Документ взят из кэша поисковой машины. Адрес оригинального документа : http://jet.sao.ru/hq/lizm/conferences/pdf/1999/2000_2_p14-p23.pdf Дата изменения: Wed Feb 24 18:49:36 2010 Дата индексирования: Tue Oct 2 09:19:33 2012 Кодировка:

T h e i n v e r s e p r o b l e m o f D o p p l e r - Z e e m a n i m a g i n g o f m a g n e t i c C P s t a r s : m a t h e m a t i c a l m o d e l a n d m e t h o d o f s o l u t i o n
D . V . V a s i l ' c h e n k o , V . V . S t e p a n o v , V . L . K h o k h l o v a
a a a b

b

D e p a r t m e n t o f C o m p u t a t i o n a l M a t h e m a t i c s a n d C y b e r n e t i c s , M o s c o w S t a t e U n i v e r s i t y , M o s c o w , 1 1 9 8 9 9 R u s s i a I n s t i t u t e o f A s t r o n o m y , R u s s i a n A c a d e m y o f S c i e n c e s , P y a t n i t s k a y a u l . , 4 8 , M o s c o w , 1 0 9 0 1 7 R u s s i a

A b s t r a c t .

W e p r e s e n t t h e f o r m u l a t i o n a n d t h e d e v e l o p m e n t o f t h e i n v e r s e p r o b l e m s o l u t i o n

m e t h o d w h i c h p e r m i t s o n e t o d e t e r m i n e s u r f a c e c h e m i c a l a n o m a l i e s a n d m a g n e t i c f i e l d c o n f i g u r a t i o n i n r o t a t i n g C P s t a r s . T h e p r o b l e m i s r e f e r e n c e d a s a n i l l - p o s e d o n e . O b s e r v e d S t o k e s p a r a m e t e r s o f a b s o r p t i o n l i n e s i n s t e l l a r s p e c t r a a r e u s e d a s i n p u t i n f o r m a t i o n . T h e p r o p o s e d m a t h e m a t i c m o d e l l e a d s t o a s y s t e m o f n o n l i n e a r i n t e g r a l e q u a t i o n s f o r d e t e r m i n i n g l o c a l a b u n d a n c e s a n d l o c a l m a g n e t i c f i e l d v e c t o r s . A n a l y t i c a p p r o x i m a t i o n s a r e u s e d t o d e s c r i b e l o c a l S t o k e s p r o f i l e s a n d m a g n e t i c f i e l d c o n f i g u r a t i o n . T h e r e g u l a r i z e d i t e r a t i o n N e w t o n a l g o r i t h m i s u s e d t o s o l v e t h e i n t e g r a l e q u a t i o n s y s t e m . T h e I , V , U , Q - i n v e r t i n g c o d e w a s t e s t e d b y n u m e r i c a l m o d e l c o m p u t a t i o n s a n d t h e q u e s t i o n o f u n i q u e n e s s o f t h e s o l u t i o n i s s t u d i e d .

T h e f o r m u l a t i o n o f t h e i n v e r s e p r o b l e m i n t e r m s o f a s y s t e m o f n o n l i n e a r i n t e g r a l e q u a t i o n s ( K h o k h l o -

1 . I n t r o d u c t i o n
T h e s t a r s w e i n v e s t i g a t e b e l o n g t o t h e t y p e o f m a g n e t i c c h e m i c a l l y p e c u l i a r A a n d s t a r s w h i c h p o s s e s s l a r g e - s c a l e s t r o n g s u r f a c e m a g n e t i c f i e l d s a n d g r e a t a t m o s p h e r i c c h e m i c a l a n o m a l i e s i n h o m o g e n e o u s l y d i s t r i b u t e d o v e r t h e i r s u r f a c e s . T o c l e a r u p t h e p h y s i c a l n a t u r e o f t h e s e s t r a n g e p e c u l i a r o b j e c t s , i t i s n e c e s s a r y t o k n o w t h e c o n f i g u r a t i o n a n d t h e v a l u e o f m a g n e t i c f i e l d s a n d s u r f a c e d i s t r i b u t i o n o f c h e m i c a l e l e m e n t s i n t h e i r a t m o s p h e r e s . T h e o n l y s o u r c e o f s u c h a k n o w l e d g e i s a n a n a l y s i s o f s p e c t r o s c o p i c a n d p o l a r i m e t r i c o b s e r v a t i o n a l d a t a o n a b s o r p t i o n l i n e s i n t h e i r s p e c t r a . S u c h a n a n a l y s i s r e q u i r e s : 1 ) f o r m u l a t i o n o f a m a t h e m a t i c a l m o d e l o f l i n e p r o f i l e f o r m a t i o n i n t h e a t m o s p h e r e o f a r o t a t i n g s t a r i n t h e p r e s e n c e o f a m a g n e t i c f i e l d b a s e d o n t h e p h y s i c a l t h e o r y o f s p e c t r a l l i n e f o r m a t i o n i n a s t e l l a r a t m o s p h e r e a n d 2 ) d e v e l o p m e n t o f a m e t h o d o f s o l u t i o n o f t h e i n v e r s e p r o b l e m t o r e c o n s t r u c t t h e l o c a l S t o k e s p a r a m e t e r s a n d t o t r a n s f o r m t h e m i n t o l o c a l a b u n d a n c e s a n d l o c a l m a g n e t i c f i e l d v e c t o r s . T h e f i r s t a t t e m p t t o s o l v e t h i s p r o b l e m w a s m a d e b y D e u t s c h ( 1 9 7 0 ) w h o u s e d s p h e r i c a l f u n c t i o n s a n d F o u r i e r a n a l y s i s t o d e s c r i b e t h e o b s e r v e d c h a n g e s o f e q u i v a l e n t w i d t h s a n d m e a n l o n g i t u d i n 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 w i t h p e r i o d o f s t a r r o t a t i o n . B u t t h e s e i n p u t d a t a d i d n o t p e r m i t u s e o f a l l i n f o r m a t i o n c o n t a i n e d i n t h e o b s e r v e d S t o k e s p r o f i l e s . T h i s i s w h y D e u t s c h e m e t h o d c o u l d n o t b e u s e d l a t e r .

v a , 1 9 7 6 , 1 9 8 6 ) m a d e i t p o s s i b l e t o d e v e l o p t h e m e t h o d o f s o l u t i o n w h i c h u s e s a l l i n f o r m a t i o n c o n t a i n e d i n S t o k e s p r o f i l e s a s i n p u t i n f o r m a t i o n ( G o n c h a r s k i i e t a l . , 1 9 7 7 , 1 9 8 2 ; V a s i P c h e n k o e t a l . , 1 9 9 6 ) . T h i s m e t h o d w a s w i d e l y u s e d l a t e r u n d e r t h e n a m e o f D o p p l e r - Z e e m a n i m a g i n g ( V o g t e t a l . , 1 9 8 7 ; B r o w n e t a l . , 1 9 9 1 ; P i s k u n o v a n d R i c e , 1 9 9 3 ; K h o k h l o v a e t a l . , 1 9 9 7 , 2 0 0 0 ) . B e l o w w e d e s c r i b e i n m o r e d e t a i l t h e p o i n t s w h i c h h a v e b e e n o m i t t e d i n o u r p r e v i o u s p u b l i c a t i o n s b e c a u s e o f l a c k o f s p a c e b u t a r e u s e f u l f o r a b e t t e r u n d e r s t a n d i n g , a n d a l s o r e m o v e d i s c r e p a n c i e s i n n o t a t i o n a n d t e r m i n o l o g y .

2 . M a t h e m a t i c a l p r o b l e m

f o r m u l a t i o n

o f t h e

A c c o r d i n g t o t h e t h e o r y o f s p e c t r a l l i n e f o r m a t i o n i n t h e a t m o s p h e r e o f a r o t a t i n g s t a r i n t h e p r e s e n c e o f a m a g n e t i c f i e l d , t h e o b s e r v e d i n t e g r a t e d l i n e p r o f i l e a t e a c h m o m e n t i s t h e s u m o f l o c a l p r o f i l e s o v e r t h e v i s i b l e s t a r h e m i s p h e r e . L o c a l p r o f i l e s d e p e n d i n g o n t h e c o o r d i n a t e s o n a n i n h o m o g e n e o u s s t a r s u r f a c e a n d i n t h e p r e s e n c e o f a m a g n e t i c f i e l d a r e t h e r e s u l t o f p o l a r i z e d l i g h t r a d i a t i v e t r a n s f e r a t e a c h p o i n t o f t h e s t a r s u r f a c e a n d m a y b e c o m p u t e d b y s o l v i n g t h e r a d i a t i o n t r a n s f e r e q u a t i o n . I t i s k n o w n t h a t t h e p r o p e r t i e s o f p o l a r i z e d r a d i a t i o n c a n b e c o m p l e t e l y d e s c r i b e d b y f o u r S t o k e s p a r a m e t e r s :

© S p e c i a l A s t r o p h y s i c a l O b s e r v a t o r y o f t h e R u s s i a n A S , 2 0 0 0


1 20

I - o v e r a l l ( f u l l ) i n t e n s i t y a t a g i v e n w a v e l e n g t h , V - p e r c e n t a g e o f c i r c u l a r p o l a r i z e d r a d i a t i o n , U a n d Q - p e r c e n t a g e o f l i n e a r l y p o l a r i z e d l i g h t ( i n s o m e s p e c i a l l y c h o s e n o r t h o g o n a l d i r e c t i o n s , u s u a l l y c o n n e c t e d w i t h t h e o r i e n t a t i o n o f t h e o p t i c a l a x i s o f l i n e a r p o l a r i z a t i o n a n a l y z e r ) . T h e s e v a l u e s d e p e n d o n t h e w a v e l e n g t h i n s i d e a n a b s o r p t i o n l i n e p r o f i l e a n d n o p o l a r i z a t i o n e x i s t s i n a d j u s t i n g c o n t i n u o u s s p e c t r u m u n l e s s t h e m a g n e t i c f i e l d e x c e e d s s o m e 1 0 k G . T o o b s e r v e a l l f o u r S t o k e s p a r a m e t e r s i n a b s o r p t i o n l i n e s o f s t e l l a r s p e c t r a o n e m u s t u s e a n a l y z e r s o f p o l a r i z e d l i g h t w h i c h i n p r i n c i p l e a r e s i m i l a r t o t h o s e u s e d f o r s t u d y i n g s u n s p o t m a g n e t i c f i e l d s a s d e s c r i b e d , f o r e x a m p l e , i n t h e b o o k b y B r a y e t a l . ( 1 9 6 4 ) . I n t h e c a s e o f a s t a r t h e i n t e g r a l e q u a t i o n s d e s c r i b i n g t h e o b s e r v e d S t o k e s p a r a m e t e r s a r e :
5

p r o c e s s e s i n s i d e t h e a t m o s p h e r e o f a s t a r , b u t o n t h e o t h e r h a n d i t m u s t b e s i m p l e e n o u g h t o p e r m i t d e v e l o p i n g a n e f f i c i e n t n u m e r i c a l a l g o r i t h m f o r s o l v i n g t h e i n v e r s e p r o b l e m . I n p r e s e n t p u b l i c a t i o n w e c o n s i d e r t h e d e t a i l s o f p h y s i c a l s u b s t a n t i a t i o n o f t h e m a t h e m a t i c a l m o d e l w e u s e d a n d d e s c r i p t i o n o f o u r m e t h o d o f s o l u t i o n f o r t h e p r o b l e m o f D o p p l e r - Z e e m a n m a p p i n g .

3 . M a t h e m a t i c a l m o d e l
3 . 1 . D e s c r i p t i o n o f l o c a l S t o k e s p a r a m e t e r p r o f i l e s i n a s t a r a t m o s p h e r e T o c o m p u t e l o c a l S t o k e s p a r a m e t e r p r o f i l e s w h i c h s t a n d i n t h e i n t e g r a n d o n t h e r i g h t s i d e o f e q u a t i o n s ( 5 - 8 ) o n e s h o u l d w r i t e d o w n t h e s o l u t i o n o f t r a n s f e r e q u a t i o n f o r e a c h p o i n t M o n t h e s t a r s u r f a c e f o r e a c h r o t a t i o n p h a s e a t w h i c h a s p e c t r u m w a s t a k e n a n d a l l t h i s s h o u l d b e d o n e f o r e a c h s t e p o f a n i t e r a t i v e p r o c e s s . T h e m e t h o d o f n u m e r i c a l s o l u t i o n o f t h e t r a n s f e r e q u a t i o n i s w e l l e l a b o r a t e d n o w , b u t i t r e q u i r e s t o o m u c h c o m p u t e r t i m e b e i n g r e p e a t e d m a n y t i m e s . T h i s

T h e o b s e r v e d p h a s e d e p e n d e n t S t o k e s p a r a m e t e r s a r e o n t h e l e f t s i d e o f e q u a t i o n s , a n d i n t h e i n t e g r a n d o n t h e r i g h t s i d e a r e l o c a l p r o f i l e s t h a t a r e d e p e n d e n t o n t h e l o c a l a b u n d a n c e s , t h e D o p p l e r s h i f t m a g n e t i c f i e l d a n d t h e a t a p o i n t M o f t h e s u r f a c e .

i s w h y w e s t a r t e d t o u s e f i n i t e - d i m e n s i o n a l a p p r o x i m a t i o n f u n c t i o n s t o p r e s e n t t h e l o c a l p r o f i l e s o f t h e S t o k e s I p a r a m e t e r a t t h e v e r y b e g i n n i n g o f o u r w o r k ( G o n c h a r s k i i e t a l . , 1 9 7 7 , 1 9 8 2 ) . I t t u r n e d o u t t o b e c o n v e n i e n t t o p r e s e n t a l l f o u r S t o k e s p a r a m e t e r s b y t h e a n a l y t i c a l s o l u t i o n s o f t r a n s f e r e q u a t i o n s f o r p o l a r i z e d l i g h t o b t a i n e d b y U n n o ( 1 9 5 6 ) a n d c o m p l e m e n t e d b y c o n s i d e r a t i o n o f t h e m a g n e t o - o p t i c a l e f f e c t i n a s t e l l a r a t m o s p h e r e b y R a c h k o v s k i i ( 1 9 6 2 ) , L a n d o l f i a n d L a n d i D e g l ' I n n o c e n t i ( 1 9 8 2 ) r e s u m e d i n t h e p a p e r b y J e ff e r i e s e t a l . ( 1 9 8 9 ) . O u r c a l c u l a t i o n s h a v e s h o w n t h a t f o r e a r l y - t y p e s t a r s t h e F a r a d e y e f f e c t i s n e g l i g i b l e , a n d t h e n t h e s o l u t i o n a p p e a r s t o b e :

T h e S t o k e s p a r a m e t e r s n o r m a l i z e d t o a n o n p o l a r i z e d c o n t i n u u m a r e u s u a l l y m e a s u r e d a n d t h e y a r e c a l l e d S t o k e s p a r a m e t e r p r o f i l e s . T a k i n g i n t o a c c o u n t t h a t R ( , 0 ) = 1 - I / I C ( M , 0 ) a n d a s s u m i n g t h a t t h e s p e c i f i c i n t e n s i t y o f t h e c o n t i n u u m d o e s n o t d e p e n d o n c o o r d i n a t e s b u t o n a n g l e o n l y , o n e o b t a i n s f o r t h e p o l a r i z a t i o n p r o f i l e s t h e f o l l o w i n g e q u a t i o n s :

w h e r e t o r ,

i s t h e n o r m a l i z i n g f a c -

i s t h e c e n t e r - t o - l i m b c o n t i n u u m v a r i a t i o n l a w . F o r t h e p u r p o s e o f n o r m a l i z i n g w e a s s u m e t h a t I c ( 0 ) = 1 , a n d u 1 ( 0 ) i n t h e i n t e g r a n d o f ( 5 - 8 ) o n e

m a y c o n s i d e r a s t h e w e i g h t i n g f a c t o r . T h e s y s t e m o f e q u a t i o n s i s t o b e s o l v e d n u m e r i c a l l y b y a p r o p e r l y c h o s e n i t e r a t i o n m e t h o d a n d fo r t h i s a l l f u n c t i o n s s h o u l d b e w r i t t e n e x p l i c i t l y , s o t h e m a t h e m a t i c a l m o d e l s h o u l d b e f o r m u l a t e d i n d e t a i l s . T h e s y s t e m ( 5 - 8 ) o f f o u r i n t e g r a l e q u a t i o n s d o e s p e r m i t d e t e r m i n a t i o n o f t h e l o c a l p r o f i l e s o f f o u r S t o k e s p a r a m e t e r s w h i c h i n t u r n a r e d e t e r m i n e d b y f o u r s c a l a r s : l o c a l a b u n d a n c e a n d t h r e e c o o r d i n a t e s o f t h e l o c a l m a g n e t i c f i e l d v e c t o r . T h e s u c c e s s i n s o l v i n g t h e p r o b l e m i s g r e a t l y d e p e n d e n t o n t h e p r o p e r c h o i c e o f m a t h e m a t i c a l m o d e l . F i r s t l y i t m u s t p r o v i d e a n a d e q u a t e d e s c r i p t i o n o f t h e L e t u s c o n s i d e r f i r s t a s i m p l e r c a s e o f m a p p i n g a b u n d a n c e a n o m a l i e s w h e r e t h e m a g n e t i c f i e l d i s s m a l l o r t h e l i n e u s e d h a s a s m a l l L a n d e f a c t o r , s o T h e s e a n a l y t i c a l s o l u t i o n s w e r e o b t a i n e d f o r a s i m p l i f i e d l i n e f o r m a t i o n m o d e l u n d e r t h e a s s u m p t i o n o f d e p t h - i n d e p e n d e n t r a t i o o f s e l e c t i v e t o c o n t i n u u m a b s o r p t i o n c o e f f i c i e n t s a n d a l s o l i n e a r d e p t h d e p e n d e n c e o f s o u r c e f u n c t i o n i n c o n t i n u u m


121

th e effect o f magneti c field i s negligible . The n th e so lutio n o f onl y equatio n (9) permit s on e t o obtai n a ma p o f chemica l anomalie s (Dopple r mapping) . I n thi s cas e al l term s whic h accoun t for th e mag neti c fiel d i n equatio n (9 ) tur n t o b e zero . Remem berin g tha t an d afte r simpl e transformation s on e obtain s for I Stoke s paramete r profile :

no t differ muc h fro m th e atmosphere s o f norma l mai n sequenc e stars , an d wel l elaborate d theoretica l atmo spher e model s suc h a s Kuruc z (1992 ) model s wit h appropriat e parameter s T an d l og g ma y b e use d i n thi s case . Th e mos t explici t wa y o f usin g theoretica l profile s for Dopple r imagin g wa s take n b y Hatze s (1991 ) an d als o b y Piskuno v an d Ric e (1993 ) wh o precalculate d an d store d i n memor y a gri d o f profile s a s a functio n o f abundanc e an d i n th e proces s o f iteration s retrieve d an d interpolate d dat a fro m tables .
ef f

wher e Not e tha t expressio n (13 ) resemble s b y it s struc tur e th e empirica l Minnaer t (1935 ) formul a use d i n ou r earlie r wor k (Goncharski i e t al. , 1977 , 1982) :

Thi s formul a wa s propose d b y Minnaer t t o ap proximat e lin e profile s i n a sola r spectrum , an d h e measure d R (th e centra l lin e depth ) directl y fro m thi s spectrum . B y th e definitio n 0 < R < 1 . Comparin g (13 ) an d (14 ) on e ma y sugges t tha t i n th e cas e o f (13 ) th e firs t facto r als o play s th e rol e o f th e lin e centra l dept h whic h depend s (nonlinearly ) o n th e numbe r o f absorbin g atoms . W e hav e foun d i t convenien t t o us e a s a n approximatin g functio n th e expression :
c c

I n ou r metho d w e als o precalculat e b y numeri ca l integratio n a gri d o f Stoke s specifi c intensit y pro file s R for a se t o f abundance s for eac h lin e w e us e for mappin g an d w e approximat e thes e profile s b y formul a (15) . Th e specia l cod e ha s bee n develope d t o choos e parameter s o f th e approximatin g functio n whic h wa s describe d i n detail s i n sectio n 2.2. 1 o f ou r pape r (Khokhlov a e t al. , 1997) .
I

I t i s possibl e t o choos e a n approximatin g functio n : expressin g th e dependenc e o f R o n
c

Thes e parameters , excep t , d o no t depen d o n co ordinate s o n th e sta r surface . Practicall y the y d o no t depen d o n th e valu e o f , (th e lin e intensity ) either . Th e dependenc e o f th e lin e profil e o n i s take n int o . Thi s functio n present s accoun t b y th e facto r a linea r o r quadrati c expansio n b y = cos ( ) , coefficient s bein g foun d fro m th e se t o f theoretica l pro file s for eac h particula r line . Th e lin e ma y becom e stronge r o r weake r fro m cente r t o limb , dependin g o n ionizatio n an d excitatio n potentials . Thi s techniqu e i s als o demonstrate d i n Fig . 2 i n th e pape r b y Khokhlo v a e t al . (1997) . I t i s clea r tha t whe n th e rol e o f a magneti c fiel d canno t b e neglected , an d i t i s necessar y t o solv e si multaneousl y th e fou r equation s (5-8) , th e proble m i s gettin g muc h mor e complicated . I n thi s cas e th e lo ca l polarizatio n profile s depen d no t onl y o n th e loca l valu e o f th e magneti c vecto r a t poin t M bu t als o o n a n instantaneou s valu e o f th e angl e betwee n thi s vecto r an d th e lin e o f sigh t a t poin t M , whic h change s dur in g th e sta r rotation . I n thi s cas e th e precalculatio n o f th e loca l profiles , retrieva l an d interpolatio n fro m table s i n th e proces s o f iteration s becom e unrealisti c eve n for bi g computers . Applicatio n o f analytica l ap proximation s i s practicall y th e onl y wa y t o solv e th e problem . T o describ e th e Stoke s profile s i n th e cas e o f mag neti c field , w e us e analytica l solution s (9-12) . I n th e = th e function s quantitie s ar e convolution s o f eac h componen t o f th e Zeema n patter n spli t b y th e loca l magneti c fiel d a t th e momen t o f phas e wit h a Voigh t profil e (th e grou p o f component s a s wel l a s th e right - an d leftpolarize d component s ar e treate d separately) . Th e Zeema n pattern s an d relativ e intensitie s o f compo nent s ar e know n fro m th e classica l physic s (for ex ample , se e Condo n an d Shortley , 1951) , an d for L S

wher e C i s th e centra l dept h o f a ver y saturate d lin e an d C i s th e properl y chose n numerica l parameter . Thes e analytica l expression s describ e wel l th e de pendenc e o f th e lin e profil e o n th e abundanc e o f a n i s proportiona l t o th e numbe r o f lin e element , a s formin g absorbin g atoms . Thi s permit s t o b e con sidere d a s on e o f th e principa l value s t o b e determine d whe n solvin g th e invers e problem . Expressio n (13) , nevertheless , canno t describ e rig orousl y th e center-to-lim b variatio n o f a loca l lin e pro file becaus e th e abov e analytica l solutio n take s int o accoun t onl y th e angl e for th e intensit y i n th e con tinuum . Beside s th e assumptio n tha t th e sourc e func tio n i s linearl y dependen t o n dept h ma y b e wron g i n th e uppe r atmospheri c layer s wher e stron g line s form . Thi s ma y caus e difficultie s an d error s whe n estimat foun d fro m (13) . S o on e in g loca l abundance s usin g naturall y need s t o connec t profile s (13 ) wit h th e pro files obtaine d b y numerica l integratio n o f th e transfe r equatio n for a mor e adequat e atmospher e model . Numerou s observationa l dat a sho w tha t th e atmo sphere s o f C P star s wit h moderat e magneti c field s d o
1 2


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coupling the y wer e calculate d b y Becker s (1969) . Al l formulae w e use d for thi s genera l cas e ar e give n i n ou r paper s (Vasil'chenk o e t al. , 199 6 an d Khokhlov a e t al. , 1997). I n conclusio n o f thi s sectio n w e not e som e state ment s whic h justif y ou r us e o f analytica l approxima tion s o f th e loca l Stoke s profiles : 1 . Th e transfe r o f ligh t o f on e o f polarizatio n state s ( + , , o r -- component ) ma y b e considere d independentl y o f eac h other . 2 . Th e intensit y profil e o f eac h polarize d Zeema n componen t i s forme d i n th e sam e wa y a s th e profil e o f intensit y o f unpolarize d light . 3 . Th e differenc e betwee n Miln-Eddingto n atmo spher e an d a mor e sophisticate d moder n compute d sta r atmospher e i s mor e importan t for paramete r an d henc e th e intensit y profile . Bu t well chose n pa rameter s o f approximatio n describe d b y Khokhlov a e t al . (1997 ) mak e th e differenc e betwee n intensit y profile s rathe r small . W e ma y refe r als o t o th e re sult s reporte d b y Hardor p (1976) , whic h sho w tha t th e numerica l solutio n o f th e transfe r equatio n for a magneticall y splitte d profil e an d analytica l formul a o f Unn o typ e lea d t o simila r results . 4 . Th e assumptio n tha t th e atmospher e mode l i s independen t o f coordinate s i s doubtfu l whe n th e mag neti c fiel d an d chemica l patche s o n th e sta r surfac e ar e strong . On e canno t b e sur e tha t th e loca l profile s calculate d b y numerica l solutio n o f th e transfe r equa tio n for on e fixe d atmospher e mode l ar e vali d for th e whol e star , n o matte r ho w the y ar e used : b y extrac tio n fro m table s o r b y analytica l approximation . 5 . W e hav e foun d tha t th e magneti c fiel d con figuratio n obtaine d fro m line s wit h differen t Zeema n pattern s i s practicall y th e same . Thi s i s a n evidenc e tha t n o gros s error s aris e du e t o ou r analytica l ap proximatio n o f loca l Stoke s profiles . 3.2 . Analytica l descriptio n o f magneti c fiel d configuratio n All measurement s o f effectiv e magneti c field s o f C P star s tha t hav e bee n mad e u p t o no w sho w tha t mos t o f the m hav e large-scal e regula r dipola r mag neti c fiel d structures , bu t no w a few star s ar e know n wit h a quadrupol e component . Fo r example , i n th e B2 V He-variabl e C P sta r H D 3777 6 th e quadrupol e componen t i s dominant , an d eve n a n octupol e compo nen t wa s suspecte d (Thompso n an d Landstreet , 1985 ; Khokhlov a e t al. , 2000 ) Thi s make s i t natura l t o searc h magneti c configu ratio n a s a n expansio n o f spherica l harmonic s o f mag neti c multipol e potential s t o a n arbitrar y hig h orde r (Bagnul o e t al. , 1996) . I n principle , thi s permit s on e t o describ e an y fiel d configuration , bu t takin g int o accoun t highe r numbe r o f multipole s on e get s insta bilit y o f th e solutio n du e t o incompletenes s o r inac -

curac y o f inpu t data . I t wa s show n (Khokhlov a e t al. , 2000 ) tha t i n th e cas e o f H D 3777 6 havin g a dominan t quadrupol e field , th e additio n o f octupole-produce d instabilit y (i n th e sens e tha t octupol e vector s derive d fro m differen t spectra l line s sprea d ove r a bi g area) . Thi s questio n i s considere d i n mor e detail s i n sectio n 5 whe n discussin g th e proble m o f uniquenes s o f th e solution .

4 . Metho d o f solutio n
Fro m th e mathematica l poin t o f vie w th e mappin g proble m lead s t o th e syste m o f integra l equations :

Her e th e unknown ment s -- functio n o f positio n an d ove r th e sta r i n equation s (17-20 )

s ar e function s o f tw o argu distributio n o f chemica l com distributio n o f magneti c hel d surface . Th e function s ca n b e writte n i n th e for m , wher e th e function s

ar e denne d i n (3.1 ) an d represen t non linea r function s o f thei r arguments . Thu s th e syste m o f integra l equation s (17-20 ) i s non-linear . I n thi s metho d w e us e parametri c representatio n o f th e magneti c fiel d o n th e sta r surfac e whic h ma y b e eithe r a displace d dipol e mode l o r decompositio n t o axiall y symmetri c spherica l harmonic s u p t o th e thir d orde r (dipole , quadrupol e an d octupol e moments) . In cludin g o f highe r order s i n ou r cod e i s als o possible . Le t u s denot e th e se t o f parameter s tha t defin e mag neti c fiel d a s h R whe n usin g th e displace d dipol e mode l an d a s h R for spherica l harmoni c decom position . Designatin g integra l operator s i n (17-20 ) a s , w e writ e i t i n a mor e compac t form :
6 9

W e wil l suppos e tha t al l th e observe d profile s belon g t o , where i s define d a s Th e valu e hal f o f th e full widt h o f th e spectra l lin e unde r inves tigation . Th e choic e o f th e spac e L i s determine d b y th e usag e o f mea n squar e metri c t o measur e th e dis crepanc y betwee n th e observe d an d synthesize d pro files . T o solv e th e invers e problem , w e hav e als o t o
2


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determin e a spac e be of functio n Z= z = (h, ) spac e T on e ha s aspects :

th e solutio n spac e T , an R x T (Z t o tak e int o
6

wil l belon g to . Le t th e d th e se t o f unknown s = R x T). To choos e accoun t th e followin g
9

· operator s F ar e t o b e define d i n Z o r it s close d subset ; · convergenc e i n spac e T shoul d guarante e a de sire d convergenc e o f th e approximat e solutions ; · th e effective algorith m o f th e solutio n o f th e invers e proble m for non-linea r integra l equation s i n spac e T exists .
x

o n circula r polarizatio n i n additio n (numerica l exper iments) . Th e propertie s o f th e operator s F : Z --- > L mak e u s us e specia l algorithm s t o solv e th e mini mizatio n proble m whic h guarante e stabilit y o f th e ob taine d approximat e solution s (Tikhono v e t al. , 1995) . Regularize d method s t o minimiz e th e discrepanc y functiona l
x 2

Her e w e us e th e Hilber t spac e L a s T , wher e i s th e surfac e o f th e uni t sphere . Thi s spher e ca n b e parameterized , fo r instance , b y longitud e an d latitud e L . Th e mai n reaso n for thi s choic e i s th e simplicit y o f th e numerica l implementatio n o f th e al gorithm . I t i s eas y t o se e tha t integra l operator s F ar e continuou s an d eve n completel y continuou s i n th e . Unfortunately , us e o f spac e pai r o f space s Z -- > L T= L doe s no t guarante e unifor m o r eve n point wis e convergenc e o f approximat e solutions . T o forc e stronge r convergenc e on e need s t o us e stronge r met whil e devel ric s i n spac e T , for exampl e T = W opin g method s o f solutio n for th e invers e problem . A t th e presen t tim e onl y wide-ban d dat a o f linea r polarizatio n measurement s ( U an d Q Stoke s param eters ) ar e available , an d ther e ar e onl y a few star s tha t provid e sufficien t dat a o n circula r polarization . Therefor e w e nee d sometime s t o solv e th e proble m o f findin g chemica l compositio n an d magneti c fiel d o f a sta r b y a n incomplet e se t o f inpu t Stoke s parameters . Th e followin g case s o f indice s X ma y b e realisti c t o describ e differen t statement s o f th e proble m whe n differen t observationa l dat a o f Stoke s parameter s ar e available :
2 x 2 2 2 2

ar e use d t o solv e non-linea r ill-pose d problems . on e o f th e mos t effective I n th e cas e T = L method s o f solvin g th e syste m o f non-linea r integra l equation s (21 ) i s th e Newton' s iterativ e metho d a s de scribe d for exampl e b y Bakushinski i an d Goncharski i (1994) . Give n th e curren t approximatio n z , th e nex t on e ca n b e calculate d usin g th e formul a
2 k

i s th e derivativ e o f th e Her e F a t poin t z , an d : i s th e operato r conjugat e t o o f positiv e number s tendin g t o zero .
x

operato r Z is th e sequenc e Operator s

exis t an d ar e contin . T o guaran uou s owin g t o th e positiv e valu e o f te e th e stabilit y o f th e approximat e solutions , th e sequenc e mus t no t decreas e to o fast . Generall y basin g o n th e investi speaking , on e ha s t o choos e gation s o f th e propertie s o f th e non-linea r operator s F , bu t i t i s well know n tha t 1/ usuall y pro vid e convergenc e an d stabilit y o f th e approximat e so lution s i f th e firs t approximatio n i s chose n sufficientl y clos e t o th e exac t solution . W e use d th e sequenc e
x

- onl y informatio n for non-polarize d spectr a i s available ; - bot h intensit y an d circula r polarizatio n pro file s ar e available ; for no w thi s i s th e mos t usua l case ; - all Stoke s lin e profile s ar e availabl e - thi s i s th e mos t favourabl e cas e t o solv e th e invers e problem . Bu t measuremen t o f th e U an d Q Stoke s lin e profile s i s a rathe r difficult technica l proble m an d th e se t o f thes e dat a ma y b e replace d b y integrate d broad-ban d linea r polarizatio n data . Ou r experiment s showe d tha t i f a sta r ha s a sufficientl y stron g magneti c field, s o tha t magneti c split tin g o f Zeema n component s i s greate r tha n rotationa l Dopple r widths , on e ca n fin d chemica l compositio n an d magneti c field eve n whe n onl y I Stoke s parame te r i s availabl e (th e cas e o f Babcoc k star , Khokhlov a e t al. , 1997) . T o solv e th e mappin g proble m for star s wit h relativel y wea k magneti c field s on e need s dat a

wher e th e valu e o f wa s foun d b y numerica l exper iments . Iterativ e algorithms , use d t o solv e ill-pose d prob lems , ar e t o b e supplie d b y th e so-calle d stoppin g rule . Thi s implie s tha t on e ha s t o tak e a s th e approximat e solutio n o f th e ill-pose d proble m th e iteratio n o f (22 ) wit h th e numbe r k , tha t correspond s t o th e observa tiona l dat a precision . Th e mor e precis e i s th e inpu t information,th e greate r numbe r o f iteration s mus t b e calculate d b y (22 ) t o ge t a stabl e approximat e solu tion . Th e metho d whic h use s (22) , (23 ) guarantee s th e stabilit y o f th e approximat e solutio n i f w e us e a stop wher e i s th e precisio n pin g rul e lik e


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o f th e observationa l dat a i n mean-squar e metric . W e use d

i s no t sufficien t enough . Hence , th e followin g modifi catio n ha s bee n develope d for furthe r usage . I t wa s decide d t o us e th e smoothin g functiona l i n th e followin g form :

an d adjuste d th e valu e o f n empirically . A s i t wa s mentione d abov e th e metho d usin g (22 24 ) give s a stabl e solutio n onl y i f th e firs t approxi matio n z i s chose n t o b e clos e t o th e unknow n exac t solution . Unfortunately , i n practic e i t i s impossibl e t o verify condition s impose d o n th e firs t approximatio n b y Bakushinski i an d Goncharski i (1994 ) du e t o th e to o comple x for m o f th e operator s F (z). Numerica l experiment s sho w tha t th e direc t applicatio n o f th e metho d (22-24 ) coul d b e successfu l onl y i f a ver y goo d firs t approximatio n i s available . T o solv e integra l equation s (21 ) i n th e cas e wher e th e qualit y o f th e firs t approximatio n z canno t b e a prior i estimated , w e use d th e regularize d iterativ e metho d i n th e form :
o o x 0

Th e additiona l ter m wit h th e gradien t o f unknow n doe s no t allo w larg e oscillation s o f th e functio n loca l abundanc e whe n minimizin g M(z). A s well i t smoothe s th e edge s o f abundanc e distribution . Reg ularizatio n o f th e zer o orde r i n th e for m o f (25 ) ma y lea d t o a mor e "detailed " distributio n especiall y i f th e inpu t informatio n erro r leve l i s underestimated . Bu t thes e detail s usuall y hav e artificia l o r computa tiona l origi n resultan t fro m th e ill-pose d natur e o f th e proble m an d canno t b e regarde d a s rea l ones . W e d o believ e tha t on e ha s t o fin d a s muc h smoot h solution s o f (21 ) a s possible . I n thi s cas e th e regularize d Newton' s metho d ca n b e writte n a s

wit h th e constan t regularizatio n paramete r a . Thi s i s jus t th e Newton' s metho d t o minimiz e smoothin g functiona l a s describe d b y Tikhono v e t al . (1995) . Th e regularizatio n paramete r wa s chose n base d o n numer ica l experiments . W e use d discrepanc y metho d a s a stoppin g rule . Accordin g t o thi s metho d on e ha s t o continu e itera tion s (22 ) o r (25 ) unti l th e discrepanc y reache s th e valu e o f th e precisio n o f inpu t information , tha t i s

Th e followin g expressio n for th e sequenc e use d ( bein g chose n empirically) :

wa s

Thi s metho d ha s prove d t o b e effective whil e solvin g a lo t o f applie d problems . Non-adherin g t o th e stop pin g rul e an d th e us e o f to o man y iteration s lead s t o instabilit y o f th e obtaine d solution s an d therefor e t o false decision s o n th e structure s o f th e surfac e distri bution s i n th e sta r atmosphere . Not e tha t th e possibilit y o f choosin g th e numbe r o f iteration s fro m conditio n (26 ) depend s o n th e ad equac y o f th e use d mathematica l mode l an d i n tur n ca n serv e t o prov e thi s adequacy . W e hav e performe d a lo t o f numerica l experiment s an d hav e investigate d th e propertie s o f method s (22) , (23) , (26 ) an d (25) , (26 ) for differen t mode l distribu tions . I t appear s tha t th e firs t combinatio n o f (22) , (23) , (26 ) allow s on e t o investigat e ver y detaile d dis tributions , bu t require s a goo d firs t approximation . Th e secon d combinatio n o f (25) , (26 ) i s no t sensibl e t o th e firs t approximation , bu t make s convergenc e to o slow . I t als o becam e clea r tha t th e convergenc e o f th e approximat e solution s i n th e metri c o f th e spac e L
2

Thi s sequenc e (29 ) assure s tha t th e regularizatio n paramete r decrease s a s th e approximatio n tend s t o th e exac t solution . W e use d a stoppin g rul e i n th e for m (26) . W e use d th e followin g finite-dimensio n approx imatio n o f th e dat a t o implemen t th e numerica l wa s metho d o f solvin g (21) . Th e functio n approximate d a s a piecewis e constan t functio n o n th e rectangula r gri d o n th e visibl e surfac e o f a sta r o f dimensio n an d wa s represente d a s th e vecto r n= n x . Inpu t dat a -- Stoke s lin e profile s -- an d eac h wer e give n for th e se t o f rotatio n phase s o f thes e profile s wa s represente d a s a se t o f value s a t th e wavelength s i n th e vicinit y o f th e centra l wave . Thus , th e inpu t dat a ar e als o th e finite lengt h dimensio n vecto r . No w w e ca n approximat e non linea r operator s F a s function s o f th e finit e numbe r o f argument s n + 6 . Th e derivative s o f th e operator s ar e approximate d a s th e matri x o f partia l derivative s an d th e conjugat e operato r ca n easil y b e foun d a s conjugatio n o f matrices .
L x


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A s w e hav e alread y mentioned , th e matri x approx imatin g i s well-conditione d du e t o th e > 0 an d w e ca n us e standar d algorithm s o f linea r algebr a t o invers e th e matri x whe n solvin g th e syste m o f linea r equations .

5 . O n uniquenes s o f th e solutio n an d testin g o f th e metho d b y mode l com putation s
Th e questio n o n uniquenes s o f th e solutio n obtaine d b y Doppler-Zeema n imagin g i s obviousl y o f a grea t importanc e an d i t shoul d b e thoroughl y studied . Th e answe r wil l certainl y depen d o n th e precisio n an d completenes s o f th e observe d inpu t information . Usin g analytica l expression s for th e magneti c fiel d a s expansio n i n serie s o f spherica l harmonic s o f mag neti c multipole s potential s ma y lea d t o a situation , whe n variou s set s o f multipole s creat e a magneti c fiel d configuratio n whic h i s simila r t o th e rea l one , thu s providin g severa l loca l minim a o f th e residual . I t i s importan t for u s i n suc h a cas e t o kno w tha t thes e configuration s ar e reall y ver y clos e t o eac h other . I n suc h a cas e on e ma y spea k no t abou t ambiguou s so lutio n bu t rathe r abou t ambiguou s mathematica l de scriptio n o f th e sam e tru e solution . Th e mos t convenien t wa y t o stud y th e proble m o f uniquenes s o f th e solution , a s i t wa s mentione d ear lier, i s numerica l testin g o f variou s models . W e hav e foun d b y suc h testin g tha t non-uniquenes s appeare d dependin g o n th e mode l whic h i s use d a s a n initia l approximation : i f i t i s take n t o b e no t to o fa r fro m a tru e model , th e solutio n converge s t o th e tru e model . I t i s convenien t t o tak e a s suc h initia l approximatio n th e homogeneou s abundanc e distributio n equa l t o th e observe d average d ove r th e sta r dis k abundanc e an d roughl y visuall y gues s estimatio n fro m th e observe d V profile s o f th e positio n an d strengt h o f magneti c poles . I f th e initia l approximatio n i s take n arbitrar y t o b e fa r fro m th e tru e model , the n th e minimizatio n o f th e residua l ma y lea d t o false minima . Belo w ar e som e example s o f suc h situations . Th e model s whic h wer e teste d ar e show n i n Tabl e 1 . Mag neti c field s wer e forme d b y combination s o f non-axia l bu t centere d dipole , quadrupol e an d octupol e wit h an d pola r strengt h H (kG) . coordinate s L , Th e testin g wa s mad e for th e angl e i = 45 ° an d V sini = 50km/s . Th e "observed " profile s o f th e
p equ

S i II I lin e 457 4 (norma l Zeema n triplet ) wer e com pute d for th e atmospher e o f a B2 V star , simila r t o H D 37776 , studie d b y u s (Khokhlov a e t al. , 2000) . Mode l Stoke s profile s wer e compute d for thre e kind s o f abundanc e distribution : a ) a spo t wit h ten-fol d underabundanc e (m2a) , b ) a spo t wit h ten-fol d overabundanc e (m2b) , c ) a highl y overabundan t stri p locate d alon g th e positio n o f th e maximu m tangentia l magneti c fiel d componen t (m2c) . Th e compute d mode l profile s wer e use d a s inpu t in formatio n t o solv e th e mode l invers e problem . I n Ta bl e 2 th e model s use d a s initia l approximation s ar e shown , an d th e solution s obtaine d ar e give n i n Ta bl e 3 . I n th e firs t colum n o f Tabl e 3 th e uppe r ro w in dicate s th e tes t number , th e nex t ro w i s th e mode l teste d fro m Tabl e 1 , an d th e lowe r ro w indicate s th e initia l approximatio n mode l for th e magneti c fiel d an d abundanc e distribution . I n th e secon d an d thir d column s solution s (multi -


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p o l e s a n d t h e i r c o o r d i n a t e s ) a r e g i v e n . I n t h e f o u r t h c o l u m n a r e t h e r e s i d u a l s a n d i n t h e fifth i s t h e t y p e o f s o l u t i o n ( I , V , U , Q - i n v e r s i o n o r I , V i n v e r s i o n ) F i g u r e s 1 -7 d e m o n s t r a t e t h e o r i g i n a l a b u n d a n c e a n d t h e m a g n e t i c f i e l d c o n f i g u r a t i o n t o g e t h e r w i t h t h o s e r e c o n s t r u c t e d b y t h e i n v e r s i o n p r o c e d u r e . T h e e x a m p l e s N o . l a n d N o . 2 s h o w t h a t i f t h e i n i t i a l a p p r o x i m a t i o n s a r e c h o s e n p r o p e r l y , t h e s o l u t i o n s c o n v e r g e t o t h e o r i g i n a l m o d e l s p r e c i s e l y e v e n w i t h o u t l i n e a r p o l a r i z a t i o n i n p u t d a t a . T h e m i n i m i z a t i o n o f t h e r e s i d u a l s w a s m a d e b y t h e I a n d V S t o k e s p r o f i l e s o n l y i n t h a t c a s e , b u t t h e U a n d Q p r o f i l e s w e r e c o m p u t e d a n d c o m p a r e d a s w e l l . T h e m i n i m i z a t i o n b y I a n d V w a s e n o u g h t o d e c r e a s e r e s i d u a l s f o r U a n d Q . I n t h e e x a m p l e s N o . 3 - N o . 5 t h e i n i t i a l a p p r o x i m a t i o n s a r e f a r t h e r f r o m t h e o r i g i n a l m o d e l , o r t h e m o d e l i s m o r e c o m p l e x . I n t h e s e c a s e s r e s i d u a l s a r e a b o u t a n o r d e r o f m a g n i t u d e l a r g e r t h a n i n t h e c a s e s N o . l a n d N o . 2 . I t i s i m p o r t a n t t o n o t e t h a t t h e y s t i l l r e m a i n s m a l l e r t h a n t h e p o s s i b l e e r r o r i n t h e a v a i l a b l e o b s e r v e d p r o f i l e s . B u t w h a t i s m o r e i m p o r t a n t , t h e s e s o l u t i o n s d e s c r i b e m a g n e t i c f i e l d c o n f i g u r a t i o n s w h i c h a r e c l o s e e n o u g h t o t h e o r i g i n a l " t r u e " m o d e l s . T h e e x a m p l e s N o . 6 a n d N o . 7 s h o w t h a t i n t h e a b s e n c e o f a n o c t u p o l e c o m p o n e n t i n t h e m a g n e t i c f i e l d c o n f i g u r a t i o n i t i s e a s i e r t o g e t a " t r u e " s o l u t i o n . T h e t e s t i n g s h o w e d t h a t t h e d i f f e r e n c e i n t h e a b u n d a n c e d i s t r i b u t i o n d o e s n o t i n f l u e n c e n o t i c e a b l y th e d e te r m in e d m a g n e tic c o n fig u r a tio n . A t le a s t tw o r e c o m m e n d a t i o n s f o l l o w f r o m t h e a b o v e r e s u l t s o f t e s t i n g , w h i c h m a y h e l p f i n d " t h e b e s t " s o l u t i o n : 1 . I f o n l y o n e s p e c t r a l l i n e o f a n e l e m e n t i s u s e d fo r D . - Z . m a p p i n g , i t i s n e c e s s a r y t o c o m p a r e t h e r e s u l t s o b t a i n e d w i t h d i f f e r e n t i n i t i a l a p p r o x i m a t i o n s . 2 . S e v e r a l l i n e s o f o n e e l e m e n t s h o u l d b e u s e d f o r m a p p i n g . A n o t h e r i m p o r t a n t t e s t s d e s c r i b e d a b o v e : f i g u r a t i o n o f m o d e l s c a r e s i d u a l i s v e r y s m a l l e r r o r s f o r r e a l s t a r s . c o n c l u s i o n f o l l o w s f r o m t h e f i n e d e t a i l s o f m a g n e t i c c o n n b e r e v e a l e d o n l y w h e n t h e -- m u c h b e l o w o b s e r v a t i o n a l

o f c h e m i c a l a n o m a l i e s . T h e l i m i t i s s e t a t t h e p r e s e n t t i m e m a i n l y b y o b s e r v a t i o n a l e r r o r s , a n d a l s o b y a p o o r k n o w l e d g e o f t h e l o c a l a t m o s p h e r e s t r u c t u r e .

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B a g n u l o S., L a n d i D e g l ' I n n o c e n t i M ., L a n d i D e g l ' I n n o c e n t i E ., 1 9 9 6 , A s t r o n . A s t r o p h y s . , 3 0 8 115 B a k u s h i n s k i i A . B . a n d G o n c h a r s k i i A . V . , 1 9 9 4 , " I I I - p o s e d p r o b l e m s : T h e o r y a n d a p p l i c a t i o n s " , K l u w e r B e c k e r s J . M . , 1 9 6 9 , A T a b l e o f Z e e m a n M u l t i p l e t s , S a c r a m e n t o P e a k O b s e r v . , 6 4 9 B o h l e n d e r D . , 1 994, P u l s a t i o n , R o t a t i o n a n d M a s s L o ss i n E a r l y - T y p e S t a r s , I A U S y m p . N o l 9 2 , e d s . : B a l o n a L .A ., e t a l . , 1 5 5 B r a y R ., L o u g h h e a d R . , 1 9 6 4 , S u n s p o t s , C h a p m a n a n d H a ll L T D , L o n d o n B r o w n S . F . , D o n a t i J . - F . , R e e s D . E . , S e m e l M ., 1 9 9 1 , A s t r o n . A s t r o p h y s . , 2 0 , 4 6 3 C o n d o n E . U . , S h o r t l e y G . H . , 1 9 5 1 , T h e T h e o r y o f A t o m i c S p e c t r a , C a m b r i d g e U n i v e r s i t y P r e s s , E n g l a n d D e u t s c h A ., 1 970, A s t r o p h y s . J . , 1 5 9 , 8 9 5 G o n c h a r s k i i A . V . , S t e p a n o v V . V . , K h o k h l o v a V . L . , Y a g o la A . G . , 1 9 77, S o v i e t A s t r o n o m y L e t t e r s , 3 ( 3 ) , 1 47 G o n c h a r s k i i A . V . , S t e p a n o v V . V . , K h o k h l o v a V . L . , Y a g o l a A . G . , 1 9 8 2 , S o v . A s t r o n . , 1 9 , 5 7 6 J e ffe r ie s J . , L i t e s B . W . , S k u m a n i c h A . , 1 9 8 9 , A s t r o p h y s . J . , 3 4 3 , 920 H a r d o r p J . , S h o r e S . N . , W i t t e m a n A . , 1 9 7 6 , P h y s i c s o f A p S t a r s , e d s . : W e i s s W . W . , J e n k n e r H . , W o o d H . J . , V i e n n a , 4 1 9 H a t z e s A . P . , 1 9 9 1 , M o n . N o t . R . A s t r o n . S o c , 2 5 3 , 8 9 K h o k h l o v a V . L . , 1 9 7 6 , A s t r o n . N a c h r i c h t . , 2 9 7 , N o . 5 , 2 0 3 K h o k h l o v a V . L . , 1 9 86 , U p p e r M a i n S e q u e n c e S t a r s w i t h A n o m a l o u s A b u n d a n c e s , e d s . : C . R . C o w l e y , M . M . D v o r e t s k y , C . M e g e s s i e r , R e i d e l P u b l . C o m p a n y , 1 25 K h o k h l o v a V . L . , V a s i l ' c h e n k o D .V ., S t e p a n o v V .V ., T s y m b a l V . V . , 1 9 9 7 , A s t r o n o m y L e t t e r s , 2 3 , 4 6 5 K h o k h l o v a V . L . , V a s i l ' c h e n k o D . V . , S t e p a n o v V .V ., R o m a n y u k I . I . , 2 0 0 0 , A s t r o n o m y L e t t e r s , 2 6 , N o . 3 , 2 1 7 K u r u c z R . L . , 1 9 9 2, R e v i s t a M e x i c a n a d e A s t r o n o m i a A s t r o f i s i c a , 2 3 , 1 81 L a n d o l f i M ., L a n d i D e g l ' I n n o c e n t i E . , 1 9 8 2 , S o l . P h y s . , 7 8 , 3 5 5 M i n n a e r t M ., 1 935, Z h . A s t r o p h y s . , 1 0 , 4 0 P i s k u n o v N . E . , R i c e J . , 1 9 9 3 , P u b l . A s t r . S o c . P a c if ic , 1 0 5 , 4 1 5 R a c h k o v s k i i D . N . , 1 9 6 2 , I z v . K r y m . A s t r o f i z . O b s . , 7 , 1 48 T h o m p s o n I . B . , L a n d s t r e e t J . D . , 1 9 8 5 , A s t r o p h y s . J . , 2 8 9 , L 9 T i k h o n o v A . N . , G o n c h a r s k i i A . V . , S t e p a n o v V . V . , Y a g o la A . G . , 1 9 9 5 , " N u m e r i c a l M e t h o d s fo r t h e s o l u t i o n o f I I I P o s e d P r o b l e m s " , K l u w e r U n n o W . , 1 9 5 6 , P u b l . A s t r o n . S o c . J a p a n , 8 , 1 08 V a s i l c h e n k o D ,V ., S t e p a n o v V . V . , K h o k h l o v a V . L . , 1 996, A s t r o n o m y L e t t e r s , 2 2 , 8 2 8 V o g t S .S ., P e n r o d G . D . , a n d H a t z e s A . P . , 1 9 8 7 , A s t r o p h y s . J . , 3 2 1 , 4 9 6

T h e r e a l e r r o r i n t h e o b s e r v e d S t o k e s p a r a m e t e r s i s d e t e r m i n e d n o t o n l y b y t h e S / N r a t i o , w h i c h n o w p r o b a b l y m a y r e a c h a v a l u e o f 5 0 0 ( w h i c h w o u l d c o i n c i d e w i t h = 0 . 0 0 2 ) , b u t a c c o r d i n g t o o u r e x p e r i e n c e i t i s l i m i t e d b y a v a l u e o f > 0 . 0 0 5 b e c a u s e o f u n r e v e a l e d b l e n d i n g b y v e r y f a i n t l i n e s a n d b y e r r o r s i n d r a w i n g t h e c o n t i n u u m . T h e n e c e s s i t y o f s t o p p i n g i t e r a t i o n s a t t h i s v a l u e o f t h e r e s i d u a l ( s e e s t o p p i n g r u l e ( 2 6 ) i n s e c t i o n ( 4 ) ) m a y n o t a l l o w t h e b e s t o f p o s s i b l e s o l u t i o n s t o b e a c h i e v e d . N e v e r t h e l e s s o u r t e s t s h o w s ( F i g u r e s 1 - 7 ) t h a t t h e s o l u t i o n s o b t a i n e d g i v e a c o r r e c t ( t h o u g h i n e v i t a b l y r o u g h ) i d e a o f m a g n e t i c f ie ld c o n f i g u r a t i o n a n d p e r m i t s t u d y i n g i t s c o n n e c t i o n w i t h t h e l o c a t i o n


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