h t h e . s u r f a c e , o r t h e i n f l u e n c e _ o f r o t a -t i o n i t s e l f may o p e r a t e t o s u p p r e s s t h e f l u x as shewn b y MR, 58 W r i g h t a n d D a v i e s . T h e c h a n g e s i n t h e r a d i u s , a n d t h e e f f e c t i v e t e m p e r a t u r e o v e r t h e s u r f a c e o f t h e s t a r p r o d u c e d i f f e r e n c e s i n t h e UEV i n -d i c e s a s t h e m a g n e t i c a x i s . i s t i l t e d w i t h r e s p e c t t c t h e o b s e r -v e r . B e c a u s e o f t h e . o b l a t e f i g u r e , o f t h e . s t a r t h e s u r f a c e " g r a v i t y " i s h i g h e r a t . a t . t h e p o l e s w h i c h g i v e s r i s e t c a h i g h e r T g a t t h e p o l e s , i n a d d i t i o n t h e s u r f a c e v i s i b l e t o t h e o b s e r v e r h a s a l a r g e r c r o s s - s e c t i o n a l a r e a when v i e w e d p o l e c n . C o n s e -q u e n t l y t h e s e two e f f e c t s , c o m b i n e , t o m a k e „ t h e . s t a r a p p e a r b r i g h t e r a n d . h o t t e r when v i e w e d p o l e on t h a n w h e n . v i e w e d e g u a t o r o n . F o r a f i e l d o f . 1 5 x 1 0 * G t h e d i f f e r e n c e b e t w e e n V . p o j e •-Ve»*,J»r i s a b o u t . 1 5 m a g n i t u d e , a n d t h e p o l e m i n u s e q u a t o r . (B-V) c h a n g e i s a b o u t . 0 0 8 m a g n i t u d e . I n t h e A V - A ( B - V ) d i a g r a m t h e l i n e r e p -r e s e n t i n g t h e . c h a n g e i n t h e . i n d i c e s f o r v a r i o u s . a n g l e s o f i n c -l i n a t i o n i s v i r t u a l l y a . s t a i g h t l i n e , a s o p p o s e d t o t h e c a s e o f r o t a t i o n ( 9 5 ) , where a . s h a l l o w c u r v e i s d e s c r i b e d b e t w e e n t h e p o l e o n an e q u a t o r on v a l u e s . S i m i l a r l y f o r an i n f e r r e d maximum o b s e r v e d p o l o i d a l f i e l d o f 1 0 s G t h e . . v a r i a t i o n s a r e 4.Vp^ = 0 . 6 7 x 1 0 - 5 mag. a n d A ( B - V ) = r 0 . 7 5 x 1 0 ~ * mag. , . h a r d l y s u f f i -c i e n t t o a c c o u n t f o r t h e o b s e r v e d v a r i a t i o n s , w h i c h a r e o f o r d e r 0 . 1 m a g n i t u d e . 59 4 . 6 T h e V a l i d i t y o f The A p p r o x i m a t i o n The e x p a n s i o n . o f t h e v a r i a b l e s was o n l y made . t o f i r s t o r d e r , a n d . a l l s e c o n d o r d e r t e r m s were n e g l e c t e d , . I n o r d e r t o s a t i s f y t h e a p p r o x i m a t i o n c o n d i t i o n s , c a l c u l a t i o n s o f t h e f i e l d s t r u c t u r e were o n l y c a r r i e d t o a maximum r a t i o o f t h e m a g n e t i c f i e l d e n e r g y t o t h e g r a v i t a t i o n a l . e n e r g y o f 1 0 $ , h o p i n g t h a t a t t h i s p o i n t t h e s e c o n d o r d e r e f f e c t s w o u l d be a b o u t ( , 1 ) 2 o r 1% a n d h e n c e s t i l l e a s i l y d i s c a r d a b l e . . F o r t h e f i e l d o f 10x10* G i n a 5 M g>star t h e h i g h e s t f o r c e e n c o u n t e r e d was a t t h e c e n t r e w h e r e 'X =F*/]j = - , 1 0 f o r f l u x p e n e t r a t i n g a n d . - . 16 f o r - t h e f l u x e x c l u d e d f i e l d . T h e . s e c o n d o r d e r c o r r e c t i o n t e r m s i n a l l t h e e q u a t i o n s i n v o l v e d , t e r m s i n c- 2 (r) o r S (r) , w h i c h a r e o f o r d e r g-2. F i g u r e .15 s h o w s t h a t f o r a f i e l d w i t h an e n e r g y . r a t i o o f 6.9%, t h e maximum, v a l u e , o f i s . o n l y . 1 1 3 and h e n c e €• 2 i s . 0 1 2 . S i n c e a l l p h y s i c a l v a r i a b l e . a r e c o n s t a n t on s u r f a c e s o f c o n s t a n t p o t e n t i a l , . n o e x p l i c i t p r o b l e m o f . t h e p e r t u r b a t i o n e x -p a n s i o n a r o s e a t t h e s u r f a c e w h e r e t h e p r e s s u r e a n d t e m p e r a t u r e go t o z e r o . B u t . i t was n e c e s s a r y t o make an e x p a n s i o n o f t h e p o t e n t i a l i n t o s p h e r i c a l a n d P a ( « ) t e r m s i n . o r d e r t o c a l c u l a t e t h e d i s t o r t i o n , . fe. (r) . . . . . . The o t h e r v a r i a b l e s c a n be l e f t a s i d e o n c e t h e p o t e n t i a l s u r f a c e s a r e k n o w n . 60 5 . CONCLUSIONS A d i p o l e m a g n e t i c f i e l d i n an u p p e r m a i n s e q u e n c e s t a r was c a l c u l a t e d by a q u i c k a n d s i m p l e m e t h o d . T h e f o r c e p r o d u c e d b y t h e m a g n e t i c f i e l d was c o n s t r a i n e d t o be c u r l f r e e . T h e f i e l d was c h o s e n t o m a t c h o n t o . a n e x t e r n a l d i p o l e , so t h a t t h e s t r e a m f u n c t i o n was made u p . o f . o n l y s p h e r i c a l a n d P ^ ( c o s 6 ) t e r m s . T h i s a l l o w e d g r e a t s i m p l i f i c a t i o n s o f t h e e q u a t i o n s o f s t e l l a r s t r u c -t u r e , i n f a c t t h e o n l y . c h a n g e i s a n a l t e r a t i o n o f t h e e f f e c t i v e mass a s a . f u n c t i o n o f r a d i u s . T h e f i e l d i t s e l f i s c a l c u l a t e d f r o m a " p s e u d o - p o l y t r o p i c ' ! . e q u a t i o n , d e p e n d e n t o n l y u p o n t h e s p h e r i c a l p a r t o f t h e m o d e l . Two t y p e s o f . f i e l d s w e r e . c a l c u -l a t e d , one w h e r e t h e f l u x p e n e t r a t e s t h e c o n v e c t i v e c o r e , t h e o t h e r where t h e f l u x i s e x c l u d e d . f r o m t h e c o r e . T h e r e s u l t i n g p e r t u r b a t i o n o f t h e s t a r , a n d . t h e c h a n g e s o f t h e f i e l d w i t h i n -c r e a s i n g s t r e n g t h a r e e x p l a i n a b l e on an i n t u i t i v e p h y s i c a l b a s i s . . When r o t a t i o n was a d d e d t o a s t a r w i t h a m a g n e t i c f i e l d , i t was f o u n d t h a t t h e e f f e c t s . w e r e a l m o s t s t r i c t l y a d d i t i v e , s i n c e t h e o n l y c o u p l i n g was t h r o u g h t h e s p h e r i c a l m o d e l . I n t h e f u t u r e , t h i s t e c h n i q u e w i l l be e x t e n d e d t o c o v e r f i e l d s w i t h t o r o i d a l a n d m i x e d p o l o i d a l - t o r o i d a l f i e l d s . B e i n g a v e r y q u i c k way t o c a l c u l a t e t h e f i e l d , i t w i l l a l s o be an e f -f i c i e n t way t o f o l l o w t h e e v o l u t i o n o f a s t a r w i t h a m a g n e t i c f i e l d . 61 6 . ACKNOWLEDGMENTS ... . I am d e e p l y g r a t e f u l f o r t h e v a l u a b l e d i s c u s s i o n s a n d a s -s i s t a n c e o f P r o f , G . G . F a h l m a n . 62 BIBLIOGRAPHY 1 . A b r a m o w i t z , M. a n d S t e g u n , I . A . , H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s , N B S , U . s . G o v e r n m e n t P r i n t i n g O f f i c e , W a s h i n g t o n , D . C . ( 1 9 6 4 ) . , 2 . A l f v e n , H . a n d F a l t h a m m a r , C . - G . , C o s m i c a l E l e c t r o j y n a j i c s , 2nd e d . , C l a r e n d o n P r e s s , - O x f o r d , 1 9 6 3 . 3 . A l l e r ,M . F . , a n d C o w l e y , C . R . , A s t r o p h j j s . J o u r n . 1 6 2 , L 1 4 5 (1970) . . . " * " " " " " . 4 . B a b c o c k , H . 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S c h w a r s c h i l d , H . , S t r u c t u r e And E v o l u t i o n o f t h e S t a r s , r p t . D o v e r , N e w Y o r k " 1965)".~ 9 9 . S c h w a r z s c h i l d , M . , A s t r o p h y s . J o u r n . 1 1 2 , 2 2 2 ( 1 9 5 0 ) . 100. S m i t h T . S . , A s t r o p h y s . J o u r n . 1 3 9 , 7 6 7 ( 1 9 6 4 ) . 101 . S t e p i e n , K . , A s t r o p h y s . J o u r n . 1 5 4 , 9 4 5 ( 1 9 6 8 ) . 102. S t i b b s , D . W . N . , M o n t h l y N o t i c e s R o y . A s t r o n . S e c . 1 1 0 , 3 9 5 (1950)7 1 0 3 . S t r i t t m a t t e r , P . A . , a n d N o r r i s , J . , A s t r o n . A s t r o p h y s . 1 5 , 2 3 9 (1 9 7 1 ) . 104 . S t o t h e r s , R , a n d C h i n , C . - W . , A s t r o p h y s . J o u r n . 1 8 0 , 9 0 1 (1973) . 1 0 5 . T a y l e r , R . J . , M o n t h l y N o t i c e s R o j . A s t r o n . S o c . 1 6 1 , 3 6 5 ( 1 9 7 3 ) . 106 . T r a s c o , J . D . , A s t r o p h y s . J o u r n . 1 6 1 , 6 3 3 ( 1 9 7 0 ) . 107. T r a s c o , J . D . , A s t r o p h y s . J o u r n . 171 ,569 ( 1 9 7 2 ) . 108. T r e h a n , S . K . , a n d U b e r o i , M . S . , A s t r o p h y s . J o u r n . 1 7 5 , 1 6 1 ( 1 9 7 2 ) . 1 0 9 . V a n d a k u r o v , Y u . V . , S o v i e t A s t r o n . AJ 16, 265 ( 1 9 7 2 ) . 1 1 0 . Van d e r B o r g h t , R . , A u s t r a l i a n J o u r n . P h y s . 2 0 , 6 4 3 (1 967) . 1 1 1 . W e i s s , N . O . , P r o c . . R o y . S o c . L o n d o n A 2 9 3 , 3 1 0 (1966) . 1 1 2 . . W e n t z e l , D . G . , A s t r o p h y s . J o u r n . 1 3 3 , 1 7 0 ( 1 9 6 1 ) . . 113. W e n t z e l , D . G . , A s t r o p h y s . J o u r n . S u p p . S e r . 5 , 1 8 7 (1966) . " ~ ~ ~ 1 1 4 . W o l f f , S . C . a n d W o l f f , R . J . A s t r o n . J o u n . 7 6 , 4 2 2 ( 1 9 7 1 ) . 68 1 1 5 . W o l t j e r , L . , A s t r o p _ h y s . J o u r n . 1 3 0 , 4 0 0 ( 1 9 5 9 ) . 1 1 6 . W o l t j e r , L . , A s t r o p h y s . J o u r n . 1 3 1 , 2 2 7 ( 1 9 6 0 ) . 1 1 7 . W o l t j e r , L . , A s t r o p b j s . J o u r n . 1 3 5 , 2 3 5 ( 1 9 6 2 ) . 1 1 8 . W r i g h t , G . A . E . , M o n t h l y N o t i c e s B o y . A s t r o n . S e c . 1 4 6 , 1 9 7 ( 1 9 6 9 ) " " 1 1 9 . W r i g h t , G . A . E. , M o n t h l y N o t i c e s ROJJ . A s t r o n . S o c . 1 6 2 , 3 3 9 ( 1 9 7 3 ) . 69 APPENDIX I I STELLAR STRUCTURE EQUATIONS By a s s u m p t i o n t h e p o t e n t i a l c a n be d e c o m p o s e d a s ( A l . 1) And a l l q u a n t i t i e s c a n be e x p a n d e d l i k e (A I . 2) where Q 0 ( r 0 ) i s t h e q u a n t i t y e v a l u a t e d on t h e l i n e P 2 _(cos9) = 0 , w h e r e r 0 i s t h e d i s t a n c e f r o m t h e c e n t r e t o t h a t p o i n t cn t h e s u r f a c e . N o t e a l l q u a n t i t i e s Q 0 ( r © ) a r e c o n s t a n t on t h e e q u i p o -t e n t i a l s u r f a c e < £ = c o n s t a n t . S u b s t i t u t i n g t h e e x p a n d e d v a r i a b l e s i n t o P o i s s o n ' s e g u a t i o n we o b t a i n . E x p r e s s i n g t h i s i n s p h e r i c a l p o l a r c o o r d i n a t e s and i n t e g r a t i n g o v e r 0 on t h e s h e l l r=rt> , t h e P, t e r m s d r o p o u t a n d , (AI. 3) ( A I . 4 ) I n t e g r a t i n g t h i s o v e r r g i v e s 70 A & m V ~~ ) ( A l . 5) (AI . 6 ) T o d e t e r m i n e t h e s h a p e o f t h e s u r f a c e s o f c o n s t a n t p o t e n -t i a l we make a s e c o n d o r d e r T a y l o r ' s e x p a n s i o n o f t h e t c t a l p o -t e n t i a l a b o u t t h e p o i n t (r , P ^ (cos&) =0) , $tr,e)~ $ ( r . , p w . o ) + ^ ^ - r 0 ) Ir-r.)* »iv (Ai. 7) We want a s u r f a c e o f c o n s t a n t p o t e n t i a l so t h a t ( A I . 8) P u t t i n g t h i s a l l t o g e t h e r w i t h t h e e x p a n d e d p o t e n t i a l we have y p ^ o . ( A I . 9 ) w h i c h i s a q u a d r a t i c e q u a t i o n i n ( r - r & ) . U s i n g t h e b i n o m i a l e x p a n s i o n w i t h t h e q u a d r a t i c e q u a t i o n we o b t a i n ( $ b » j £ ) r Which c a n be w r i t t e n as 71 ( i l . 1 1 ) T o f i r s t o r d e r t h i s i s s i m p l y The v o l u m e i n s i d e a n e g u i p o t e n t i a l s u r f a c e , V J i s e a s i l y o b t a i n e d ( A l . 12) V f = f j J f'cfr^ df. ( A I . 1 4 ) E v a l u a t i n g t h i s w i t h t h e r a d i u s g i v e n t o s e c o n d o r d e r by ( A I . 1 1 ) g i v e s , 3 ^ s 5 ( A l . 15) N o t e t h a t t o f i r s t o r d e r V =4 / 3 r , 3 , and t h a t ( A I . 1 6 ) T h e a r e a o f t h e s u r f a c e ^ = c o n s t a n t i s a l s o e a s i l y d e t e r -72 mined ( A I . 1 7 ) ( A I . 1 8 ) { A l . 19) T h e L a r s o n - D e m a r q u e s c h e m e f o r t h e s o l u t i o n c f t h e s t e l l a r s t r u c t u r e e g u a t i i o n s r e q u i r e s a s t h e i n d e p e n d e n t v a r i a b l e t h e mass f r a c t i o n i n s i d e t h e v o l u m e w i t h r a d i u s r , h e r e g i v e n by t h e mass i n s i d e t h e e q u i p o t e n t i a l s u r f a c e w i t h r a d i u s r a l o n g t h e l i n e P =0. T h i s mass M ^ i s , £ ( A I . 2 0 ) ( A I . 2 1 ) 73 • o ( A I . 22) r ( A I . 2 3 ) T o d e v e l o p t h e p e r t u r b e d e q u a t i o n s o f s t e l l a r s t r u c t u r e , we f o l l o w t h e f o r m a l i s m o f t h e J 2 m e t h o d . T h e d e p e n d e n t v a r i a b l e w i l l be r „ . F i r s t t h e r a d i a t i v e t r a n s f e r e q u a t i o n , £ * r s vr, ( A I . 2 4 ) T h i s i s e a s i l y i n t e g r a t e d o v e r a s u r f a c e o f c o n s t a n t |> , s i n c e a l l q u a n t i t e s e x c e p t t h e g r a d i e n t a r e c o n s t a n t o n | > = c c n s t a n t . ( A I . 2 5 ) L f i s d e f i n e d as t h e t o t a l e n e r g y r e l e a s e d i n s i d e the v o l u m e V £ , ( A I . 2 6 ) A l t h o u g h t h e m o d e l s d i s c u s s e d h e r e do n o t have c i r c u l a t i o n , t h e J 2 m e t h o d e a s i l y d e a l s w i t h i t , i n f a c t i t d r o p s o u t o f t h e 74 s t r u c t u r e e q u a t i o n s . T o show t h i s , t h e a b o v e e x p r e s s i o n i s r e -w r i t t e n u s i n g t h e e n e r g y b a l a n c e e g u a t i o n o f FRS (11) (AI.27) T h e n w i t h G a u s s ' T h e o r e m a n d t h e s i m p l i f y i n g e x p r e s s i o n o f I R S ( A l . 28) l u t s i n c e c « T + $ i s c o n s t a n t on c o n s t a n t ^ s u r f a c e s , a nd B U T - biiiue <-^>. t h e mass i n s i d e an e g u i p o t e n t i a l s u r f a c e must be c o n s t a n t , t h i s s i m p l i f i e s t o ( A I . 2 9 ) Now t h e e g u a t i o n o f r a d i a t i v e e q u i l i b r i u m , w i t h a l i t t l e r e a r r a n g i n g becomes ( A I . 3 0 ) The o t h e r e q u a t i o n s a r e r e c a s t i n t o t h e J 2 f c r m a t a n a l o -g o u s l y , w i t h g r e a t e r e a s e . T h e e q u a t i o n o f h y d r o s t a t i c e q u i l i -b r i u m , \7 i ) r s ( A l . 31) i n t e q r a t e d o v e r a s u r f a c e o f c o n s t a n t p o t e n t i a l , a n d 75 P o i s s o n ' s e g u a t i o n w i t h G a u s s ' T h e o r e m i s u s e d t o t r a n s f o r m t h e i n t e g r a l o v e r t h e g r a v i t a t i o n a l p o t e n t i a l t o one o v e r t h e d e n s i -t y . (AI .32) The e g u a t i o n o f mass c o n s e r v a t i o n i s s i m p l y , 1 ( A l . 33) A n d s i m i l a r l y e n e r g y c o n s e r v a t i o n i s . d 3 , y , r / , r J ^ A r i f ( A l . 34) T o e v a l u a t e t h e m o d i f y i n g f a c t o r s , t h e f ' s , we s i m p l y n e e d t o e v a l u a t e a few i n t e g r a l s . The f i r s t i s ( A l . 35) S i n c e dH i s a v e c t o r p e r p e n d i c u l a r t o t h e c o n s t a n t p o t e n t i a l s u r f a c e , \7r a n d d ^ a r e p a r a l l e l . H e n c e d ^ - c a n be w r i t t e n K/VJ ( A l . 36) On s u r f a c e s o f c o n s t a n t p o t e n t i a l , t h e d i s t o r t i o n f a c t o r £ { r 0 ) i s c o n s t a n t , s o t h a t t h e g r a d i e n t <7r i s . 7 6 ( A I . 3 7 ) T o f i r s t o r d e r /4 e P L (AI .38) Now f i s •fl ( A I . 3 9 ) S u b s t i t u t i n g t h e v a r i o u s q u a n t i t i e s a n d i n t e g r a t i n g g i v e s I-I 3 ( A I . U O ) T h e f a c t o r f o r t h e p e r t u r b i n g p o t e n t i a l i s «. ( A I . 4 1 ) U s i n g t h e e x p r e s s i o n s d e v e l o p e d a b o v e , we h a v e 77 ( A l . 4 2) W i t h t h i s t h e e q u a t i o n o f h y d r o s t a t i c e q u i l i b r i u m i s ( A I . 43) we d e f i n e So t h e h y d r o s t a t i c e q u a t i o n i s now. ( A I . 44) ( A I . 4 5 ) T h i s use o f ^ r e s t r i c t s u s t o p o t e n t i a l s t h a t g i v e a b o u n d e d /V a t t h e o r i g i n . T h e c o n d i t i o n f o r t h e v a l i d i t y o f a l l t h e s e e x p a n s i o n s i s t h e n III *«/ ( A I . 4 6 ) T h i s \ f a c t o r i s t h e o n l y e f f e c t , t o f i r s t o r d e r , t h a t we s e e o f t h e p e r t u r b i n g f o r c e , so t h a t t h e m o d i f i c a t i o n t o a c a l -c u l a t i o n o f s t e l l a r s t r i c t u r e i n v o l v e s o n l y c o m p u t i n g a n e f f e c -t i v e mass 78 ( A I . 4 7 ) P u t t i n g t h i s a l l t o g e t h e r we h a v e , i n t h e J 2 f o r m a t I 4dP ^ (/+>) 1 7 f o ^ ( A I . 4 8 ) ( A I . 49) ( A I . 50) ( A I . 5 1 ) ( A I . 5 2 ) 79 UJ ( A I . 5 3 ) ( A I . 5 4 ) ( A I . 5 5 ) 80 APPENDIX III THE D I S T O R T I O N TERMS T h e v a r i a b l e s . e x p a n d e d a s ( A I . 2 ) a r e s u b s t i t u t e d i n t o s t e l l a r s t r u c t u r e e q u a t i o n s , a n d t h e f i r s t o r d e r t e r m s a r e e q u a -t e d . From t h e e q u a t i o n o f s t a t e we o b t a i n *f° ~ f o P0 (AII.1) T h e r a d i a l c o m p o n e n t o f t h e e q u a t i o n o f h y d r o s t a t i c e q u i l i -b r i u m g i v e s ( A l l . 2 ) And t h e t a n g e n t i a l c o m p o n e n t E q u a t i n g P 2 t e r m s i n P o i s s o n ' s e q u a t i o n ( A l l . 3) J . l_ ( r 1 ^ ) ' ^ ***** ( A l l . 4 ) We u s e t h i s s e t o f e q u a t i o n s t o o b t a i n an e q u a t i o n i n d%. a l o n e , (A I I . 5) No a t t e m p t was made t o s o l v e f o r T ^ , fi, o r P ^ , s i n c e a l l 81 t h e i n f o r m a t i o n a b o u t t h e d i s t r i b u t i o n o f T , f , a n d P i s g i v e n o n c e t h e e g u i p o t e n t i a l s u r f a c e s a r e k n o w n . T h e s e a r e d e t e r m i n e d by € , a f u n c t i o n o f *fx a n d w h i c h a r e now k n o w n . S i n c e we want t h e p o t e n t i a l t o match an e x t e r n a l s o l u t i o n o f L a p l a c e ' s e q u a t i o n w h i c h h a s a g e n e r a l s o l u t i o n 0= 1 4. P~(A) . t h i s g i v e s t h e b o u n d a r y c o n d i t i o n on T- Rjt , 4 r-( A I I . 6) ( A l l . 7) ( A l l . 8) a t t h e c e n t r e a l l f o r c e s v a n i s h , so (6l{0)=0, a n d / 2 £ ' ( 0 ) = 0 82 APPENDIX I I I l THE LABSOH-DEMAHO.UE EQUATIONS T h e L a r s o n - D e m a r q u e (47) v a r i a b l e s a r e : {kill. 1) L $ ( A I I I . 2) ( A I I I . 3) ( A I I I . 4) T h e d e n s i t y i s e l i m i n a t e d f r o m a l l e q u a t i o n s by u s i n g t h e e q u a t i o n o f s t a t e , ( A I I I . 5 ) w i t h ( A I I I . f r ) 83 The mass f u n c t i o n , d e f i n e d a s . ( A I I I . 7) f o r u p p e r main s e q u e n c e s t a r s w i t h r a d a t i v e s u r f a c e , r e d u c e s f r o m L D ' s g e n e r a l e x p r e s s i o n t o 3 fU)-- [ I - (i--x>'] . The o p a c i t y was c a l c u l a t e d f r o m w k e r « "Kc. - . / ? * ( A I I I . 8) ( A I I I . 9 ) Q u a n t i t i e s u s e d i n t h e c a l c u l a t i o n a r e "6 4.o* 7<t \ - J - * , ( A I I I . 1 0 ) 84 T h e e n e r g y g e n e r a t i o n r a t e i s g i v e n by where e = £x. -fix' f 1 -" * > << •= /-7* */o< /X ) 2 e ' f 4 v/ -t„ -A, - 1 + .02/4 - t ' 2 / i 3" -•• - ' > • osrs 3 j ^ t ' , f i ( A I I I .11) y7 ^ / r - t f 3 1 - ^ ( A I I I . 1 2 ) The mean m o l e c u l a r w e i g h t i s d e f i n e d a s 85 ( A I I I . 1 3 ) W i t h t h e s e s u b s t i t u t i o n s , t h e s p h e r i c a l p a r t c f t h e e q u a -t i o n s o f s t e l l a r s t r u c t u r e become ( A I I I . 14) ^k'd^ 7—. — , 3 L*- L o j ( A I I I . 15) ( A I I I . 1 6 ) ( A I I I . 1 7 ) ( A I I I . 1 8 ) 86 ( A I I I . 19) ( A I I I . 2 0 ) (A I I I .21) ( A I I I . 2 2 ) T h e 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 a r e t a k e n and t h e d i f -f e r e n c e c o e f f i c i e n t s a r e f o r m e d i n t h e same m a n n e r a s L D . The b o u n d a r y c o n d i t i o n s a t t h e c e n t r e a r e as i n L D , w i t h a few m i n o r m o d i f i c a t i o n s . F o r s m a l l x S i n c e l i m s=0 , a n d l i m q = 0 , a t t h e c e n t r e , n« •- Pit -A. J h l I''1 (A I I I . 2 3 ) ( A I I I .24) 87 I ^ ( A I I I . 2 5 ) 0 0 , = { vo, ( A I I I . 2 6 ) Dv/, , 2 0Q, ( A I I I . 27) ( A I I I . 28) D P, ( A I I I . 2 9) F o r a r a d i a t i v e s u r f a c e t h e r a d i a t i v e z e r o (T-K), P-*0 at s u r f a c e ) b o u n d a r y c o n d i t i o n s a r e u s e d b o u n d a r y c o n d i t i o n s . A s shown by S c h w a r z s c h i l d ( 9 8 ) , t h e s e a r e a p p r o p r i a t e . T h e i n c l u s i o n o f r a d i a t i o n p r e s s u r e i n t r o d u c e s seme m i n o r p r o b l e m s . The e f f e c t i v e p o l y t r o p i c i n d e x i s 88 p1 * T ( A I I I . 30) I n t h e s u r f a c e l a y e r s o f a s t a r i n r a d i a t i v e e q u i l i b r i u m •>• T r . S u b s t i t u t i n g t h i s i n t o t h e a b o v e e x p r e s s i o n g i v e s f o r t h e s u r f a c e v a l u e o f (n+1) )«u ~ p * * v (i- fi)-( A I I I . 3 1 ) T h e u s e d i n t h i s e x p r e s s i o n was s e t e q u a l t o t h e (3^ o f t h e p r e v i o u s l a y e r . T h e o u t e r b o u n d a r y c o n d i t i o n s t h e n a r e DS = P ' D T = 0*0 H*? (»+0 ( A I I I . 3 2 ) cr ( A I I I . 33) 89 D P - 0 , ( A I I I . 34) Du =.o, ' ( A I I I . 3 5 ) D V - ITT ' ( A I I I . 3 6 ) P Q - - P V ( A I I I . 3 7 ) 90 APPENDIX IH I S I GREY ATMOSPHERE T h e r e a r e two b a s i c e q u a t i o n s f o r a g r e y a t m o s p h e r e . R a -d i a t i v e t r a n s f e r ( I V . 1) w h e r e : i s c»*i-* e o-t **\e «>vj |e t t * * ) t e n t k e l i » e s t k e P l a n c k On , •< 3 T h e a b s o l u t e f l u x d e t e c t e d b y the o b s e r v e r i s , L ( I V . 2) where i i s t h e a n g l e b e t w e e n t h e l i n e o f s i g h t a n d t h e a x i s o f s y m m e t r y o f t h e s t a r , a n d t h e i n t e g r a l i s d o n e o v e r t h e o b s e r -v a b l e s u r f a c e . T h e d i a g r a m b e l o w e x p l a i n s t h e g e o m e t r y 9 1 T h e t e m p e r a t u r e d i s t r i b u t i o n T e ( 6 ) i s f o u n d f r o m V o n Z e i p e l ' s T h e o r e m , whch s t a t e s F < 13/, ( I V . 3 ) where g i s t h e l o c a l e f f e c t i v e a c c e l e r a t i o n . S i n c e F= T e S and where t h e t o t a l p o t e n t i a l i s g i v e n b y , T ( I V . 4) I n s p h e r i c a l c o o r d i n a t e s we h a v e 92 '3> (IV. 5) The surface of the star i s given by (IV. 6) C o n s e q u e n t l y t h e g r a v i t y t c f i r s t o r d e r i s r 0 ( l u l l ) ' (IV.7) T h e e f f e c t i v e t e m p e r a t u r e a v e r a g e i s d e f i n e d a s T.v e (IV.8) So ue have for the temperature d i s t r i b u t i o n . ( IV . 9) T o d e t e r m i n e , t h e a n g l e b e t w e e n t h e o b s e r v e r ' s l i n e o f s i g h t a n d t h e n o r m a l t o t h e s u r f a c e , we n o t e t h a t t h e s u r f a c e i s g i v e n t o f i r s t o r d e r b y . 9 3 ^ / +€. P,. ( I V . 1 0 ) T h e n o r m a l t o t h e s u r f a c e i s g i v e n by !\7r B| ( I V . 1 1 ) C a l c u l a t i n g t h i s o u t we f i n d V ^ , + fiP, A 1 i P. 1 ( I V . 1 2 ) a n d \ 1 / +€ R. ( I V . 1 3 ) T h e r e f o r e t h e n o r m a l i s ( I V . 1 4 ) Now yf< c a n be d e t e r m i n e d s i m p l y f r o m t h e d o t p r o d u c t b e -94 tween t h e v e c t o r t o t h e l i n e o f s i g h t a n d t h e n o r m a l v e c t o r . T h e v e c t o r t o t h e o b s e r v e r i s 0 ( I V . 1 5 ) s o we h a v e ( I V . 1 6 ) M o t e t h a t yiA i s an e v e n f u n c t i o n i n ft, Now t o i n t e g r a t e t h i s i n t e n s i t y o v e r t h e v i s i b l e p a r t o f t h e s t a r s s u r f a c e . The f l u x i s t h e n g i v e n by ( I V . 17) T h e n u m e r i c a l s o l u t i o n s a r e b a s e d on a number o f q u a d r a t u r e f o r m u l a s f o r d o i n g t h e i n t e g r a t i o n s . O v e r 6 l e g e n d r e - G a u s s i s u s e d , o v e r ^ C h e b y s h e v - G a u s s , a n d o v e r "C L a g u e r r e - G a u s s . The a b o v e f o r m u l a f o r t h e f l u x b e c o m e s ( I V . 1 8 ) where Ui ' Z « ' » * S <rf Py*, i k e U pgly>o *\a ( irf 95 S i m i l a r l y t h e i n t e n s i t y i s g i v e n by / ( I V . 19) where Yi=T* Z e ' B * * of L/j l V e , f e polj^o *f I t i s n e c e s s a r y t o make a s i m p l e t r a n s f o r m a t i o n b e t w e e n t h e c o o r d i n a t e s i n t h e o b s e r v e r ' s s y s t e m t o one i n t h e s t a r s s y s t e m . T h e s e a r e ( p r i m e d g u a t i t i e s a r e i n s t a r ' s s y s t e m ) C O S . 0 ' = C o « © ^ + J^^** 0 t o c f ( c o s < . ^ j ' - • ( I V . 2 0 ) ttoty & t » t -M a g n i t u d e s a r e c o m p u t e d f r o m ( I V . 2 1 ) ( I V . 2 2 ) 96 T h e s e m a g n i t u d e s were t a k e n a t r o u g h l y t h e c e n t r e s c f t h e V a n d B bands (V=5530A a n d B=4350A a b o u t ) . The V and . ( E - V ) i n d i c e s we're t h e n c a l c u l a t e d t o s e e what t h e o b s e r v a t i o n a l e f f e c t s of t h e m a g n e t i c f i e l d m i g h t b e . 97 APPENDIX V i COMPUTER PROGRAMS T h e f i r s t p r o g r a m l i s t e d i s t h e c a l c u l a t i o n o f s t e l l a r s t r u c t u r e . As i n p u t i t r e q u i r e s a n i n i t i a l m o d e l on u n i t 3 and t h e s p e c i f i c a t i o n o f c e r t a i n NAMEL 1ST v a r i a b l e s : MO t h e mass o f t h e m o d e l , X1 , Z1 t h e . a b u n d a n c e s o f h y d r o g e n a n d m e t a l s r e s p e c t i v e l y , E R C P , e t c . T h e maximum a l l o w e d r e l a t i v e c h a n g e i n P , T , S a n d Q f r o m o n e m o d e l t o t h e n e x t , HO t h e m a g n e t i c f i e l d i n g a u s s . T h e r e a r e a l s o a v a r i e t y o f p a r a m e t e r s t o c e n t r e ! t h e amount o f o u t p u t , t h e c a l c u l a t i o n o f c e r t a i n q u a n t i t i e s , and a f e w v a r i a b l e p a r a m e t e r s i n t h e s t e l l a r m o d e l . A l l o f t h e s e h a v e d e f a u l t v a l u e s . T h e m a g n e t i c f i e l d i s f o u n d by c a l l i n g t h e s u b r o u t i n e B F I E L D . T h e main p r o g r a m a l s o r e q u i r e s : STEMP t o c a l c u l a t e e f f e c t i v e s u r f a c e t e m p e r a t u r e s , ENGEN c a l c u l a t e s t h e e n e r g y g e n e r a t i o n r a t e , SMAX f i n d s t h e . m a x i m u m o f a r a t i o , L A G I N S i s a . L a g r a n g i a n i n t e r p o l a t i o n , s u p p l i e d by D r . . A . J . B a r n a r d , FLUXPO c a l c u l a t e s t h e f l u x f o r a p o l o i d a l f i e l d , TANB c a l c u l a t e s t h e t a n g e n t i a l c o m p o n e n t o f t h e m a g n e t i c f i e l d a t a n y p o i n t i n t h e e g u a t o r i a l p l a n e , KAPPA i s an o p t i o n a l o p a c i t y t a b l e l o o k up (not u s e d ) , ENPOLE c a l c u l a t e s t h e f i e l d e n e r g y f o r a p o l o i d a l m a g n e t i c f i e l d , FH c a l c u l a t e s t h e m a g n e t i c e n e r g y i n a mass s h e l l , FG c a l c u l a t e s t h e g r a v i t a t i o n a l e n e r g y i n a mass s h e l l . 98 N e x t t h e - s u b r o u t i n e COLOUR c a l c u l a t e s t h e g r e y a t m o s p h e r e . I t i s c a l l e d f r o m STEMP a n d r e q u i r e s i n p u t f r o m t h e i n i t i a l N A M E L I S T s t a t e m e n t o f t h e number o f i n c l i n a t i o n a n g l e s t o be a v a l u a t e d and t h e i n i t i a l B and Y i n d i c e s . A l l i n t e g r a l s a r e a p p r o x i m a t e d by s i x t e e n t h o r d e r p o l y m n o m i a l s . T h e r e a r e t h r e e v e r s i o n s o f t h e s u b r o u t i n e B F I E L D . The f i r s t c a l c u l a t e s , t h e f i e l d f o r t h e f l u x p e n e t r a t i n g t h e c o r e , t h e s e c o n d f o r t h e f l u x e x c l u d e d f r o m t h e c o r e , a n d t h e l a s t t h e f l u x p e n e t r a t e s t h e c o r e a n d t h e m o d e l i s i n u n i f o r m r o t a t i o n . A l l o f t h e s e u s e a . R u n g e - K u t t a r o u t i n e p r o v i d e d by t h e UBC C o m p u t i n g C e n t r e t o f i n d a p a r t i c u l a r s o l u t i o n , t o t h e p s e u d o -p o l y t r o p i c e q u a t i o n f o r t h e m a g n e t i c f i e l d . T h i s r o u t i n e DRK r e g u i r e s a n a u x i l i a r y r o u t i n e AUXRK t o s p e c i f y t h e d i f f e r e n t i a l a q u a t i o n . T h e B F I E L D . f o r t h e r o t a t i n g m a g n e t i c . s t a r c a l l s EROTAT and FROT t o c a l c u l a t e t h e e n e r g y o f r o t a t i o n . 99 S C STAR (1 ,767 ) TO * S I N K * 9 N 0 C C C T H I S IS A MODEL OF A MAGNETIC STAR C NANG=N UMBER OF I N C L I N A T I O N ANGLES COLOUR IS TO E V A L U A T E C COL (162) ARE THE B AND V I N D I C E S OF I N I T I A L STAR , I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION AX (100) , AS (100) , AQ (100) , A P (100) , AT (100) , ABETR (100) ,ARHO ( 1 100) , A N 1 (100) , A F X (100) , A L A M (100) R E A L * 8 MAG (100) , DMAG ( 1 0 0 ) , P O T 2 (100) DIMENSION SOLY(9) R E A L * 8 KAPPA (100) DIMENSION A ( 8 , 4 , 100) , B (4 , 100) , A U X (6400) , B A D X (400) , I P (400) DIMENSION I P A S S (5) , P A S S (5) , C O L (2) R E A L * 4 OP (30, 10) INTEGER T I T L E (20) R E A L * 8 K A P S , K A P T , K A P , L 1 , H 0 , L O G L O , L O G T O R E A L * 4 T M E , S C L O C K C OUTPUT=T FOR F I N A L MODEL OF I T E R A T I O N SEQUENCE ON UNIT 7 C OLDMOD=T I F REREAD I N I T I A L MODEL FOR NEW S E R I E S OF I T E R A T I O N S C INTER=TRUE FOR DUMP OF D E T A I L E D MODEL PARAMETERS ONTO UNIT 0 C DUMP=T TO OUTPUT A MODEL WHICH DID NOT CONVERGE L O G I C A L C O N V G , M A G O N , Q U I T , O U T P U T , O L D M O D , I N T E R , S H O R T , D U M P , P R E V L O G I C A L SECOND,MA G O U T , C U R F I T COMMON / D A T / SECOND COMMON / M A I N B / MAG ,DMAG , A L A M , S O L Y , H O , MO , OMEGA, I C V C T 1 , N M A S S , P A S S , I P A S S , S H O R T , M A G O U T , I C O N U M , I C V G 2, M C V C T , C U R F I T COMMON / M A I N R K / A N 1 , A B E T R , A T COMMON / A L L / A S , A R H O , N P COMMON / M A S S / AFX COMMON / C M A I N / C O L , S I N I , N A N G COMMON / E T C / W T , Q U I T C MASS F U C T I O N S FM(Y)= (1. D O - ( 1 . DO-Y) * * S I G ) **3 FP(Y) = 3 . D 0 * S I G * ((1 . D O - (1 . D O - Y ) * * S I G ) **2) * ( 1 . DO-Y) * * ( S I G - 1 . DO) N A M E L I S T / P A R A M / M O , L 1 , A L , X 1 , Z 1 , N P , E R C P , E R C T , E R C S , E R C Q , I C M A X 1 , A L P H A , B E T A , H 0 , M A G O N , P C O N , S U R F C , O U T P U T , C O N V G , N M A S S , 2 P A S S , I C O N U M , I P A S S , O L D M O D , I N T E R , S H O R T , D U M P , M A G O U T 3 , N A N G , O M E G A , C O L , M C V C T , S I N I , C U R F I T , L O G L O , L O G T O C OPACITY T A B L E ON UNIT 1 C REWIND 1 C R E A D ( 1 , 5 0 0 1 ) NOP C5001 FORMAT (12) C READ (1 ,5000) ( ( O P ( I , J ) J=1 , 10) , 1 = 1 , N O P ) C5000 F O R M A T ( E 8 . 3 , F 8 . 0 , 8 E 8 . 3 ) REWIND 7 O L D M O D = . F A L S E . S E C O N D = . F A L S E . 1001 CONTINUE REWIND 3 DO 67 1=1 ,5 PASS ( I )=0 . DO 67 I P A S S ( I ) =0 C INPUT PARAMETERS AND COMPUTE CONSTANTS A L = 1 . D - 2 NP=81 E R C Q = 5 . D - 3 i ERCT=0.DO ERCP=O.DO ERCS=O.DO PCON=10.DO P R E V = . F A L S E . D O H P = . F A L S E . SHORT=. F A L S E . . C O S V G = . T R U E . M A G O U T = . F A L S E . H0=1.D5 NMASS=0 SIB 1=0.DO Q M E G A = 1 . 1 1 1 1 1 1 1 D - U L O G L 0 = 0 . D 0 LOGT0=0 .D0 MCVCT=0 NANG=5 CUR F I T = . F A L S E . I N T E R = . F A L S E . ICMAX=5 OUTPUT=. F A L S E . C O M V G = . F A L S E . MAGON=. T R U E . S U R F C = 1 . D - 5 C ICONUM=# OF TIMES B F I E L D C A L L E D FOR C A L C U L A T I N G S T R U C T U R E . C HOTE THAT B F I E L D C A L L E D ONCE MORE FOR F I N A L OUTPUT ICONUM=1 READ (3 ,PARAM) READ (3 ,3000) (AX (I) , A S (I) , AQ (I) , AP (I) , AT (I) ,1=1 ,NP) WRITE (6 , P ARAM) I F (OLDMOD) GO TO 1002 C C I N I T I A L I Z A T I O N C 1000 CONTINUE READ ( 5 , 2 3 0 5 , E N D = 9 9 9 ) T I T L E 2305 FORMAT (20A4) Q U I T = . F A L S E . NT= 1 READ (5 ,PARAM) IF(OLDMOD) GO TO 1001 1002 O L D M O D = . F A L S E . NP1=NP-1 ML=5 NU=5 LCM=2*ML+NU LC=LCM+1 LK=ML+NU+1-LC L T = 4 * N P 1 * L C I F ( E R C T . E Q . O . D O ) ERCT=ERCQ I F ( E R C P . E Q . Q . D O ) ERCP=ERCQ I F ( E R C S . E Q . O . D O ) ERCS=ERCQ ETA=ALPHA T H E T A = B E T A - 1 . 5 D 0 * A L P H A C FROM SACKMANN-ANAND RKO = . 2 4 5 D 0 * ((1 . D 0 + X1) / ( 2 . D 0 - Z 1 ) ) * * . 6 7 D 0 KAPS = . 1 9 5 D 0 * ( 1 . D0+X1) SIG= (4 .D0+ALPHA + BETA) / (1 .DO + ALPHA) I F ( S H O R T ) GO TO 71 WRITE (6 ,2200) W R I T E ( 6 , 2 2 0 3 ) WRITE (6 ,2204) (AX (I) , AS (I) , AQ (I) , A P (I) , AT (I) , 1 = 1, NP) 2203 FORMAT (' 1 • , 1 2 X , » X • , 2 0 X , • S • , 2 0 X , 1 Q ' , 2 0 X , » P ' , 2 0 X , « T • / ) 2204 FORMAT ( 1 X , 5 F 2 1 . 8 ) 71 P A L T = P C O N / 1 0 . D 0 C O N A = . 0 1 2 2 5 3 9 2 5 6 7 D 0 * M 0 * P A L T C O N B = . 0 2 8 2 8 9 0 7 2 0 4 D 0 * M 0 * M 0 * P A L T C O N U = 4 1 . 5 0 2 5 5 3 9 3 D 0 * M 0 / L 1 * P A L T C O N R P = 7 . 9 7 4 1 5 7 4 6 1 D - 4 * P A L T C O N E = . 5 1 7 9 6 8 7 5 0 0 * M 0 / L 1 Y 1 = 1 . D O - (X1+Z1) WT= 1. D O / (2. D 0 * X U . 7 5 D 0 * Y 1 + . 5 D 0 * Z 1 ) C O N R H O = 3 8 . 3 1 0 8 9 7 4 2 / P A L T * W T C I N I T I A L L O G I C ICVG=0 IF (CONVG) ICVG=1 I F ( P R E V . A N D . M A G O N ) GO TO 900 ICffCT=101 DO 66 1=1,100 ALAM (I) =0. DO ARHO (I) =0 .DO ABETR (I) = 1. DO 66 A N 1 ( I ) = 0 . D 0 ITOT=0 1 CONTINUE ITOT=ITOT+1 ICOUNT=1 C c c C SQPT SQPT SQPT C c c 2 CONTINUE DO 10 1=1,NP DO 10 J=1 ,4 DO 10 K=1,8 10 A (K , J , I) =0 . DO C C CENTRAL BOUNDARY CONDITIONS ASSUMING CONVECTION C 1=1 S 3 = S I G * S I G * S I G X=AX(I) S=AS (I) Q=AQ(I) P=AP (I) T=AT(I ) T 3 = T * * . 3 3 3 3 3 3 3 3 3 3 333333 T32=T*DSQRT (T) KAPPA (1) =0. DO B E T R = 1 , D 0 - C O N R P * T 3 2 / P ABETR (1) =BETR RHO=CONRHO*BETR*P*T32 ARHO(1)=RHO AN 1 (I) = 4 . D 0 - 1 . 5 D 0 * B E T R * B E T R / (4 . DO-3 . DO * B E T R ) A F X ( I ) = 0 . D O D S = ( C O N A * 3 . D 0 / ( W T * B E T R * P * T 3 2 ) ) * * . 3 3 3 3 3 3 3 3 3 3 333333 DS=SIG*DS C A L L E N G E N ( P , T , E P , E C , X 1 , Z 1 , W T , R H O , T 3 , T 3 2 ) 102 E=EP+EC DUK=CONE*AL*S3 D U P = 3 . D 0 * E P * D U K DUC=3. D 0 * E C * * D U K DQ=DUK*E DV= 2 . D 0 * D Q DC=0.D0 DR=0.D0 DU=DUP+DUC DT=0.D0 DP=0.D0 I F ( S H O R T ) GO TO 72 WRITE (6 ,4001 ) WRITE (6 ,4002 ) 4001 F O R M A T ( ' 1 ' , 1 0 X , ' E N E R G Y GENERATION C H E C K ' / / ) 4 002 FORMAT ( 5 X , • E P ' , 1 0 X , « E C ' , 1 0 X , • D S • , 8 X , • D Q ' , 8 X , » D P • , 8 X , ' D T • , 7 X, 'ENR' 1 , 7 X , ' E N C , 7 X , »KAP » , 6 X , ' BETR * , 5 X , » A L P H A ' , 6 X , ' B E T A ' ) WRITE (6 ,400 3) E P , E C , D S , D Q , D P , D T 72 A ( 3 , 1,1) = . 5 D 0 * D S / ( P * B E T R ) A ( 4 , 1 , 1 ) = . 7 5 D 0 * D S / T * { ( 2 . D 0 * B E T R - 1 . DO) / B E T R ) A ( 3 , 2 , I ) = - . 5 D 0 * D U / ( P * B E T R ) BTF=1 . 5D0* (1 . DO-BETR) / B E T R TRH 1 = . 8 3 3 3 3 3 3 3 3 3333333+5 .231 D O / T 3 - B T F TRM2=.8333333 3333 3 3 3 3 3 + 2 3 . 5 6 6 D 0 / T 3 - B T F A ( 4 , 2 , 1 ) =- . 5 D 0 * ( D U P * T R M 1 + D U C * T R M 2 ) / T DX= 1. D O / ( A X (1+1)-AX (I) ) A ( 3 , 3 , 1 ) =-DX A ( 4 , 4 , 1 ) =-DX SL=S QL=Q PL=P TL=T DXL=DX DSL=DS DQL=DQ DPL=DP DTL=DT C C S H E L L S BETWEEN CENTRE AND S U R F A C E C NP1=NP-1 DO 100 1=2,NP1 X=A X (I) X2=X*X S=AS(I ) S2=S*S S4=S2*S2 Q=AQ(I) P=AP (I) T=AT (I) T 3 = T * * . 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 D 0 T32 = T * D S Q R T (T) B E T R = 1 . D 0 - C O N R P * T 3 2 / P ABETR (I) =BETR RH0=C0NRHO*BETR*P*T32 ARHO (I) = RHO AFX (I) =FM (X) FPE=FP(X) AM A SS= (1 .DO+ALAM (I) ) * A F X (I) D S = C O N A * F P E / ( W T * S 2 * P * T 3 2 * B E T R ) EC=O.DO EP=O.DO I F (T . L E . .5D0) GO TO 19 C A L L ENGEN (P, T , E P , E C , X 1, Z 1, WT, RHO, T 3 , T32) 19 CONTINUE E=EP+EC D U E = C O N E * F P E * ( 1 . D 0 + A L / X 2 ) DUP=DUE*EP DUC=DUE*EC DU=DUP + DUC D V = 2 . D 0 * A L * Q / ( (X2 + AL) *X) DQ=DU-DV C OPACITY USING KRAMERS FORMULA KAPT= RKO*(RHO**AL PHA/T**BETA) KAP=KAPS+KAPT KAPPA (I) =KAP C O P A C I T Y USING T A B L E LOOK UP C I F ( T . L E . 2 . D - 3 ) G O TO 68 C C A L L K A P P A ( O P , R H O , T , K A P , K A P 0 , A L P H A , B E T A , N T ) C EIA=ALPHA C T H E T A = B E T A - 1 . 5 D 0 * A L P H A C K A P T = K A P - . 2004 DO* (1 . DO +X 1) C SIG= (4 .DO+ALPHA+BETA) / (1 . DO+ALPHA) C RKO=KAP0 C68 CONTINUE E N R = C O N U * A M A S S * T 3 2 * ( 1 . D 0 + A L / X 2 ) / ( K A P * Q * P ) ENC = 4 . D 0 - 1 . 5 D 0 * B E T R * B E T R / (4 . DO-3 . DO *BETR) D T B = - C O N B * A M A S S * F P E / ( S 4 * P * T 3 2 ) I F ( E N C . G T . E N R ) GO TO 18 DT= D T B / E N R DR=DT EN1 = ENR I F ( I C V C T . G T . I ) I C V C T = I GO TO 17 18 DT=DTB/ENC DC= DT EN 1 =ENC 17 AN1 (I)=EN1 D P = P / T * D T * ( E N 1 - 2 . 5 D 0 ) I F ( S H O R T ) GO TO 73 W R I T E ( 6 , 4 0 0 3 ) E P , E C , D S , D Q , D P , D T , E N R , E N C , K A P , B E T R , A L P H A , B E T A 4003 FORMAT (1X, 2 D 1 1 . 3, 4D10 . 3 , 6 F 1 0 . 4) C C C THE D I F F E R E N C E EQUATION C O E F F I C I E N T S 73 D X = 1 . D 0 / ( A X (1+1)-AX (I) ) IM = I - 1 K=4 11 CONTINUE A ( K + 1 , 1 , I M ) = D X L + D S / S A ( K + 2 , 1 , I M ) = 0 . D 0 A (K + 3 ,1 , I M ) = . 5 D 0 * D S / ( P * B E T R ) A (K + 4 , 1 , I M ) = . 7 5 D 0 * D S / T * (2. D 0 * B E T R - 1 . DO) / B E I R A ( K + 1 , 2 , I M ) =0. DO A (K + 2 , 2 , I M ) = D X L + . 5 D 0 * D V / Q A ( K + 3 , 2 , I M ) = - . 5 D 0 * D U / ( P * B E T R ) B T F = 1 . 5 D 0 * ( 1 . D O - B E T R ) / B E T R T R M 1 = . 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 D 0 + 5 . 2 3 1 D 0 / T 3 - B T F T R M 2 = . 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 D 0 + 2 3 . 5 6 6 D 0 / T 3 - B T F A (K + 4 , 2 , I M ) = - . 5 D 0 * ( D U P * T R M l + D O C * T R M 2 ) / T A (K + 1 , 3 , I M ) = 2 . D 0 * D P / S 1 0 I F ( E N R - E N C ) 2 0 , 2 0 , 2 1 20 A (K + 2 , 3 , I M ) =0 .D0 DBR= 2 . D O * ( E N 1 - 4 . D O ) * ( E N 1 - 4 . D O - B E T R ) * ( 1 . D O - B E T R ) / ( B E T R * B E T R * EN 1 ) A(K + 3 , 3 , I H ) = D X L + 1 . 2 5 D 0 * D T * D B R / T A ( K + 4 , 3 , I M ) = 1 . 2 5 D 0 * D P / T - 1 . 8 7 5 D 0 * P * D T * D B R / (T*T) GO TO 22 21 A (K + 2 , 3 , I M ) = 1 . 2 5 D 0 * P * D R / ( T * Q ) T K A P 1 = E T A * K A P T / K A P / B E T R A (K + 3 , 3 , IM)=DXL + 1 . 2 5 D 0 * D R * (1 . DO+TKAP 1 ) / T T K A P 2 = K A P T / K A P * ( T H E T A * E T A * B T F ) A(K + 4 , 3 , I M ) = 1 . 2 5 D 0 * D P / T - 1 . 2 5 D 0 * D R * P * (1 .5 +TKAP2) / ( T * T ) 22 CONTINUE A ( K + 1 , 4 , I M ) = 2 . D 0 * D T / S I F (ENR-ENC) 3 0 , 3 0 , 3 1 30 A (K + 2 , 4 , I M ) = 0 . D 0 A (K + 3 , 4 , IM) =. 5 D 0 * D C / P * ( 1 . D 0 - D B R ) A(K + 4 , 4 , I M ) = D X L + . 7 5 D 0 * D C / T * (1 .D0 + DBR) GO TO 32 31 A ( K + 2 , 4 , I M ) = - . 5 D 0 * D R / Q A (K + 3 , 4 , I M ) = - . 5 D 0 * D R * T K A P 1 / P A (K + 4 , 4 , I M ) = D X L + . 5 D 0 * D R * (3 . DO + T KAP2 ) / T 32 CONTINUE I F ( I M . E Q . I ) GO TO 40 IM=I K=0 DXP=DXL DXL=-DX GO TO 11 40 CONTINUE IM=I-1 C R I G H T HAND S I D E S B(1 , I M ) = - D X P * ( S - S L ) +. 5 D 0 * ( D S + DSL) B ( 2 , I M ) = - D X P * ( Q - Q L ) + . 5 D 0 * (DQ+DQL) B(3 , IM) = - D X P * ( P - P L ) +. 5 D 0 * (DP+DPL) B ( 4 , I M ) = - D X P * ( T - T L ) + . 5 D 0 * (DT+DTL) SL=S QL=Q PL=P TL=T DXL=DX DSL=DS DQL=DQ DPL=DP DTL=DT 100 CONTINUE C C BOUNDARY CONDITIONS FOR A R A D I A T I V E SURFACE C I=NP X=AX(I) S=AS (I) S2=S*S S4=S2*S2 Q=AQ(I) P=AP (I) T=AT(I ) K A P P A ( N P ) = 0 . DO ABETR (I) =BETR ARHO (I) =0. DO A F X ( I ) = 1 . D 0 EN1 = B E T R * S I G + 4 . DO* (1 . DO - B E T R ) ENR=EN1 AN1 (I)=EN1 AMASS=1 .DO+ALAM (I) CONK=RKO*(WT* BETR*CONR H O ) * * A L P H A CQ=CONU* (1 .DO+AL) / C O N K POW = 1. DO / (1 . DO + ALPHA) P 2 = 1 . D 0 / S I G DT=- ( 3 . D 0 * S I G * C O N B * A M A S S / ( E N 1 * ( A M A S S * C Q / ( E N 1*Q) ) * *POW*S4) ) * *P2 DS=- . 4 3 3 1 6 8 1 7 3 7 / M 0 * E N 1 * S 2 * D T / (WT*AM A S S * BETR) DP=O.DO DUP = O.DO DUC=0.D0 DU=O.DO DV=2. D 0 * A L * Q / ( 1 . DO+AL) DQ=-DV I F (SHORT) GO TO 7 4 W R I T E ( 6 , 4 0 0 3 ) E P , E C , D S , D Q , D P , D T , E U R , E N C , K A P T , B E T R 74 A ( 5 , 1 , N P 1 ) = D X L + D S / S A ( 6 , 1 ,NP1) =0 .D0 A (5 , 2 , H P 1 ) =0. DO A ( 6 , 2 , N P 1 ) = D X L + . 5 D 0 * D V / Q A ( 5 , 3 , N P 1 ) = 2 . D 0 * D P / S A ( 6 , 3 , N P 1 ) =0 .D0 A (5 , 4 , N P 1 ) =2. D 0 * D T / S A ( 6 , 4 , N P 1 ) = - . 5 D 0 * D T / Q B (1 , NP1) = - D X L * ( S - S L ) +. 5 D 0 * (DS + DSL) B (2 ,NP1) = - D X L * (Q-QL) + . 5 D 0 * (DQ+DQL) B (3 ,NP1) = - D X L * ( P - P L ) +. 5 D 0 * (DP+DPL) B ( 4 , N P 1 ) = - D X L * ( T - T L ) + . 5 D 0 * (DT + DTL) C C S O L V E D I F F E R E N C E EQUATIONS FOR CORRECTIONS C DO 150 L = 1 , L T 150 A O X ( L ) = 0 . D 0 DO 200 1=1,NP1 DO 200 J=1 ,4 NR=4*I+J-4 BAUX (NR) =B ( J , I) KS=1 KF=8 I F ( I . E Q . 1 ) KS=3 I F ( I . E Q . N P 1 ) KF=6 DO 200 K = K S , K F NC=4*I+K-6 KK=NC*LC+NR-NC+LK AUX (KK) = A (K, J , I) 200 CONTINUE NEQ=4*NP1 C TME = SCLOCK (0 .0 ) C A L L D G B A N D ( A U X , B A U X , N E Q , M L , N U , 1 , I P , D E T , N C N ) C T « E = S C L O C K ( T H E ) I F ( D E T . N E . 0 .DO) GO TO 250 WRITE ( 6 , 2 3 0 0 ) 2300 FORMAT ( ' 1 ' , 1 0 X , ' D E T E R M I N A N T IS Z E R O ' ) GO TO 1000 250 CONTINUE DO 340 1=1,NP1 AS (I + 1) = AS (1+1)+BAUX (4*1-1) 1 0 6 AQ (1 + 1) =AQ (1+1) +BAUX (4*1) A P ( I ) = A P (I)+BAUX (4*1-3) AT (I) =AT (I) +BAUX(4*1 -2) 340 CONTINUE C C C OUTPUT OUTPUT OUTPUT I F ( S H O B T ) GO TO 82 83 WRITE (6 ,2306 ) T I T L E 2306 FORMAT ( ' 1 ' , 9 X , 2 0 A 4 / / ) WRITE ( 6 , 2 0 0 0 ) MO 2000 FORMAT (10X, T H I S IS A ' , F 7 . 2 , ' SOLAR MASS M O D E L ' / / ) RLUM=L 1*AQ ( N P ) / ( 1 .DO+AL) WRITE (6 ,2001) BLUM 2001 FORMAT ( 1 0 X , ' T H E LUMINOSITY OF T H I S MODEL I S ' , F 1 0 . 4 , ' SOLAR U N I T S ' / V ) WRITE (6 ,2010) A L P H A / B E T A , S I G 2010 FORMAT(1 O X , ' K R A M E R S O P A C I T Y : ALPHA= * , F 8 . 4 , • , B E T A = • , F 8 . 4 , • , SIGM 1 A = » , F 8 . 4 / / ) TCNT=AT(1) *1 . D 7 PCNT = A P ( 1 ) * (TCNT) * * 2 . 5 D 0 / P C O N W R I T E ( 6 , 2 0 0 8 ) AS(NP) , TCNT , PC NT , ARHO (1) 2008 FORMAT ( 1 0 X , ' R = » , F 1 0 . 8 , 1 0 X , ' T C E N T R E = • , F 1 0 . 0 , 1 0 X , * P C E N T R E = ' , D 1 3 . 6 , 1 1 0 X , ' R H O C E N T R E = * , F 1 0 . 7 / / ) C WRITE (6 ,2004) TME 2004 F O R M A T ( 1 0 X , ' E X E C U T I O N TIME FOR S O L U T I O N OF MATRIX I S ' , F 1 0 . 4 / / ) T0= ( R L U M * 1 . 1 1 4 6 D 4 - 1 5 / (AS (NP) * A S (HP) ) ) * * . 25D0 WRITE (6 ,3500) TO 3500 FORMAT ( 1 0 X , ' T H E E F F E C T I V E S U R F A C E TEMPERATURE I S » , F 9 . 2 , ' D E G R E E S ' V / / ) HRT=DLOG10(TO) HLOGL=DLOG10(RLUM) H R L = 4 . 7 3 4 D 0 - 2 . 5 D 0 * H L O G L WRITE (6 ,2206) H R L , H R T , H L O G L 2206 F O R M A T ( 1 0 X , »M BOL 0= ' , F1 0 . 6 , 5 X , • LOG (T E F F ) = • , F10 . 6 , $ 10X, « L O G ( L ) = ' , F 1 0 . 7 / / / ) I F ( L O G L 0 . E Q . 0 . D 0 ) GO TO 2408 D L O G L - H L O G L - L O G L O DL0GT=HR T - L O G TO WRITE (6^2409) D L O G L , D L O G T 2409 FORMAT (1 O X , ' C H A N G E I N LOG (L) • , D1 2 . 5 , 10X , ' CH ANGE IN LOG (T E F F ) « , D 1 2 . 2408 CBV = COL ( 1 ) - C O L (2) WRITE (6 ,2207) COL (2) , C B V 2207 F O R M A T ( 1 0 X , ' T H E COLOURS OF THE I N I T I A L MODEL A R E : V= • , F 1 0 . 6 , 1 O X , # ' B - V = ' , F 1 0 . 6 / / / ) WRITE (6 ,2205) I T O T 2205 F O R M A T ( 1 0 X , ' T H I S I S T H E ' , 1 3 , ' I T E R A T I O N ' ) I F (MAGON) W R I T E ( 6 , 2 2 1 1 ) HO 2211 F O R M A T ( 1 0 X , ' H 0 = ' , F 9 . 0 ) I F (OMEGA. N E . 1. 111 1111 D-4) WRITE (6 ,2212) OMEGA 2212 F O R M A T C - ' , 9 X , ' O M E G A = » , D 1 2 . 5 ) WRITE (6 ,2200 ) 2200 FORMAT (• 1 ' , 1 0 X , ' T H E NEW S T A R T I N G MODEL F O L L O W S ' / / ) WRITE (6 ,PARAM) WRITE (6 ,2201) 2201 F O R M A T ( 6 X , ' F X • , 1 1 X , • S ' , 1 1 X , • Q ' , 1 1 X , • P ' , 1 1 X , • T ' , 1 2 X , • R H O • , 9 X , 1 ' N + 1 ' , 1 0 X , ' L A M B D A ' , 1 O X , ' K A P P A ' / / ) WRITE ( 6 , 2 2 02) (AFX (I) , A S (I) , AQ (I) , AP (I) , AT (I) , ARHO (I) , AN 1 (I) , AL AM 1 (I) , KAPPA (I) ,1=1, NP) 2202 F 0 R M A T ( 1 X , 5 F 1 2 . 6 , D 1 3 . 5 , F 1 2 . 6 , F 2 0 . 1 6 , F 1 2 . 6 ) I F ( Q U I T . A N D . S H O R T ) GO TO 84 1 0 7 C C I N T E R M E D I A T E OUTPUT I N T E R M E D I A T E OUTPUT I N T E R M E D I A T E OUTPUT 82 I F ( ( . N O T . I N T E R ) . O R . ( I T O T . G T . 2 ) ) GO TO 860 WRITE (0 ,2401) H O , M O , W T , I C V C T , N M A S S , N P , Q U I T WRITE (0 ,2400 ) (AFX (I) ,MAG (I) ,DMAG (I) , ALAM (I) , AN1 (I) , ABETR (I) , AT (I) , 1 AS (I) , ARHO (I) , 1= 1,NP) 2400 F O R M A T ( 9 D 2 5 . 1 6 ) 2401 FORMAT (6D25. 1 6 , 3 I 3 , L 1 ) W R I T E ( 0 , 2 4 0 2 ) P A S S , I P A S S 2402 F O R M A T ( 5 D 2 5 . 16 ,5110) 860 CONTINUE C C CONVERGENCE T E S T S CONVERGENCE T E S T S C A L L SMAX ( B A O X , 4 0 0 , 0 , A Q , N P , 1 , R M A X C Q , I X Q ) I F ( R M A X C Q . G T . E R C Q ) GO TO 310 C A L L SMAX ( B A U X , 4 0 0 , - 1 , A S , N P , 1 , R M A X C S , I X S ) I F ( R M A X C S . G T . E R C S ) GO TO 310 C A L L SMAX ( B A U X , 4 0 0 , - 3 , A P , N P , 0 , R M A X C P , I X P ) I F ( R M A X C P . L E . E R C P ) GO TO 308 BXP= DABS (BAUX ( 4 * I X P - 3 ) ) I F ( B X P . L T . S U R F C ) GO TO 308 GO TO 310 308 CALL S M A X ( B A U X , 4 0 0 , - 2 , A T , N P , 0 , R M A X C T , I X T ) I F (RMAXCT. L E . ERCT) GO TO 320 BXT=DABS (BAUX ( 4 * I X T - 2 ) ) I F ( B X T . L T . S U R F C ) GO TO 320 C C F I N A L L O G I C L O G I C L O G I C L O G I C C 310 CONVG=. F A L S E . ITOT=ITOT+1 ICOUNT=ICOUNT+1 I F ( I C O U N T . L E . I C M A X ) GO TO 2 WRITE (6 ,2301) 2301 F O R M A T ( ' 1 ' , 1 0 X , ' C O N V E R G E N C E TOO SLOW, NO NEW OUTPUT MODEL PRODUCED T ) I F ( . N O T . D U M P ) GO TO 9 9 9 . WRITE (7 , PARAM) WRITE (7 ,3000 ) (AX (I) , A S (I) , A Q ( I ) , A P ( I ) , A T (I) , 1=1,NP) 999 STOP 320 CON VG=. T R U E . ICVG=ICVG+1 I F ( I C V G . L E . I C O N U M ) GO TO 850 Q U I T = . T R U E . I F (SHORT) GO TO 83 84 I F ( . N O T . O U T P U T ) GO TO 870 WRITE (7 ,PARAM) WRITE (7 ,3000 ) (AX (I) , AS (I) , AQ (I) , AP (I) , AT (I) , 1= 1 , NP) 3000 F 0 R M A T ( 5 D 2 5 . 1 6 ) 870 CONTINUE I F (MAGON) GO TO 900 GO TO 1000 850 I F ( . N O T . M A G O N ) GO TO 1 C c . c C THE C A L C U L A T I O N OF THE MAGNETIC F I E L D STRUCTURE C 900 CON V G = . F A L S E . 1 0 8 C A L L B F I E L D I F ( . N O T . Q U I T ) GO TO 1 C c C E F F E C T I V E S U R F A C E TEMPERATURES C A L L S T E M P ( R L U M , T 0 , A S (NP)) WRITE (6 ,3125) 3125 FORMAT(* 1') PREV=. T R U E . S E C O N D = . T R U E . GO TO 1000 END SUBROUTINE S T E M P ( R L U M , T O , S ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION PASS (5) , C O L ( 2 ) , B ( 2 0 ) , V (20) DIMENSION ANG (20) COMMON / C H A I N / C O L , S I M I , N A N G COMMON / C T E M P / D P O T 2 , H L A M , R O T , E S2=S*S DEN=1.D0+HLAM-ROT CP= (1 . D O / (1 . DO + E) **2 + DPOT2) / D E N CE= (1 . D 0 / ( 1 . D O - , 5 D 0 * E ) **2 + 1 . 5 D 0 * ( H L A M - R O T * ( 1 - . 5D0*E) ) - . 5D0*DPOT2) $ / D E N T P O L = T 0 * C P * * . 2 5 D 0 T E Q U = T 0 * C E * * . 2 5 D 0 WRITE (6 ,3120 ) T 0 , T P O L , T E Q U 3120 FORMAT ( ' 0 ' , ' T H E S U R F A C E TEMPERATURE I S ' , F 1 9 . 1 2 , ' DEGREES ON P2 = 0 ' 1 / 2 8 X , F 1 8 . 1 2 , » DEGREES AT THE P O L E ' / 2 8 X , F 1 8 . 1 2 , * DEGREES AT T H E EQU 2 A T O R ' ) RD=1.5D+2*E T D I F F = T P O L - T E Q U WRITE (6 ,200) T D I F F , R D 200 FORMAT ( 1 X , ' T E M P E R A T U R E D I F F E R E N C E 3 ' , F 1 2 . 5 , 1 0 X , ' R A D I U S D I F F E R E N C E P 1 0 L E - E Q U A T O R = ' , F 1 2 . 8 , • % • ) WRITE (6 ,100) DPOT2 100 FORMAT (1 X , ' (DPOT2/DR) / (GM/R*R) =DPOT2= ' , D15. 7 ) WRITE (6 ,300) D P O T 2 , H L A M , R O T , E 300 F 0 R M A T ( 1 X , 4 D 2 5 . 1 5 ) I F (NANG. E Q . 0) RETURN I F ( S I N I . G E . O . D O ) A N G ( 1 ) = S I N I WRITE (6 ,400) 400 F O R H A T C - » , 9 X , ' T H E UBV COLOURS A R E • / ' - ' , ' I D E G R E E S ' , 7 X , » V ' , 2 0 X , # » B - V ' / / ) C A L L COLOUR ( S , T O , B , V , A N G ) DO 20 K=1,NANG 20 B(K) =B ( K ) - V ( K ) 10 W R I T E ( 6 , 5 0 0 ) (ANG ( K ) , V ( K ) , B ( K ) , K = 1 , N A N G ) 500 F O R M A T ( 1 X , F 6 . 2 , 2 D 2 0 . 1 0 ) RETURN END SUBROUTINE E N G E N ( P , T , E P , E C , X , Z , W T , R H O , T 3 , T 3 2 ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) Q=DSQRT (RHO) / T 3 2 R H O H = R H O / ( T 3 * T 3 ) Y = 1 . D O - (X+Z) A L = 3 9 . 3 3 6 6 3 1 D 0 - 4 6 . 4 1 6 D 0 / T 3 + 2 . D O * (DLOG ( Y ) - D L O G ( X ) ) DA= DEXP (AL) D G = D A * ( D S Q R T ( 1 . D 0 * 2 . D 0 / D A ) - 1 . D O ) E F 1 1 = 1 . D 0 + . 7 9 D - 2 * Q E F 7 1 = 1 . D 0 + 3 . 1 6 D - 2 * Q G11 = 1 .037D0+ . 3 3 D - 1 * T G71=1.DO E P T = 1 3 . 0 0 3 5 8 0 D 0 - 1 5 . 6 9 3 D 0 / T 3 £ P H = D E X P (EPT) E P S = E P H * R H 0 H * E F 1 1 * G 1 1 * X * X WTH=36.656236D0-4 7 . 6 2 3 D 0 / T 3 WTH=WTH+DLOG(EF71)+DL0G(G71) H T H = W T H - ( . 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 * D L O G ( T ) ) 1 WTH=WTH+DLOG(X)-DLOG(1.DO+X) DW=DEXP(WTH) D G 1 = 1 . D O - D G DGW1=DG/(1 .DO + DW) GW1=1.96D0*DGW1 GW2=1.46D0*DGW1*DH EP1 = EPS*DG1 EP2=EPS*GW1 EP3=EPS*GW2 EP=EP1+EP2+EP3 EF141=1 . DO+5. 3 3 D - 2 * Q G 1 4 1 = . 9 9 5 D 0 - 8 . D - 3 * T E C T = 6 2 . 7 0 6 8 7 6 D 0 - 7 0 . 6 9 7 D 0 / T 3 ECH=DEXP (ECT) X U = . 5 8 5 D 0 * Z I F ( T . L T . 1 . 6 D 0 ) X 1 4 = . 1 8 8 D 0 * Z E C = E C H * E F 1 4 1 * G 1 4 1 * X * X 1 4 * R H O H RETURN END SUBROUTINE S M A X ( X N , N N , K N , X D , N D , K D , Q , I X ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION X N ( N N ) , X D ( N D ) Q=0.D0 NF=ND- 1 DO 10 1=1,NF QQ=DABS(XN (4*I + K N ) / X D (I+KD) ) I F ( Q Q . L E . Q ) GO TO 10 Q=QQ IX=I CONTINUE RETURN END SUBROUTINE L A G I N S ( X V , F V , X , F , N , N L , N M I N , N M A X , N D ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION XV (ND) , FV (ND) DIMENSION C V ( 1 0 ) DO 10 K= NMIN, NMAX M=K I F ( X . L T . X V (K) ) GO TO 20 10 CONTINUE 20 NMIN=M N = t ! - N L / 2 - 1 ENTRY L A G I N C ( X V , F V , X , F , N , N L , N D ) DO 60 1=1,NL C V ( I ) = 1. DO DO 50 J=1 , N L I F ( J . E Q . I) GO TO 50 C V ( I ) =CV (I) * ( X - X V (J+N) ) / (XV (I + N) - X V (J+N) )' 50 CONTINUE 60 CONTINUE ENTRY LAGINT (XV , F V , F , N , N L , ND) r — <J m DO 80 1=1,NL 80 F=F+CV(I) * F V (I + M) RETURN END SUBROUTINE F L U X P O ( N P , H O , R C , R O , F L U X ) I M P L I C I T R E A L * 8 ( A - H , R ) COMMON / L A G / N L , N M I N , N M A X EXTERNAL TANB NL=4 NMIN=NL/2+1 NMAX=NP- ( N L - 1 ) / 2 SRC = SNGL (RC) SR0 = SNGL(R0) F L U X = H 0 * 3 . 0 4 3 6 7 D * 2 2 * D B L E ( S Q U A N K ( T A N B , S R C , S R O , . 0 0 1 , T O L , T F ) ) RETURN END FUNCTION TANB (R) R E A L * 8 A S ( 1 0 0 ) , A R H O ( 1 0 0 ) , W T , H T ( 1 0 0 ) , D T A N B , D R R E A L * 8 HR(100) L O G I C A L QUIT COMMON / A L L / A S , A R H O , N P COMMON / H F L U X / H T , H R COMMON / L A G / N L , NMIN, N.MAX DR = DBLE(R) C A L L LAGINS ( A S , H I , D R , D T A N B , N , N L , N M I N , N M A X , N P ) TANB=SNGL (DTANB) *R RETURN END SUBROUTINE K A P P A { O P , R H O , T , K A P , K A P O , A L P H A , B E T A , N T ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 4 OP (30, 10) R E A L * 4 T L , R L O G T L = T * 1 0 . RLOG=SNGL(DLOG10(RHO)) DO10 I = N T , 3 0 I F ( T L . L T . O P (1 ,1 ) ) GO TO 20 10 CONTINUE 20 NT=I NR= I FIX ( R L O G - O P (NT, 2) ) +3 ALPHA=DLOG1 0 (DBLE (OP ( N T , NR+1) / O P (NT , NR) ) ) NR1 = I F I X (RLOG-OP ( N T - 1 , 2) ) +3 BETA=DLOG1 0 (DBLE (OP (NT-1 ,NR1 ) / O P (NT , NR) ) ) BET A = - B E T A / DLOG10 (DBLE (OP ( N T - 1 , 1) / O P (NT, 1) ) ) KAP0=OP (NT , NR) * (. 1*OP (NT,1 ) ) * * B E T A / (OP (NT ,2 ) * * A L P H A ) KAP=KAP0* ( R H O * * A L P H A ) / ( T * * B E T A ) RETURN END SUBROUTINE ENPOLE ( B , D B , H 0 , M O , E R , E G ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 MO DIMENSION B (100) , DB(100) , HR (100) , HT (100) , RATIO (100) , AS (100) 1 , A F X ( 1 0 0 ) ,ARHO (1 00) REAL*4 S Q U A N K , T O L , F I F , F G , F H , S O COMMON / E N E R G / RATIO COMMON / L A G / N L , N M I N , N M A X COMMON / H F L U X / H T , H R COMMON / A L L / A S , A R H O , N P COMMON / M A S S / AFX EXTERNAL F H , F G RATIO (1) =0 .D0 SS0 = AS (NP) C CONSTANT = R / ( 2 4 * P I * G * M S U N ) C = . 6 9 6 0 1 3 6 4 D - 1 7 * H 0 * H 0 / M 0 S2=SS0*SS0 S 2 2 = S 2 / 2 . D O NP1=NP-1 RATIO (1) =3. D 0 * H R ( 1 ) * H R (1) DO 10 1=2,NP1 H R ( I ) = - S 2 * B ( I ) / ( A S ( I ) * A S (I) ) HT (I) =S22*DB (I) / A S (I) RATIO (I) = (HR (I) * H R ( I ) +2. D0*HT (I) * H T (I) ) 10 CONTINUE H R ( N P ) = 1 . D 0 • H T ( N P ) = . 5 D 0 BAT 10 (NP) = 1 .5D0 NL=U NMIN=NL/2+1 NHA X = N P - ( N L - 1 ) / 2 S0 = SNGL (SSO) E H = D B L E ( S Q U A N K ( F H , 0 . , S O , . 0 0 0 0 1 , T 0 L , F I F ) ) NMIN=NL/2+1 S0=S8GL(SSO) EG= D B L E (SQUANK ( F G , 0 . , S O , 1. E - 7 , T 0 L , F I F ) ) E R = C * E H / E G RATIO (1) =0. DO RATIO ( N P ) = 0 . D 0 DO 20 1=2,NP1 20 RATIO (I) = C * R A T I O ( I ) * A S ( I ) / (AFX (I) *ARHO (I) ) RETURN END FUNCTION F H ( S ) R E A L * 8 R A T I O (100) , A S (100) , A R H O (100) , R A T , S S COMMON / A L L / A S , A R H O , N P COMMON / E N E R G / RATIO COMMON / L A G / N L , N M I N , N M A X SS=DBLE (S) C A L L L A G I N S ( A S , R A T I O , S S , R A T , N , N L , N M I N , N M A X , N P ) F H = S N G L ( R A T * S S * S S ) RETURN END FUNCTION F G ( S ) R E A L * 8 A S ( 1 0 0 ) ,ARHO (100) , AFX (1 00 ) , S S , MR, RHO COMMON / A L L / A S , A R H O , N P COMMON / M A S S / AFX COMMON / L A G / N L , N M I N , N M A X SS=DBLE(S) C A L L LAGINS ( A S , A F X , S S , M R , N , N L , N M I N , N M A X , N P ) C A L L L A G I N T ( A S , A R H O , R H O , N , N L , N P ) FG=SNGL(MR*RHO*SS) RETURN END $C * S K I P 112 $C STAR (768) TO *SINK*c&NOCC SUBROUTINE COLOUR ( S O , T E O , B , V , A N G ) C S I N I I S SIN (ANG OF MAGNETIC A X I S WRT L I N E OF S I G H T ; I . E . 0=POLE ON) C I N T E G R A T I O N DONE WITH Z A X I S D E F I N E D AS L I N E OF SIGHT C THETA=ANGLE BETWEEN POINT ON S U R F A C E OF STAR AND L I N E OF S I G H T C PHI=ANGLE AROUND Z A X I S C BY SYMMETRIES 0<=THETA<=PI AND 0<=PHK=PI/2; I . E . HALF OF V I S I B L E SURFA C THETA PRIME I S ANGLE WRT TO SYMMETRY AXIS (MAGNETIC AXIS) OF STAR C I . E . I F S I N I = 0 , THEN THETA=THETA P R I M E . I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION COSPHI(8) ,SINT (8) , SINPHI ( 8) I N T E G E R S THETA , P H I , T H E T A 2 DIMENSION H L G N D R ( 8 ) , H L A G 0 E ( 8 ) DIMENSION ALGND (8) , A L A G (8) DIMENSION B(20) , V (20) ,COL(2) ,ANG(20) L O G I C A L SECOND COMMON / D A T / SECOND COMMON / C T E M P / D P O T 2 , H L A M , R O T , E COMMON / C M A I N / COL,SINI,NANG DATA ALGND / . 0 9 5 0 1 2 5 0 9 8 3 7 6 3 7 D O , . 2 8 1 6 0 3 5 5 0 7 7 9 2 5 8 DO, f . 4 5 8 0 1 6 7 7 7 6 5 7 2 2 7 D O , . 6 1 7 8 7 6 2 4 4 4 0 2 6 4 3 0 0 , . 7 5 5 4 0 1 4 0 8 3 5 5 0 0 3 D O , # . 865631202387831 D O , . 9 4 4 5 7 5 0 2 3 0 7 3 2 3 2 D O , . 9 8 9 4 0 0 9 3 4 9 9 1 6 4 9 D O / DATA HLGNDR / . 1 8 9 4 5 0 6 1 0 4 5 5 0 6 8 D 0 , . 1 8 2 6 0 3 4 1 5 0 4 4 9 2 3 D 0 , # . 1 6 9 1 5 6 5 1 9 3 9 5 0 0 2 D 0 , . 1 4 9 5 9 5 9 8 8 8 1 6 5 7 6 D O , . 1246 28971255533 DO, # . 0 9 5 1 5 8 5 1 1 6 8 2 4 9 2 D 0 , . 0 6 2 2 5 3 5 2 3 9 3 8 6 4 7 D 0 , . 0 2 7 1 5 2 4 5 9 4 1 1 7 5 4 D 0 / DATA ALAG / . 1 7 0 2 7 9 6 3 2 3 0 5 D 0 , . 9 0 3 7 0 1 7 7 6 7 9 9 DO, # 2 . 2 5 1 0 8 6 6 2 9 8 6 6 D 0 , 4 . 2 6 6 7 0 0 1 7 0 2 8 8 D 0 , 7 . 0 4 5 9 0 5 4 0 2 3 9 3 D 0 , # 1 0 . 7 5 8 5 1 6 0 1 0 1 8 1 D 0 , 1 5 . 7 4 0 6 7 8 6 4 1 2 7 8 D 0 , 2 2 . 8 6 3 1 3 1 7 3 6 8 8 9 D 0 / DATA HLAGUE / 3 . 6 9 1 8 8 5 8 9 3 4 2 D - 1 , 4 . 1 8 7 8 6 7 8 0 8 1 4 D - 1 , # 1 . 7 5 7 9 4 9 8 6 6 3 7 D - 1 , 3 . 3 3 4 3 4 9 2 2 6 1 2 D - 2 , 2 . 7 9 4 5 3 6 2 3 5 2 3 D - 3 , # 9 . 0 7 6 5 0 8 7 7 3 3 6 D - 5 , 8 . 4 8 5 7 4 6 7 1 6 2 7 D - 7 , 1 . 0 4 8 0 0 1 1 7 4 8 7 D - 9 / PI2 = 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 D 0 / 2 . DO P O S N E G = - 1 . DO T E R E D 4 = . 7 5 D 0 * T E 0 * * 4 / (1 .DO+HLAM-ROT) S2=S0*S0 NLAGUE=8 NLGNDR=8 I F ( S E C O N D ) GO TO 2 C ALGND=COS (THETA) AND P I 2 * A L G N D = P H I DO 1 K=1,NLGNDR SINT (K) = DSQRT (1 . D O - A L G N D (K) * A L G N D ( K ) ) COSPHI (K) =DCOS ( P I 2 * A L G N D (K) ) S I N P H I (K) =DSIN ( P I 2 * A L G N D (K) ) 1 CONTINUE 2 DO 5 K=1,NANG I F ( N A N G . E Q . 1) GO TO 100 FR= (K-1) / D F L O A T (NANG- 1) SINI=DSIN (PI2*FR) ANG (K)=90. D0*FR 100 C 0 S I = D S Q R T ( 1 . D 0 - S I N I * S I N I ) FLUXB=0. DO F L U X V = 0 . D 0 DO 10 THETA=1 ,NLGNDR DO 20 T H E T A 2 = 1 , 2 POSNEG=-POSNEG DO 30 PHI=1 ,NLGNDR TPHI=SINT (THET A) * S I N P H I ( P H I ) / (SINT (THETA) * C O S P H I (PHI) * C O S I $ - P O S N E G * A L G N D (THETA). * S I N I ) C P H I P = 1 . D O / D S QRT{1 .DO+TP H I * T P H I ) I F ( T P H I . L T . O . D O ) C P H I P = - C P H I P CTHEP=+POSNEG*ALGND (THETA) * C O S I + S I N T (THETA) * C O S P H I (PHI) * S I N I C 0 2 - C T H E P * C T H E P P 2 = ( 3 . D 0 * C O 2 - 1 . D O ) * . 5 D 0 S I 2 = 1 . D 0 - C O 2 STH=DSQRT(SI2) AMO = - (1 . D0+3. D 0 * E * C O 2 ) * S T H * C P H I P * S I N I + ( 1 . D O - 3 . D O * E * S I 2 ) * C T H E P * C O S I AMU=DABS (AMU) TET = T E R E D 1 * (1 . D O / (1 . D0+E*P2) * * 2 + 1 . 5 D 0 * ( H L A M - R O T * (1 .D0+E*P2) ) * S T H # +DPOT2*P2) H N K T B = . 1 1 9 6 8 1 1 2 7 6 D + 1 9 / T E T E 3 = ( 1 . D 0 + E * P 2 ) * * 3 FIUB=O.DO F I U V = O . D O DO 50 I L A G = 1 , B L A G U E T E M P = A L A G ( I L A G ) * A M U + . 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 DO ARG4 = (HNKTB/TEMP) * * . 25D0 D E H B = D E X P ( A R G 4 ) - 1 . D O D E N V = D E X P ( A R G 4 * . 7 8 6 6 1 8 4 4 4 8 4 D 0 ) - 1.DO F I U B = F I U B + H L A G U E ( I L A G ) / D E N B 50 FIUV=FIUV + HLAGUE (ILAG) / D E N V F I U B = F I U B * A M U FIUV=FIUV*AMU C B I S C A L C U L A T E D AT 4350A AND V AT 5 5 3 0 A FLUXB= FLUXB+HLGNDR (THETA) *HLGNDR (PHI) * F I U B * E 3 FLUXV=FLUXV+HLGNDR (THETA) *HLGNDR (PHI) * F I U V * E 3 30 CONTINUE 20 CONTINUE 10 CONTINUE C COL F I N D S D I F F E R E N C E FROM I N I T I A L BV COLOURS B ( K ) = - 2 . 5D0*DLOG10 ( S 2 * F L U X B ) - C O L (1) C C O N S T A N T = - 2 . 5 * L O G 1 0 ( ( 4 3 5 0 / 5 5 3 0 ) * * 3 ) V (K) =-2. 5D0*DLOG10 ( S 2 * F L U X V ) - COL (2) +. 7817690578D0 5 CONTINUE RETURN END $C * S K I P $C NORM TO * S I N K * 3 N 0 C C SUBROUTINE B F I E L D I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 AS (100) , ARHO (100) ,AN1 (100) , MAG (100) /DMAG (100) 1 , E P S (100) , ALAM (10 0) , P O T 2 (100) , P P (100) , PH (100) , A FX (1 00) 2 , H R ( 1 0 0 ) , H T (100) , R A T I O (100) R E A L * 8 MO L O G I C A L Q U I T , M A G O U T , S H O R T , C U R F I T DIMENSION I P A S S (5) , P A S S (5) DIMENSION SOLY (9) , S O L F (9) , S O L Q ( 9 ) DIMENSION CONPAT(25) COMMON / M A I N B / MAG ,DMAG , A L A M , S O L Y ,HO , MO , OMEGA, ICVCT 1 , N M A S S , P A S S , I P A S S , S H O R T , M A G O U T , I C O N U M , I C V G 2 , M C V C T , C U R F I T COMMON / A L L / A S , A R H O , N P COMMON / R K B / C O N 1 , C O N 2 , R H O A V , S C COMMON / H F L U X / H T , H R COMMON / M A S S / AFX COMMON / E T C / W T , Q U I T COMMON / L A G / N L , N M I N , N M A X COMMON / E N E R G / RATIO COMMON / C T E M P / D P O T 2 , H L A M , R O T , E SURF=AS(NP) S2=SURF*SURF S3=SURF*S2 R H O A V = 1 . 4 0 8 3 7 6 6 6 9 * M 0 / S 3 DO 910 1=1,9 910 S O L Y ( I ) = 0 . D 0 SOLY (1) = AS (2) - A S (1) M A G ( 1 ) = 0 . D 0 DMAG(1)=0 .D0 P O T 2 ( 1 ) = 0 . D 0 P P ( 1 ) = 0 . D 0 PH( 1) =0 .D0 SOLY (2)=SOLY (1) * S O L Y (1) SOLY (4) = SOLY (2) SOLY (6) =SOLY (2) SOLY (8) =SOLY (2) SOLY (3) =2. D0*SOLY (1) SOLY (5) =SOLY (3) SOLY (7) =SOLY (3) SOLY (9) =SOLY (3) MAG (2)=SOLY (8) DMA G (2) =SOLY (9) POT2 (2) =SOLY (8) PP (2) =SOLY (8) PH (2)=SOLY (8) CON1=4 .917521412D0 C L A M = . 6 9 6 0 3 1 6 4 D - 1 7 * ( S O R F * * 4 * H 0 * H 0 / ( R H O A V * M 0 ) ) CON2=CON1*CLAM NP1 = NP-1 NL=4 NMIN=NL/2+1 NMA X=NP- ( N L - 1 ) / 2 DO 920 J = 2 , N P 1 I=J +1 H=AS (J + 1) - A S (J) ' 1 1 5 C A L L D B K ( S O L Y , S O L F , S O L Q , H , 9 , 1 ) MAG (I) = SOLY (2) DMAG (I) =SOLY (3) I F ( . N O T . Q U I T ) GO TO 920 P H ( I ) =SOLY (4) P P ( I ) = S O L Y (6) POT2 (I) = SOLY(8) 920 CONTINUE BCB=- (MAG (NP) +SURF*DMAG (NP) ) / (3 . D 0 * S U R F * S U R F ) CNORM=DABS (MAG (NP) «-BCB*AS (NP) * A S (NP) ) DO 930 1=1,NP MAG (I) = (BCB*AS (I) * A S (I) +MAG(I) ) / C N O R M DMAG (I) = ( 2 . D 0 * B C B * A S (I) +DMAG (I) ) / C N O R M 930 CONTINUE CLAM=CLAM/CNORM CONPAT (ICVG)=CLAM ALAM (1) = 2 . D 0 * S 3 * R H O A V * C L A H * (BCB + 1 . D 0 ) / (CNORM*ARHO (1) ) DO 940 1=2,NP 940 ALAM (I) =CLAM*DMAG (I) * A S (I) * A S (I) / A F X (I) I F ( ( . N O T . QUIT) . A N D . SHORT) RETURN H R ( 1 ) = - (BCB + 1 .DO) / C N O R M * S U R F * S U R F HT (1) = - H R (1) C A L L E N P O L E ( M A G , D M A G , H 0 , M O , E R , E G ) SC=0.DO C A L L FLUXPO ( N P , H O , S C , S U R F , F L U X ) WRITE (6 ,3200) 3200 F O R M A T ( ' 1 ' , 1 0 X , ' T H E STREAM F U N C T I O N FOR T H E MAGNETIC F I E L D ' / / ) WRITE (6 ,3250) C L A M , C N O R M , F L U X 3250 F O R M A T ( 1 X , » C L A M = » , D 1 0 . 3 , 1 O X , • C N O R M = • , D 1 0 . 3 , 1 0 X , » F L U X = ' , D 1 0 . 3 ) WRITE (6 ,3208) ER, HO 3208 F 0 R M A T ( 1 X , * ENERGY R A T I O = « , D 1 0 . 3 , 1 O X , ' H 0 = « , F 9 . 0 , • G A U S S • / / ) WRITE (6 ,3210) 3210 F O R M A T ( 1 2 X , « S ' , 1 9 X , ' B ' , 1 9 X , ' D B / D S * , 1 5 X , ' H R A D » , 1 5 X , ' H T A N ' , 1 5 X , 1 "ENERGY R A T I O ' / / ) WRITE ( 6 , 3 2 2 0 ) ( A S ( I ) , M A G ( I ) , DMAG (I) , H R ( I ) , HT (I) , R A T 10(1) , 1=1,NP) 3220 F O R M A T ( 1 X , 5 F 2 0 . 6 , D 2 0 . 6 ) I F (. NOT. QUIT) RETURN C c c C THE P2 TERM OF THE G R A V I T A T I O N P O T E N T I A L B C P O T = - (3. DO* (PP (NP) +BCB*PH (NP) ) +SURF* (SOLY (7) +BCB*SOLY (5) ) ) C2=CN0RM*CN0RM BCPOT=BCPOT/ (3 . DO *POT2 (NP) +SURF*SOLY (9) ) / C 2 EPS (1) =- (BCPOT+ (1 .DO+BCB) / C 2 - C L A M ) * R H O A V * S 3 / A R H O (1) DO 950 1=2,NP POT2 ( I)=BCPOr*POT2(I ) • (PP ( I ) + B C B * P H (I) ) / C 2 950 EPS (I) = -AS (I) * (POT2 (I) - C L A M * H A G (I) ) / A F X (I) WRITE (6 ,4000) 4000 F O R M A T ( ' - » , • T H E PATH OF CLAM W A S : ' ) WRITE (6 ,3999) ( I , CONPAT (I) , 1=1, ICONUM) 3999 F O R M A T ( 1 X , I 5 , D 2 0 . 5 ) WRITE (6 ,3110) 3110 F O R M A T ( ' 1 • , 9 X , ' I ' , 1 0 X , ' E P S • , 1 7 X , ' L A M B D A ' , 1 8 X , ' S • , 1 5 X , ' P O T 2 ' / / ) WRITE (6 ,3100 ) (I , EPS (I) , ALAM (I) , AS (I) , POT2 (I) , 1=1 , NP) 3100 F O R M A T ( 1 X , I 1 0 , 2 D 2 0 . 5 , F 2 0 . 5 , D 2 0 . 5 ) E=EPS(NP) DPOT2=S2* ( B C P O T * S O L Y (9) + (SOLY (7) +BCB*SOLY (5) ) / C 2 ) HLAM=ALAM (NP) ROT = 0 . D 0 1 1 6 I F (MAGOUT) W R I T E ( 7 , 1 9 9 9 ) HO,MO 1999 FORMAT(1 O X , • N O R M A L MAGNETIC F I E L D : H 0 = • , F 9 . 0 , 1 0 X , ' F O R M 0 = ' , F 4 . 1 ) IF(MAGOUT) WRITE ( 7 , 2 0 0 0 ) (AS (I) , MAG (I) , DMAG (I) , 1= 1, NP) , 2000 FORMAT (3D25. 16) RETURN END SUBROUTINE A U X R K ( Y , F ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION ABETR (100) DIMENSION Y (9) , F (9 ) , ARHO (100) , AS (100) , AN 1 (1 00) , A T (1 00) L O G I C A L Q U I T COMMON / R K B / C O N 1 , C O N 2 , R H O A V , S C COMMON / M A I N R K / A N 1 , A B E T R , A T COMMON / A L L / A S , A R H O , N P COMMON / L A G / N L , N M I N , N M A X COMMON / E T C / W T , Q U I T X=Y(1) X2=X*X F(2) =Y(3) C A L L L A G I N S ( A S , A R H O , X , D E N , N , N L , N M I N , N M A X , N P ) F(3) = 2 . D 0 * Y (2) / X 2 + D E N / R H O A V * X 2 I F ( Q U I T ) GO TO 10 DO 5 1=4,9 5 F ( I ) = 0 . D 0 RETURN 10 CONTINUE C NOTE THAT POT2 I S A D I M E N S I O N L E S S P O T E N T I A L C A L L L A G I N T ( A S , A B E T R , B E T , N , N L , N P ) C A L L L A G I N T ( A S , A N 1 ,EN1 , N , N L , N P ) C A L L L A G I N T ( A S , A T , T , N , S L , N P ) F(4) =Y(5) F ( 6 ) = Y ( 7 ) F ( 8 ) = Y ( 9 ) I F ( T . L E . . 0 D 0 ) GO TO 50 C A L C = D E N * W T / T * ( ( 3 . D 0 * B E T - 4 . D O ) / E N 1 + 1 . D O ) GO TO 60 50 C A L C = 0 . D 0 60 C P 0 T = C 0 N 1 * C A L C CB=CON2*CALC F (5 ) =-2. D0*Y ( 5 ) / X + 6 . D 0 * Y (4) / X 2 - C P O T * Y (4) +CB*X2 F(7) = - 2 . D 0 * Y ( 7 ) / X + 6 . D 0 * Y (6) / X 2 - C P O T * Y (6) +CB*Y (2) \ F ( 9 ) = - 2 . D0*Y ( 9 ) / X + 6 . D 0 * Y ( 8 ) / X 2 - C P O T * Y (8) RETURN END $C * S K I P 117 $C CON TO * S I N K * 9 N 0 C C SUBROUTINE B F I E L D I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 AS (100) , ARHO (100) , AN 1 (100) , MAG (100) ,DMAG (100) 1, EPS (100) , ALAM (100) , P O T 2 (100) , P P (100 ) , PH (100 ) , AFX (100) R E A L * 8 MO L O G I C A L Q U I T , M A G O U T , S H O R T , C U R F I T DIMENSION HR (100) , H T ( 1 0 0 ) , R A T I O ( 1 0 0 ) DIMENSION SOLY (9) , SOLF (9) , S O L Q (9 ) DIMENSION C O E F ( 3 , 3) , RHS (3) DIMENSION I P A S S (5) , P A S S (5) DIMENSION CONPAT(25) COMMON / M A I N B / M A G , D M A G , A L A M , S O L Y , H O , M O , O M E G A , I C V C T 1 , N M A S S , P A S S , I P A S S , S H O R T , M A G O U T , I C O S U M , I C V G 2 , M C V C T , C U R F I T COMMON / A L L / A S , A R H O , N P COMMON / R K B / C O N 1 , C O N 2 , R H O A V , S C 3 COMMON / H F L U X / HP , HR COMMON / M A S S / AFX COMMON / E T C / W T , Q U I T COMMON / L A G / N L , N M I N , N M A X COMMON / E N E R G / RATIO COMMON / C T E M P / D P O T 2 , H L A M , R O T , E C MCVCT MOVES T H E F I T T I N G POINT INTO THE INTERIOR N X T = I C V C T + ( N M A S S + 1 ) / 2 - M C V C T N B = I C V C T - N M A S S / 2 - M C V C T C W = A S ( N X T ) - A S (NB) S C = A S ( I C V C T ) S C 3 = S C * S C * S C SURF=AS (NP) S3=SURF**3 R H O A V = 1 . 4 0 8 3 7 6 6 6 9 * M 0 / S 3 DO 910 1=1 ,9 910 S O L Y ( I ) = 0 . D 0 ICVCT1=ICVCT+1 DO 10 I = 1 , I C V C T 1 P P ( I ) =0 .D0 P H ( I ) = 0 . D 0 MAG (I) =0 .D0 DMAG (I) =0. DO P0T2 (I) =0 .D0 ALAM ( I )=0 . DO 10 E P S ( I ) = 0 . D 0 SOLY (1) = AS ( ICVCT) SOLY (3) = 3 . D 0 * S C DMAG ( ICVCT)=SOLY (3) C WHEN P 0 T 2 , B ARE SMALL POT2 GOES AS S * * 2 - S C * * 5 / S * * 3 SOLY (5) =5. D 0 * S C SOLY (7) =SOLY (5) SOLY (9) =SOLY (5) C C O N 1 = 4 * P I * G * H * R * R / ( K * 1 D 7 ) C CLAM C O N S T A N T = 2 / ( 3 * 1 6*PI) * R / ( G * M ) MASS OF STAR C 0 N 1 = 4 . 9 1 7 5 2 1 4 1 2 C L A M = . 6 9 6 0 3 1 6 4 D - 1 7 * ( S U R F * * 4 * H 0 * H 0 / ( R H O A V * M 0 ) ) C HO IS THE RADIAL MAGNETIC F I E L D AT THE POLE COH2=CON1*CLAM NP1 = NP-1 1 1 R NL=4 1 1 8 NMIN=NL/2+1 NMAX=NP- (NL-1) / 2 DO 920 J = I C V C T , N P 1 I=J + 1 H= (AS ( I ) - A S (J) ) / 2 . DO C A L L DRK ( S O L Y , S O L F , S O L Q , H , 9 , 2 ) MAG (I) =SOLY (2) DMA G (I) = SOLY (3) I F ( . N O T . Q U I T ) GO TO 920 P H ( I ) =SOLY (4) P P ( I ) = S O L Y (6) P 0 T 2 (I) = S O L Y ( 8 ) 920 CONTINUE BCB=- (MAG (NP) +SURF*DMAG (NP) ) / (3 . D 0 * S U R F * S U RF) CNORM=DABS (MAG (NP) +BCB*(AS (NP) * A S ( N P ) - S C 3 / A S (NP) )) DO 930 I = I C V C T , N P S2= AS (I) * AS (I) MAG (I) = ( B C B * ( S 2 - S C 3 / A S ( I ) ) + MAG (I) ) / C N O R M DMAG (I) = (BCB* (AS (I) * 2 . D0+SC3/S2) +DMAG (I) ) /CNORM 930 CONTINUE C CURVE F I T T I N G TO C O N V E C T I V E CORE I F ( . N O T . C U R F I T ) GO TO 50 W2=CW*CW W3=H2*CW 84=W3*CW W5=W4*CW C O E F ( 1 , 1 ) = W 3 C O E F ( 2 , 1 ) = 3 . D 0 * W 2 C O E F ( 3 , 1 ) = 6 . D 0 * C f f C O E F ( 1 , 2 ) =W4 COEF ( 2 , 2 ) = 4 . D0*M3 C O E F ( 3 , 2 ) = 1 2 . D 0 * H 2 COEF (1 , 3 ) =W5 C O E F ( 2 , 3 ) =5.D0*W4 COEF ( 3 , 3 ) = 2 0 . D0*W3 RHS(1) =MAG (NXT) RHS (2) =DMAG (NXT) RHS(3) = 2 . D 0 * M A G (NXT) / (AS (NXT) * A S (NXT) ) +ARHO (NXT) *AS (NXT) * A S (NXT) 1 / (RHOAV*CNORM) C A L L DSOLTN ( C O E F , R H S , 3 ,3 , D E T ) DO 12 I=NB,NXT X=AS(I) - A S ( N B ) X3=X*X*X MAG (I) =RHS(1) * X 3 * R H S (2) *X3*X+RHS ( 3 ) * X 3 * X * X 12 DMAG (I) =3. D0*RHS {1) * X * X + 4 . D0*RHS (2) *X3 + 5 . D 0 * R H S (3) * X 3 * X 50 CLAM=CLAH/CNORM CONPAT (ICVG)=CLAM DO 940 I = N B , N P 940 ALAM (I) = CLAM*DMAG (I) * A S (I) * A S (I) / A F X (I) I F ( ( . N O T . Q U I T ) . A N D . S H O R T ) RETURN H R ( 1 ) = - ( B C B + 1 . DO) /CNORM H T ( 1 ) = - H R ( 1 ) C A L L ENPOLE (MAG, DMAG, HO, MO, E R , EG) C A L L F L U X P O ( N P , H O , S C , S U R F , F L U X ) H R I T E (6 ,3200) 3200 F 0 R M A T ( » 1 ' , 1 0 X , » T H E STREAM F U N C T I O N FOR T H E MAGNETIC F I E L D WITH CO 1 N V E C T I O N V / ) WRITE (6 ,3250) C L A M , C N O R M , N M A S S , N B 119 3250 F O R M A T ( 1 X , * C L A M = ' , D 1 0 . 3 , 1 0 X , * C N O R M = • , D 1 0 . 3 , 1 0 X , ' N U M B E R OF MASS SHE I L L S IN CURRENT ZONE= » ,13 ,1 O X , • BEGINNING AT # ' , I 3 ) WRITE (6 ,321 1) F L U X , E R , HO 3211 F O R M A T ( 1 X , ' F L U X = « , D 1 0 . 3 , 1 0 X , • E N E R G Y R A T10=• , D 1 0 . 3 , 1 0 X , • H 0 = » , F 9 . 0 , • 1 G A O S S ' / / ) WRITE (6 ,3210 ) 3210 FORMAT (12X, ' S • , 19X, ' B ' , 19X, ' D B / D S ' , 15X, • H R A D ' , 1 5 X , ' H T A N ' , 1 5 X , 1 •ENERGY R A T I O ' / / ) WRITE ( 6 , 3 2 2 0 ) (AS (I) , MAG (I) , DM AG (I) , HR (I) , H T (I) , RATIO (I) , I = N B , N P ) 3220 F O R M A T ( 1 X , 5 F 2 0 . 6 , D 2 0 . 6 ) I F ( . N O T . Q U I T ) RETURN C C THE P2 TERM OF T H E G R A V I T A T I O N P O T E N T I A L C BC IS 3*POT2+SURF*POT2=0 C B C P 0 T = - ( 3 . D O * (PP(NP) +BCB*PH (NP) )+SURF* (SOLY (7) +BCB*SOLY (5) ) ) C2=CNORM*CNORM B C P O T = B C P O T / ( 3 . D 0 * P O T 2 (NP) +SURF*SOLY (9) ) DO 950 I = N B , N P POT2 (I) = (BCPOT*POT2 (I) +PP (I) +BCB*PH (I) ) / C 2 950 EPS (I) =-AS (I) * (POT2 (I) ^ C L A M * M A G (I) ) / A F X (I) E P S ( N B ) = 0 . D 0 E=EPS (NP) DPDT2=S2* ( B C P O T * S O L Y ( 9 ) +SOLY(7) +BCB*SOLY (5) ) / C 2 ROT=0.D0 HLAM=ALAM (NP) WRITE (6 ,4000 ) 4000 FORMAT (•-' , T H E PATH OF CLAM W A S : ' ) W R I T E ( 6 , 3 9 9 9 ) ( I , C O N P A T (I) ,1=1 , ICONUM) 3999 FORMAT (1X, 15, D20 . 5) WRITE ( 6 , 3 1 1 0 ) 3110 F O R M A T ( • 1 • , 9 X , • I • , 1 0 X , ' E P S ' , 1 7 X , ' L A M B D A » , 1 8 X , • S ' , 1 5 X , ' P O T 2 ' / / ) WRITE ( 6 , 3 1 0 0 ) (I , E P S (I) , A L A M (I) , AS (I) , P O T 2 (I) , 1=1, NP) 3100 FORMAT ( 1 X , 1 1 0 , 2 0 2 0 . 5 , F 2 0 . 5 , D 2 0 . 5 ) I F (MAGOUT) W R I T E ( 7 , 1 9 9 9 ) HO,MO 1999 F O R M A T ( 1 X , ' C O N V E C T I V E CORE MAGNETIC F I E L D : H 0 = • , F 9 . 0 , 10X, ' F O R M0= ' $ , F 4 . 1 ) I F (MAGOUT) W R I T E ( 7 , 2 0 0 0 ) (AS (I) , MAG (I) , DMAG (I) , 1= 1 , NP) 2000 F O R M A T ( 3 D 2 5 . 1 6 ) RETURN END SUBROUTINE A U X R K ( Y , F ) I M P L I C I T R E A L * 8 ( A - H , O ^ Z ) DIMENSION ABETR (100) DIMENSION Y (9) , F (9) , ARHO (100) , AS (100) , AN 1 (100) , AT (100) L O G I C A L Q U I T COMMON / R K B / C O N 1 , C O N 2 , R H O A V , SC3 COMMON / M A I N R K / A N 1 , A B E T R , A T COMMON / A L L / A S , A R H O , N P COMMON / L A G / N L , N M I N , N M A X COMMON / E T C / W T , Q U I T X=Y (1) X2=X*X F(2) =Y(3) C A L L LAGINS ( A S , A R H O , X , D E N , N , N L , N M I N , N M A X , N P ) F ( 3 ) = 2 . D 0 * Y ( 2 ) / X 2 * D E N / R H O A V * X 2 I F ( Q U I T ) GO TO 10 DO 5 1=4,9 5 F ( I ) = 0 . D 0 RETURN 10 CONTINUE C A L L L A G I N T ( A S , A B E T R , B E T , N , N L , N P ) C A L L LAGINT ( A S , A N 1 , E N 1 , N , N L , N P ) C A L L L A G I N T ( A S , A T , T , N , N L , N P ) F (4 )=Y (5) F ( 6 ) = Y ( 7 ) F ( 8 ) = Y ( 9 ) I F ( T . L E . . 0 D 0 ) GO TO 50 C A L C = D E N * W T / T * ( ( 3 . D 0 * B E T - 4 . D O ) / E N 1 + 1 . D 0 ) GO TO 60 50 CALC=0.DO 60 CPOT=CON1*CALC CB=CON2*CALC F(5) = - 2 . D 0 * Y ( 5 ) / X + 6 . D 0 * Y ( 4 ) / X 2 - C P O T * Y (4) +CB* ( X 2 - S C 3 / X ) F (7 ) =-2. D0*Y ( 7 ) / X + 6 . D0*Y (6) / X 2 - C P O T * Y (6) +CB*Y (2) F(9) = - 2 . D 0 * Y ( 9 ) / X + 6 . D 0 * Y ( 8 ) / X 2 - C P O T * Y (8) RETURN END $C * S K I P 121 $C ROT TO * S I N K * S N O C C SUBROUTINE B F I E L D I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 AS (100) , A R H O ( 1 0 0 ) ,AN1 (100) ,MAG (100) , DMAG (100) 1 , E P S (100) , ALAM (100) , POT2 (100) , P P (100) , PH (100) , AFX (1 00) 2,HR (100) , H T (100) , R A T I O (100) R E A L * 8 MO L O G I C A L Q U I T , M A G O U T , S H O R T , C U R F I T DIMENSION I P A S S (5) , P A S S (5) DIMENSION SOLY(9) , S O L F (9) , S O L Q (9 ) DIMENSION CONPAT(25) COMMON / M A I N B / M A G , D M A G , A L A M , S O L Y , H O , M O , O M E G A , I C V C T 1 , N M A S S , P A S S , I P A S S , S H O R T , M A G O U T , I C O N U M , I C V G 2 , M C V C T , C U R F I T COMMON / A L L / A S , A R H O , N P COMMON / R K B / C O N 1 , C O N 2 , R H O A V , S C COMMON / H F L U X / H T , H R COMMON / M A S S / AFX COMMON / E T C / W T , Q U I T COMMON / L A G / N L , N M I N , N M A X COMMON / C T E M P / D P O T 2 , H L A M , R O T , E COMMON / E N E R G / R A T I O SURF=AS(NP) S2=SURF*SURF S3=SURF*S2 02=OMEGA*OMEGA R H O A V = 1 . 4 0 8 3 7 6 6 6 9 * M 0 / S 3 C CROT C O N S T A N T 3 2 / 3 * R S U N * * 3 / ( G * M S U N ) C R O T = 1 . 6 9 4 7 5 1 7 2 8 D 6 * O 2 / M 0 C R 0 T 2 = C R 0 T / 2 . D 0 CON1=4 .917521412D0 C L A M = . 6 9 6 0 3 1 6 4 D - 1 7 * ( S U R F * * 4 * H 0 * H 0 / ( R H O A V * M 0 ) ) C0N2=C0N1*CLAM NP1=NP-1 NL=4 NMIN=NL/2+1 N M A X = N P - ( N L - 1 ) / 2 DO 910 1=1 ,9 910 S O L Y ( I ) = 0 . D 0 S O L Y ( 1 ) = A S ( 2 ) - A S ( 1 ) MAG (1)=0. DO D M A G ( 1 ) = 0 . D 0 POT2(1) = 0 . D 0 PP( 1) =0 .D0 P H ( 1 ) = 0 . D 0 SOLY (2) =SOLY (1) * S O L Y (1) SOLY (4)=CLAM*SOLY (2) SOLY (6) =CLAM*SOLY (2) SOLY (8) =CLAM*SOLY (2) SOLY (3) = 2 . D 0 * S O L Y (1) SOLY (5) =CLAM*SOLY (3) SOLY (7) =CLAM*SOLY (3) SOLY (9) = CLAM*SOLY (3) MAG (2) =SOLY (2) DMAG (2)=SOLY (3) POT2(2) =SOLY(8) P P ( 2 ) = S 0 L Y (8) 1 22 PH{2) =SOLY (8) DO 920 J = 2 , N P 1 1=J +1 H=AS (J + 1) - AS (J) C A L L D R K ( S O L Y , S O L F , S O L Q , H , 9 , 1 ) MAG (I) = SOLY (2) DMAG (I) =SOLY (3) I F ( . N O T . QUIT) GO TO 920 P H ( I ) =SOLY (4) P P ( I ) = S O L Y (6) POT2 (I) = S O L Y ( 8 ) 920 CONTINUE BCB=- (MAG (NP) +SURF*DMAG (NP) ) / (3 . DO * S U R F * S U RF) CNOEM= DABS (MAG (NP) +BCB*AS (NP) * A S (NP) ) DO 930 1=1,NP MAG (I)= (BCB*AS (I) * A S (I) +MAG (I) ) /CNORM DMAG (I) = ( 2 . D 0 * B C B * A S (I) +DMAG (I) ) / C N O R M 930 CONTINUE CLAM=CLAM/CNORM CONPAT (ICVG) =CLAM C R O T C = C R O T * S 3 * R H O A V / A R H O (1) ALAM (1)=2. D 0 * S 3 * R H O A V * C L A M * ( B C B * 1. DO) / (CNORM*ARHO (1) ) - C R O T C DO 940 1=2 ,NP 940 ALAM (I)= (CLAM*DMAG (I) - C R O T + AS (I) ) * A S (I) * A S ( I ) / A F X (I) I F ( ( . N O T . Q U I T ) . A N D . S H O R T ) RETURN HR(1)=-(BCB+1.DO)/CMORM*SURF*S0RF H T ( 1 ) = - H R (1 ) C A L L E N P O L E ( M A G , D M A G , H O , M O , E R M , E G ) C A L L E R O T A T ( M O , O M E G A , E G , E R R ) ER= ERM + ERR SC=0 .D0 C A L L F L U X P O ( N P , H O , S C , S U R F , FLUX) WRITE ( 6 , 3 2 0 0 ) 3200 FORMAT (• 1 •, 10X, T H E STREAM FUNCTION FOR THE MAGNETIC F I E L D 1 / / ) WRITE (6 ,3250) C L A M , C N O R M , F L U X 3250 FORMAT ( 1 X , ' C L A M = ' , D 1 0 . 3 , 1 0 X , ' C N O R M = « , D 1 0 . 3 , 1 0 X , * F L U X = • , D 1 0 . 3 ) W R I T E ( 6 , 3 2 0 8 ) E R M , E R R , H O 3208 F O R M A T ( 1 X , • M A G ENERGY R A T I O = » , D 1 0 . 3 , 1 0 X , • R O T E RATIO • , D 1 0 . 3 , 1 0 X , * $ • H 0 = « , F 9 . 0 , ' G A U S S ' / / ) RLAM=CLAM/CROT W R I T E ( 6 , 3 2 0 9 ) C R O T , R L A M , O M E G A 3209 F O R M A T ( 1 X , ' C R O T = ' , D 1 2 . 5 , 1 0 X , ' L A M MAG/LAM R O T = ' , D 1 2 . 5 , $ 1 0 X , « O M E G A = * , D 1 2 . 5 / / ) WRITE (6 ,3210 ) 3210 F O R M A T ( l 2 X , ' S ' , 1 9 X , » B ' , 1 9 X , ' D B / D S • , 1 5 X , • H R A D ' , 1 5 X , « H T A N ' , 1 5 X , 1 ' M A G ENERGY R A T I O ' / / ) WRITE (6 ,3220 ) ( A S ( I ) , M A G ( I ) , DMAG (I) , H R ( I ) , HT (I) , RAT 10 (I) , 1= 1 , NP) 3220 F O R M A T ( 1 X , 5 F 2 0 . 6 , D 2 0 . 6 ) I F ( . N O T . QUIT) RETURN C C c C THE P2 TERM OF THE G R A V I T A T I O N P O T E N T I A L C2=CNORM*CNORM C B 2 = B C B / C 2 - C R O T 2 / C L A M * C N O R M B C P O T = - (3. DO* (PP(NP) / C 2 + C B 2 * P H ( N P ) ) +SURF* (SOLY ( 7 ) / C 2 + C B 2 * S O L Y (5) ) ) B C P O T = B C P O T / ( 3 . D 0 * P O T 2 (NP)+SURF*SOLY (9) ) EPS (1 )= - ( (BCPOT + 1 . D 0 / C 2 - 1 . D 0 / C N O R M + C B 2 ) * C L A M * C N O R M + CROT2) $ * R H O A V * S 3 / A R H O ( 1) DO 950 1=2,NP 2 POT 2 (I) = B C P O T * P O T 2 ( I ) +PP (I) / C 2 + C B 2 * P H (I) 950 EPS (I) = -AS (I) * (POT2 (I) - C L A M * M A G (I) * C R O T 2 * A S (I) * A S (I) ) / A F X (I) R A T E Q U = 1 . 5 D 0*CROr * S 3 * ( 1 - . 5 D 0 * E P S ( H P ) ) * * 3 WRITE (6 ,3111) RATEQU 3111 FORMAT ( • - ' , ' R O T A T I O N A L F O R C E / G R A V FORCE AT EQUATOR I S : » , D 1 2 . 5 ) WRITE (6 ,4000) 4000 F O R M A T ( • - ' , ' T H E PATH OF CLAM W A S : ' ) WRITE (6 ,3999) ( I , CONPAT (I) ,1=1, ICONUM) 3999 F O R M A T ( 1 X , I 5 , D 2 0 . 5 ) WRITE (6 ,3110) 3110 F O R M A T ( ' 1 • , 9 X , » I » , 1 0 X , » E P S « , 1 7 X , • L A M B D A ' , 1 8 X , ' S ' , 1 5 X , • P O T 2 ' / / ) WRITE ( 6 , 3 1 0 0 ) ( I , EPS (I) , ALAM (I) , A S ( I ) , P O T 2 (I) ,1 — 1 , N P ) 3100 F O R M A T ( 1 X , I 1 0 , 2 D 2 0 . 5 , F 2 0 . 5 , D 2 0 . 5 ) E=EPS (NP) DPOT2=S2* ( B C P O T * S O L Y ( 9 ) +SOLY(7) / C 2 + C B 2 * S O L Y (5) ) HLAM=CLAM*DMAG(NP)*S2 BOT=CROT*S3 I F (MAGOUT) WRITE{7 ,1999) H O , M O , O M E G A 1999 F O R M A T ( 1 X , ' R O T A T I O N ' , 2 7 X , F 9 . 0 , 1 0 X , ' F O R M 0 = • , F 4 . 1 , 1 O X , $ ' O M E G A = • , D 1 2 . 5 ) I F ( MAGOUT) W R I T E ( 7 , 2 0 0 0 ) (AS (I) , M AG (I) , DMAG (I) , 1=1, NP) 2000 F O R M A T ( 3 D 2 5 . 1 6 ) RETURN END SUBROUTINE A U X R K ( Y , F ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) DIMENSION ABETR (1 00) DIMENSION Y (9) , F ( 9 ) , ARHO (100) , AS (100) , A N 1 (100) , A T ( 1 0 0 ) L O G I C A L Q U I T COMMON / R K B / C O N 1 , C O N 2 , R H O A V , S C COMMON / M A I N R K / A N 1 , A B E T R , A T COMMON / A L L / A S , A R H O , N P COMMON / L A G / N L , N M I N , N M A X COMMON / E T C / W T , Q U I T X=Y(1) X2=X*X F ( 2 ) = Y ( 3 ) C A L L L A G I N S ( A S , A R H O , X , D E N , N , N L , N M I N , N M A X , N P ) F(3) = 2 . D 0 * Y (2) / X 2 + D E N / R H O A V * X 2 I F ( Q U I T ) GO TO 10 DO 5 1=4,9 5 F ( I ) = 0 . D 0 RETURN 10 CONTINUE C POT2 IS A D I M E N S I O N L E S S P O T E N T I A L C A L L L A G I N T ( A S , A B E T R , B E T , N , N L , N P ) C A L L L A G I N T ( A S , A N 1 , E N 1 , N , N L , N P ) C A L L L A G I N T ( A S , A T , T , N , N L , N P ) F(4) =Y(5) F ( 6 ) = Y ( 7 ) F(8) =Y(9) I F ( T . L E . . 0 DO) GO TO 50 C A L C = D E N * W T / T * ( ( 3 . D 0 * B E T - 4 . D O ) / E N 1 + 1 . DO ) GO TO 60 50 C A L C = 0 . D 0 60 CPOT=CON1*CALC CB=CON2*CALC F ( 5 ) = - 2 . D0*Y (5) / X + 6 . D 0 * Y (4) / X 2 - C P O T * Y (4) +CB*X2 F(7) = - 2 . D 0 * Y ( 7 ) / X + 6 . D 0 * Y ( 6 ) / X 2 - C P O T * Y (6) +CB*Y (2) F ( 9 ) = - 2 . D 0 * Y ( 9 ) / X + 6 , D 0 * Y (8) / X 2 - C P 0 T * Y (8) RETURN END SUBROUTINE E R O T A T ( M O , O M E G A , E G , E R R ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 HO R E A L * 1 * S Q U A N K , T O L , F I F , F R O T , S O DIMENSION A S ( 1 0 0 ) , A R H O ( 1 0 0 ) COHMON / A L L / A S , A R H O , NP COMMON / L A G / N L , N M I N , N M A X EXTERNAL FROI C C O N S T A N T = 1 / 3 * R S U N * * 3 / ( G * M S U N ) C=8 .473758642D+5*OMEGA*OMEGA/MO NL=4 SMIN=NL/2+1 N M A X = N P - ( N L - 1 ) / 2 S0 = SNGL (AS (NP) ) E R O T = D B L E ( S Q U A N K ( F R O T , 0 . , S O , 0 . 0 , T O L , F I F ) ) ERR=C* E R O T / E G RETURN END FUNCTION F R O T ( S ) R E A L * 8 AS (100) , ARHO(100) , S S , D F COMMON / A L L / A S , A R H O , N P COMMON / L A G / N L , N M I N , N M A X SS=DBLE (S) C A L L L A G I N S ( A S , A R H O , S S , D F , N , N L , N M I N , N M A X , N P ) F R O T = S N G L ( D F ) * S * * 4 RETURN END $C * S K I P 

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