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Exploring th e magneti c fiel d structur e o f A p stars using Stokes and U Zeeman signatures*
G.A . Wad e

Astronomy Department , Universit y of Toront o at Mississauga, Mississauga, Ontario , Canad a L5L 1C6

Abstract . Thi s pape r describe s a n ongoin g programm e aime d a t obtainin g an d modelin g high-resolutio n spectropolarimetri c observation s o f magneti c A p star s i n al l 4 Stoke s parameters . Th e ultimat e goa l o f thi s effort i s t o produc e detaile d map s o f th e surfac e magneti c field s an d chemica l abun danc e distribution s o f thes e objects . Herei n w e revie w ou r instrumentation , observations , analysi s procedure s an d preliminar y mode l comparisons . W e furthermor e discus s ne w result s concernin g th e impac t o f anomalou s dispersio n o n spectru m synthesi s for magneti c A p stars , th e potentia l o f 4-Stoke s timeserie s for constrainin g simpl e model s o f th e magneti c fiel d geometr y an d chemica l abundanc e distributions , an d th e outloo k for detaile d reconstruction s o f th e surfac e fiel d an d abundanc e structures .

1 . Introductio n
Investigator s hav e recentl y achieve d considerabl e suc cess explorin g th e magneti c fiel d structur e o f A p an d an d V Zeema n spectropo B p star s usin g Stoke s larimetri c observation s (e.g . Mathy s & Hubrig , 1997 ; Bagnul o & Landolfi , 1999 ; Landstree t & ; Mathys , 2000 ) a s wel l a s Stoke s Q an d U ne t broadban d lin ea r polarisatio n observation s (e.g . Wad e e t al. , 1996 ; Lero y e t al. , 1996 ; Bagnul o e t al. , 2000) . Thes e da t a hav e demonstrate d th e valu e o f observation s ob taine d i n multipl e Stoke s parameter s for constrainin g th e topologie s o f th e surfac e magneti c field s o f thes e stars . A t th e sam e time , th e detaile d exploitatio n o f thes e dat a ha s bee n limite d b y th e approximat e na tur e o f th e observationa l diagnostics , a s well a s thei r often schemati c o r incomplet e theoretica l interpreta tion . For example , al l o f th e commonl y employe d di agnostic s ignor e th e effects o f chemica l abundanc e in homogeneities . Th e goa l o f thi s programm e i s t o provid e a mor e physicall y accurat e an d theoreticall y interpretabl e wa y t o constrai n th e magneti c fiel d structur e an d chemica l abundanc e distribution s o f A p stars . W e ac complis h thi s b y takin g th e observatio n an d model in g bac k t o th e plac e wher e th e Zeema n effect occurs : withi n individua l spectra l lines . I n thi s pape r w e de scrib e th e instrumental , observationa l an d numerica l solution s w e hav e adopte d an d develope d i n orde r t o obtai n an d mode l hig h resolution , hig h signal-to-nois e

rati o measurement s o f th e spectra l line s o f magneti c A p star s i n al l fou r Stoke s parameters .

2 . Th e MuSiCo S spectropolarimete r
Th e 4 Stoke s paramete r observation s wer e obtaine d usin g th e MuSiCo S (MUlti-SIt e Coordinate d Spec troscopy ) spectropolarimete r mounte d o n th e 2 metr e Bernar d Lyo t telescop e a t th e Pi c d u Mid i observa tory . Th e spectropolarimete r consist s o f a table-to p echell e spectrograp h (Baudran d & ; Bohm , 1992 ) fed b y a doubl e optica l fibr e directl y fro m a dedicate d Cassegrain-mounte d polarisatio n analysi s uni t (Do nat i e t al. , 1999) . I n on e singl e exposure , thi s appa ratu s allow s th e acquisitio n o f a stella r spectru m i n a give n polarisatio n stat e (Stoke s V , Q o r U ) through ou t th e spectra l rang e 45 0 t o 66 0 n m wit h a resolvin g powe r o f abou t 35000 . I n Fig . 1 w e sho w th e optica l layou t o f th e polar isatio n analysi s unit . I n norma l operation , starligh t enter s th e analyse r a t th e Cassegrai n focus . Th e bea m re the n ma y optionall y pas s throug h a rotatabl e tarde r (i n th e cas e o f Stoke s V observations ) o r no t (i n th e cas e o f Stoke s Qor U observations) . Th e bea m the n intersect s a Savart-typ e beamsplitte r whic h sep arate s th e stella r ligh t int o tw o beam s whic h ar e re spectivel y polarise d alon g an d perpendicula r t o th e instrumenta l referenc e azimuth . Th e analyse d beam s ar e the n focall y reduce d t o a n apertur e o f f /2. 5 for fibre , whic h trans injectio n int o th e doubl e 5 0 por t th e ligh t t o th e spectrograph . Th e uni t i s com pact , abou t 3 0 c m hig h b y 3 0 c m wid e b y 1 5 c m dee p

* Based o n observation s obtaine d usin g th e MuSiCo S spec tropolarimete r a t th e Pi c d u Mid i observatory , France .

© Specia l Astrophysica l Observator y o f th e Russia n AS , 200 0


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i n Fig . 2 , for th e coo l A p sta r Corona e Boreali s (Wad e e t al. , 1999a) . Thes e represen t th e firs t high qualit y measurement s o f stella r spectra l lin e Zeema n linea r polarisatio n eve r obtained , an d confir m bot h th e genera l weaknes s an d complexit y o f Stoke s Q an d U profiles . A mor e extensiv e atla s o f th e circula r an d linea r polarisatio n Zeema n signature s i n th e spectru m o f Cr B i s provide d b y Wad e (1999) .

4 . Least-square s deconvolutio n
Whil e Zeema n linea r polarisatio n i s clearl y detecte d i n strong , magneticall y sensitiv e spectra l line s i n man y o f ou r highes t qualit y spectra , i t i s ofte n onl y marginall y detecte d i n strong , sensitiv e line s i n th e muc h mor e commo n moderat e S/ N spectra , an d i s es sentiall y never detecte d i n weaker , les s sensitiv e lines . Th e line s i n whic h linea r polarisatio n i s detecte d (for exampl e FeII492.39 3 n m an d FeII501.84 4 nm , bot h o f multiple t 42 ; Fig . 2 ) ten d t o b e th e stronges t an d mos t magneticall y sensitiv e meta l line s i n th e visibl e spectrum . I n orde r t o increas e th e absolut e S/ N o f ou r linea r polarisatio n observation s w e exploite d th e informa tio n containe d i n th e many spectra l line s i n ou r echell e spectra . A s i s discusse d b y Donat i e t al . (1997) , Least Square s Deconvolutio n (o r LSD ) i s a cross-correlatio n procedur e designe d for th e detectio n an d measure men t o f suc h wea k polarisatio n signature s (of orde r ~ 0.0 1 - 1 % full amplitude ) i n stella r spectra . LS D take s advantag e o f th e fact , for wea k magneti c fields, tha t th e shapes o f spectra l line s (an d associate d polar isatio n features ) ar e approximatel y th e sam e fro m on e spectra l lin e t o another . Donat i e t al . (1997 ) develo p th e procedur e for extractio n o f Stoke s V signature s o f activ e late-typ e stars . Wad e e t al . (1999a ) showe d tha t LS D ca n als o b e use d for extractin g mea n linea r polarisatio n Zeema n signatures , an d tha t deconvolu tio n procedure s appl y whic h ar e analogou s t o tha t develope d b y Donat i e t al . (1997) . I n Fig . 3 w e illustrat e th e wea k line , wea k fiel d homomorphis m o f whic h LS D take s advantage . W e als o sho w tha t thi s homomorphis m break s dow n i n th e cas e o f stron g fields . Thi s i s discusse d furthe r i n Sect . 6.1 .

(excludin g th e calibratio n facilities) , an d i s affixed di rectl y t o th e TB L Cassegrai n bonnette . Thi s arrange men t reduce d differentia l mechanica l deformatio n t o withi n abou t 1 0 , an d allow s for accurat e projec tio n o f th e orthogonall y polarise d image s ont o th e fibres. A complet e polarimetri c exposur e consist s o f a se quenc e o f 4 subexposures , betwee n whic h th e retarde r (for circula r polarisatio n Stoke s V ) o r th e instrumen t itsel f (for linea r polarisation s Stoke s Q an d U ) i s ro tate d b y ± 90° . Thi s ha s th e effect o f exchangin g th e beam s withi n th e whol e instrument , an d i n partic ula r switchin g th e position s o f th e tw o orthogonall y polarise d spectr a o n th e CCD . Thi s observin g proce dur e shoul d i n principl e suppres s al l first-orde r spu riou s polarisatio n signature s dow n t o a leve l o f abou t 0.01 % (Donat i e t al. , 1997) . Th e detaile d optica l char acteristic s o f th e polarisatio n analyse r an d th e spec tropolarimetri c observin g procedure s ar e describe d b y Donat i e t al . (1999) .

3 . Observation s
W e hav e employe d th e MuSiCo S spectropolarimete r throughou t a tota l o f 5 0 night s (allocate d i n 199 7 Feb. , 199 8 Feb. , an d 199 9 Jan. ) t o obtai n bot h circu la r an d linea r polarisatio n (i.e . 4 Stoke s parameter ) observation s o f 1 2 magneti c A p stars . Fo r 7 o f thes e star s full rotationa l phas e coverag e ha s bee n achieved . Th e full lis t o f star s for whic h complet e Stoke s obser vation s wer e obtaine d i s show n i n Tabl e 1 . Circula r polarisatio n Zeema n signature s ar e de tecte d i n essentiall y al l spectra l line s o f al l target s observed , wit h a typica l relativ e amplitud e o f ~ 1% . W e furthermor e detec t muc h weake r linea r polarisa tio n Zeema n signature s i n th e strongest , mos t mag neticall y sensitiv e spectra l line s o f severa l o f thes e objects . A n exampl e o f suc h a detectio n i s show n

5 . LS D mea n Zeema n signature s
LS D wa s employe d t o extrac t mea n Zeema n signa ture s fro m al l Stoke s spectr a o f A p stars , a s well a s fro m numerou s Stoke s observation s o f sharp-line d standar d stars . 5.1 . Standar d star s Cool , sharp-line d nonmagneti c star s wer e selecte d a s standard s i n orde r t o diagnos e spuriou s contribution s


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t o th e lin e polarisatio n du e t o smal l change s i n th e positio n o r shap e o f th e observe d spectru m eithe r dur in g o r betwee n exposures . Th e 4-subexposur e observ in g procedur e w e employe d (describe d b y Donat i e t al. , 1997 ) shoul d i n principl e reduc e al l spuriou s sig nature s i n spectra l line s dow n t o a level o f aroun d 0.01% . I n practice , w e fin d (a s di d Donat i e t al. , 1997)

tha t episodi c signature s produced , althoug h thes polarisatio n spectr a an d wit h th e rotatio n o f th e

severa l time s large r ca n b e e appea r onl y i n ou r linea r ar e though t t o b e associate d polarimete r module .

Example s o f LS D profile s o f Arcturu s (exhibit in g n o spuriou s signature s i n Stoke s V dow n t o a leve l o f 0.002% , a s well a s detecte d spuriou s signa -


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ture s wit h maximu m full amplitud e 0.03 % i n Stoke s Q an d U ) ar e show n i n Fig . 4 . Becaus e suc h spu riou s signature s scal e bot h wit h lin e sharpnes s an d centra l depth , the y ar e predicte d t o b e muc h weake r for ou t typica l A p targets . Wad e e t al . (1999a ) sho w tha t potentia l spuriou s signature s ar e negligibl e for all programm e stars .

5.2 . Magneti c A p star s Th e magneti c A p star s for whic h 4 Stoke s paramete r observation s hav e bee n obtaine d ar e liste d i n Tabl e 1 . Mos t o f thes e target s wer e selecte d base d o n suc cessful detectio n o f broadban d linea r polarisatio n b y Lero y (1995) . The y ar e furthermor e selecte d t o hav e relativel y lo w v sin i , an d ar e typicall y quit e bright , bot h criteri a selecte d i n orde r t o improv e th e poten -


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t i a l fo r d e t e c t i t u r e s . F i n a l l y , p a r t i c u l a r H e t i v e l y l o w l i n e t i p l e x p o t e n t i a U sin t e r s a r e m o s t o f rotation s t a r s .

n g t h e w e a k l i n e a r w e h a v e t e n d e d t o p e c u l i a r s t a r s ) b e c d e n s i t i e s a n d t h e r e f l f o r L S D . r t i t

p a a o

o l a r i s a t i o n s i g n a v o i d h o t s t a r s ( i n u s e o f t h e i r r e l a r e t h e l o w e r m u l eo f h e o f

a g a t i n g ( d u e t o r o t a t i o n a l m o d u l a t i o n ) t h r o u g h t h e p r o f i l e s o b t a i n e d a t s u c c e s s i v e r o t a t i o n a l p h a s e s . T h e p r o f i l e s i n F i g s . 5 a n d w e r e e x t r a c t e d u s i n g fu ll l i n e m a s k s , c o n t a i n i n g i n f o r m a t i o n a b o u t s p e c t r a l l i n e s o f a l l r e l e v a n t c h e m i c a l e l e m e n t s . H o w e v e r , L S D c a n a l s o b e u s e d o n s u b s e t s o f s u c h m a s k s w h i c h a r e r e s t r i c t e d t o s i n g l e c h e m i c a l e l e m e n t s . F i g . 7 s h o w s p r o f i l e s fo r t h e A 4 p s t a r 5 3 C a m o b t a i n e d fo r F e a n d T i s u b m a s k s . L a n d s t r e e t ( 1 9 8 8 ) m o d e l l e d t h e c h e m i c a l a b u n d a n c e d i s t r i b u t i o n s o v e r t h e s u r f a c e o f t h i s s t a r , a n d f o u n d F e t o b e d i s t r i b u t e d e s s e n t i a l l y h o m o g e n e o u s l y , w h i l e T i s h o w e d a v e r y s t r o n g s u r f a c e v a r i a t i o n . T h i s i s c l e a r l y c o n f i r m e d b y t h e s e p r o f i l e s : w h i l e t h e F e S t o k e s I p r o f i l e s s h o w r e l a t i v e l y l i t t l e v a r i a t i o n , t h e T i p r o f i l e s v a r y f r o m n e a r - d i s a p p e a r a n c e t o b e i n g s t r o n g e r t h a n t h e m e a n F e p r o f i l e . T h e V , Q a n d U p r o f i l e s a l s o s h o w s i g n i f i c a n t v a r i a b i l i t y . T h e m o r p h o l o g y o f t h e F e V , Q a n d U p r o f i l e s a r e s u b s t a n t i a l l y d i f f e r e n t f r o m t h o s e o f T i ( t h e d i f f e r e n c e i n t h e n o i s e l e v e l a s i d e ; t h i s r e s u l t s f r o m t h e r e l a t i v e l y s m a l l n u m b e r o f T i l i n e s i n t h e s p e c t r u m a s c o m p a r e d t o F e ) . T h i s i s l i k e l y a r e s u l t o f t h e v e r y d i f f e r e n t s a m p l i n g o f t h e m a g n e t i c f i e l d d i s t r i b u t i o n b y t h e s e t w o e l e m e n t s d u e t o t h e i r d i s s i m i l a r s u r f a c e d i s t r i b u t i o n s . L S D p r o f i l e s c a n a l s o b e m e a s u r e d t o p r o v i d e m o r e c o n v e n t i o n a l d i a g n o s t i c s o f t h e m a g n e t i c f i e l d , s u c h a s t h e m e a n l o n g i t u d i n a l m a g n e t i c f i e l d a n d n e t l i n e a r p o l a r i s a t i o n ( D o n a t i e t a l . , 1 9 9 7 ; W a d e e t a l . , 1 9 9 9 b ) . T h e m e a s u r e m e n t s a r e o f v e r y h i g h p r e c i s i o n , a n d

g L S D , s i g n a t u c o n s i s t e n t l y d e o u r t a r g e t s . T h a l v a r i a t i o n o f

e s i n a l l f o u r S t o k e s p a r a m e c t e d i n t h e s p e c t r a l l i n e s s h a s a l l o w e d u s t o f o l l o w t h e s e p r o f i l e s f o r a n u m b e r

T h e L S D p r o f i l e v a r i a t i o n s fo r t h e s l o w l y - r o t a t i n g -1 5 k m s , P r o t = 1 8 . 4 9 d ) , c o o l F O p s t a r (vsin i C r B a r e s h o w n i n F i g . 5 . T h e S t o k e s Q a n d U s i g n a t u r e s i n t h e s p e c t r a l l i n e s o f t h i s s t a r a r e e s p e c i a l l y s t r o n g , a n d a r e i n f a c t d e t e c t e d i n m a n y i n d i v i d u a l s p e c t r a l l i n e s a t m o s t r o t a t i o n a l p h a s e s . T h e l i n e a r p o l a r i s a t i o n s i g n a t u r e s o f C r B a r e t h e c l e a r e s t o f a n y s t a r o b s e r v e d i n t h i s p r o g r a m m e . C o r r e s p o n d i n g l y , t h e L S D p r o f i l e s s h o w w e l l - d e f i n e d , h i g h S / N s i g n a t u r e s w h i c h a r e o b s e r v e d t o v a r y c l e a r l y t h r o u g h o u t t h e r o t a t i o n a l c y c l e . I n F i g . 6 t h e L S D S t o k e s p r o f i l e v a r i a t i o n s o f t h e -1 m o r e r a p i d l y r o t a t i n g ( v s in i ~ = 1 4 k m s , P r o t = C V n a r e s h o w n . N o t e t h e c l e a r 5 . 4 7 d ) A O p s t a r D o p p l e r d i s t o r t i o n o f t h e m e a n S t o k e s I p r o f i l e s ( d u e t o i n h o m o g e n e o u s d i s t r i b u t i o n s o f t h e c h e m i c a l e l e m e n t s , i n p a r t i c u l a r F e ) . T h e v a r i a t i o n s o f S t o k e s V , Q a n d U a r e w e a k e r a n d m o r e c o m p l e x t h a n t h o s e o f C r B ; o n e c a n f u r t h e r m o r e o b s e r v e f e a t u r e s p r o p -


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na l polarise d transfe r calculation . O n th e othe r hand , CP U cycle s ar e no w extremel y affordable , an d furthermor e w e expecte d magneto-optica l effects t o hav e ver y importan t effects o n calculate d Stoke s Q an d U profiles . W e hav e therefor e substantiall y modifie d Landstreet' s origina l lin e synthesi s cod e t o includ e magneto-optica l effects . Th e resultan t cod e (know n a s ZEEMAN2 ) i s describe d i n som e detai l b y Wad e e t al . (2000) . 6.1 . Impac t o f magneto-optica l effect s T o examin e th e impac t o n th e Stoke s profil e calcu lation s o f includin g magneto-optica l effects , w e hav e performe d severa l comparison s o f profile s calculate d bot h wit h magneto-optica l effects turne d off, an d wit h magneto-optica l effects turne d on . Ou r initia l con cer n wa s th e impac t o n Stoke s Q an d U profiles , an d magneto-optica l effects clearl y hav e importan t conse quence s for th e inferre d linea r polarisatio n amplitud e an d orientation . However , w e wer e surprise d t o not e tha t thes e effects als o have , for saturate d lines , a ver y dramati c impac t o n Stoke s I profile s a s well . A s i s show n i n Fig . 9 , magneto-optica l effects ca n desatu rat e stron g lines , substantiall y increasin g th e slop e o f th e curve-of-growth , an d resultin g i n saturate d line s whic h ca n b e muc h deepe r tha n obtaine d ignorin g magneto-optica l effects . Suc h a n effect ma y well b e responsibl e for th e inabilit y o f Landstree t e t al . (1989 ) t o reproduc e th e cor e depth s o f stron g line s o f Fe , Ti , C r an d S i i n th e spectru m o f th e strongly-magneti c sta r H D 215441 . I t probabl y als o affect s somewha t th e abundanc e distributio n determine d b y Landstree t (1988 ) for th e A p sta r 5 3 Cam . Thi s i s quit e impor tant , a s Landstreet' s mode l o f 5 3 Ca m ha s bee n use d for a numbe r o f importan t test s o f th e diffusion theor y (e.g . Babe l & Michaud , 1991) . Thi s remarkabl e desaturatin g effect o n Stoke s I profiles , a s wel l a s th e importan t impac t o n Stoke s Q an d U profiles , ha s bee n verifie d usin g tw o othe r independen t polarise d spectru m synthesi s code s (a s par t o f a large r stud y b y Wad e e t al. , 2000) . A sim ila r increas e i n th e equivalen t widt h o f Stoke s I pro file s upo n th e inclusio n o f magneto-optica l effects wa s note d b y Landolf i e t al . (1989) . Thes e result s d o no t appea r t o b e consisten t wit h th e statemen t b y Vasilchenk o e t al . (1996 ) tha t magneto-optica l effects hav e negligibl e impac t o n th e Stoke s profile s o f hotte r A p stars .

provide a powerfu l tes t o f th e accurac y o f th e LS D procedur e (Wad e e t al. , 1999b) . I n Fig . 8 w e com pare measurement s o f th e longitudina l magneti c fiel d CV n extracte d fro m LS D F e profiles , a s wel l a s of net linea r polarisatio n measurement s extracte d fro m LSD profile s o f th e A2 p sta r 7 8 Virginis , wit h previ ously publishe d observation s o f thes e quantities . Th e agreemen t betwee n th e previousl y publishe d measure ments an d thos e obtaine d fro m LS D profile s i s excel lent. Thi s indicate s tha t LS D produce s profile s whic h have magneti c diagnosti c valu e a s goo d a s o r bette r than an y othe r dat a currentl y available .

6 . Modelin g Stoke s profile s
Modeling th e lin e profile s o f non-magneti c star s re quires th e solutio n o f th e unpolarise d transfe r prob lem, whic h involve s a singl e first-order differentia l equation. Treatmen t o f th e spectra l line s o f magneti c stars, o n th e othe r hand , require s th e solutio n o f th e polarised transfe r problem , involvin g a se t o f fou r coupled first-orde r differentia l equations , on e for eac h Stokes parameter . Landstree t (1988 ) focuse d considerabl e energ y de veloping a cod e t o solv e th e polarise d transfe r prob lem i n orde r t o synthesis e spectra l lin e profile s o f mag netic stars . Becaus e o f th e cos t o f computin g cycle s a t th e tim e th e cod e wa s written , a s well a s becaus e suc h effects wer e though t t o hav e relativel y smal l conse quences for Stoke s I profiles , magneto-optica l effects (o r anomalou s dispersion ) wer e ignore d i n th e origi -

6.2 . Modelin g LS D profile s Becaus e LS D profile s ar e i n fac t calculate d usin g th e profile s o f man y individua l spectra l lines , i t i s no t im mediatel y eviden t tha t i t shoul d b e possibl e t o repro duc e thei r shape s o r variation s usin g th e parameter s o f an y singl e spectra l line . A simpleminde d exami -


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natio n o f Fig . 3 suggest s that , eve n i f suc h a n ap proac h ma y b e appropriat e i n th e cas e o f wea k field s (wher e th e LS D mode l seem s t o b e apply) , i t seem s ver y unlikel y t o produc e reasonabl e result s i n th e cas e o f stron g field s (wher e th e LS D mode l clearl y break s down) . Thi s i s i n fac t exactl y wha t wa s show n b y Wad e (1998 ) for th e specia l cas e o f a Zeema n triplet . Wad e calculate d syntheti c spectr a i n th e fou r Stoke s param eter s for specifie d magneti c fiel d configurations , an d extracte d fro m thos e spectr a LS D profile s whic h wer e subsequentl y fi t usin g a Zeema n triple t lin e profil e model . Spectr a wer e calculate d for mea n fiel d modul i o f 0.5 , 2 , 4 , 8 an d 1 2 kG . Whil e th e Stoke s I an d V LS D profile s wer e reasonabl y wel l reproduce d b y 4 kG , th e Stoke s Q an d th e triple t mode l for U LSD profile s wer e observe d t o exhibi t larg e difference s fro m th e triple t mode l for fiel d modul i a s lo w a s 2 kG , an d mor e strikingl y th e agreemen t wa s ob serve d t o degrade wit h increasin g projecte d rotationa l velocity . A fa r mor e sophisticate d stud y o f th e shape s o f LS D profiles , a s wel l a s th e outloo k for reproducin g the m usin g relativel y simpl e models , i s currentl y un derwa y (Shorlin , 2001) . Whil e muc h o f thi s wor k i s stil l i n progress , on e conclusio n tha t ca n b e mad e a t thi s poin t i s tha t a simpl e triple t lin e profil e mode l doe s no t accuratel y reproduc e th e LS D profile s o f A p star s wit h fields stronge r tha n a few kG . A s thi s comprise s mos t o f th e star s observe d b y Wad e e t al . (1999a) , modelin g o f thei r Stoke s profile s (a t leas t initially ) mus t con centrat e o n th e profile s o f individua l spectra l lines .

6.3 . Compariso n wit h publishe d model s One application of the new data is to confront the prediction s o f magneti c fiel d model s whic h hav e bee n published previously and which are based on less sophisticate d dat a sets . Wad e e t al . (1999a ) mak e suc h comparison s wit h CrB , 5 3 Cam , 4 9 Ca m an d model s o f th e A p star s H D 71866 . I n Fig . 1 0 w e reproduc e thei r result s for 5 3 Cam , a compariso n o f th e observe d profile s o f F e I I 492.39 3 n m an d thos e predictio n assumin g th e mag neti c field/abundanc e distributio n mode l for thi s sta r reporte d b y Landstree t (1988) . W e firs t poin t ou t th e acceptabl e fi t t o Stoke s I usin g Landstreet' s mode l withou t magneto-optica l effects (fine curves) , an d th e clearl y improve d fi t whe n magneto-optica l effects ar e include d (heav y curves) . Thi s indicate s tha t anoma lou s dispersio n ha s a n importan t impac t o n th e mor pholog y o f FeI I 492.39 3 nm . Secondly , w e not e tha t whil e th e shape s o f th e Stoke s V signature s ar e ap proximatel y reproduce d b y th e model , th e amplitud e o f Stoke s V i s consistentl y overestimated . Finall y w e addres s th e mos t strikin g aspec t o f thi s compari son : th e mode l substantiall y overestimate s th e am plitud e o f Stoke s Q an d U a t every phase, an d a t som e phase s thi s overestimatio n i s a t leas t 500% . A simila r disagreemen t betwee n th e observe d an d calcu late d Stoke s Q an d U profile s ha s als o bee n encoun tere d b y Bagnul o & ; Wad e (i n progress ) usin g a non axisymmetri c multipola r mode l (Bagnul o e t al. , 1999) determine d usin g variou s magneti c moments , a s well a s broadban d linea r polarisatio n measurements . Thi s ma y well provid e evidenc e tha t th e magneti c fiel d o f 5 3 Ca m i s substantiall y mor e comple x tha n ha s pre -


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v i o u s l y b e e n s u p p o s e d .

6 . 4 . R e c o v e r y o f 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 h e m o s t m e t r i c d a o f t h e m a s u r f a c e c h exploratio e x c i t i n g p o t e n t i a l o f t h e n e w s p e c t r o p o l a r i t a i s t h e i r a b i l i t y t o c o n s t r a i n n ew m o d e l 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 , a s w e l l a s t h e m i c a l i n h o m o g e n e i t i e s , o f A p s t a r s . T h i n o f t h i s p o t e n t i a l i s i n i t s e a r l y s t a g e s . s e s

T o d a t e , a f e w n u m e r i c a l e x p e r i m e n t s h a v e b e e n c o n d u c t e d t o e x p l o r e t h e p o t e n t i a l o f t h e n e w d a t a f o r c o n s t r a i n i n g m a g n e t i c a n d a b u n d a n c e m o d e l s . F o r e x a m p l e , t h e a u t h o r ( W a d e , i n p r o g r e s s ) h a s e x a m i n e d t h e a b i l i t y o f t h e Z e e m a n s i g n a t u r e s t o c o n s t r a i n l o w o r d e r a x i s y m m e t r i c m u l t i p o l a r m o d e l s o f t h e m a g n e t i c f i e l d , i g n o r i n g a n d i n c l u d i n g t h e e f f e c t s o f a b u n d a n c e n o n u n i f o r m i t i e s . T h e r e s u l t s o f t h e s e t e s t s a r e v e r y e n c o u r a g i n g , s u g g e s t i n g t h a t a r e p r e s e n t a t i v e a x i s y m m e t r i c f i e l d c o n f i g u r a t i o n c a n b e r e c o v e r e d e v e n i n t h e p r e s e n c e o f s u b s t a n t i a l n o n a x i s y m m e t r i c f i e l d c o n t r i b u t i o n s a n d / o r c h e m i c a l a b u n d a n c e n o n u n i f o r m i t i e s a s l a r g e a s s e v e r a l d e x . I n a d d i t i o n , r e c o v e r y o f t h e a x i s y m m e t r i c c o n f i g u r a t i o n c a n b e a c h i e v e d u s i n g h i g h s i g n a l - t o - n o i s e r a t i o o b s e r v a t i o n s o b t a i n e d a t f e w e r t h a n 5 r o t a t i o n a l p h a s e s . P i s k u n o v ( t h e s e p r o c e e d i n g s ) a n d K o c h u k o v ( t h e s e p r o c e e d i n g s ) d e s c r i b e t h e M a g n e t i c D o p p l e r I m a g i n g t e c h n i q u e , e s s e n t i a l l y t h e a p p l i c a t i o n o f t h e D o p p l e r I m a g i n g p r o c e d u r e t o f o u r S t o k e s p a r a m e t e r o b s e r v a t i o n s . T h i s t e c h n i q u e i s a i m e d a t r e c o n s t r u c t i n g s i m u l t a n e o u s l y b o t h t h e v e c t o r m a g n e t i c f i e l d d i s t r i b u t i o n s a n d t h e c h e m i c a l a b u n d a n c e d i s t r i b u t i o n , w i t h o u t m a k i n g a n y a p r io r i a s s u m p t i o n s a b o u t t h e s t r u c t u r e o r d i s t r i b u t i o n o f e i t h e r . T h i s t e c h n i q u e i n v o l v e s a v e r y l a r g e n u m b e r o f f r e e p a r a m e t e r s , a n d t h e r e f o r e r e q u i r e s a s u b s t a n t i a l l y l a r g e r n u m b e r o f o b s e r v a t i o n s i n o r d e r t o o b t a i n a r e l i a b l e m o d e l . O n t h e o t h e r h a n d , t h e r a n g e o f p o s s i b l e m o d e l c o n f i g u r a t i o n s i s m u c h g r e a t e r t h a n f o r a t e c h n i q u e s u c h a s t h a t d e s c r i b e d a b o v e , a n d s o w i l l n o d o u b t a l l o w f o r m u c h b e t t e r r e p r o d u c t i o n o f t h e o b s e r v a t i o n s ( a l t h o u g h w h e t h e r o r n o t t h e r e s u l t a n t m o d e l s a r e c o n g r u e n t w i t h r e a l i t y i s a l w a y s a n o t h e r q u e s t i o n ! ) . I n a d d i t i o n t o t h e s e n u m e r i c a l e x e r c i s e s , a f e w a t t e m p t s h a v e b e e n m a d e t o a c t u a l l y f i t t h e n e w d a t a u s i n g v a r i o u s m o d e l s . T h e a u t h o r ( W a d e , i n p r o g r e s s ) i s i n t h e p r o c e s s o f e m p l o y i n g t h e a x i s y m m e t r i c m o d e l i n g p r o c e d u r e d e s c r i b e d a b o v e t o r e p r o d u c e t h e C o r o n a e B o r e a l i s . I n S t o k e s p r o f i l e v a r i a t i o n s o f F i g . 1 1 w e s h o w t h e o b s e r v e d S t o k e s p r o f i l e v a r i a t i o n s o f C a I 6 1 6 . 2 i n t h e s p e c t r u m o f C r B , c o m p a r e d w i t h t h e b e s t - f i t a x i s y m m e t r i c m a g n e t i c d i p o l e + l i n e a r q u a d r u p o l e + o c t u p o l e m o d e l o b t a i n e d v i a l e a s t - s q u a r e s . W h i l e t h e f i t t o S t o k e s V , Q a n d U i s c l e a r l y o n l y a p p r o x i m a t e , t h e m o d e l d o e s r e c o v e r v a l u e s o f t h e i n c l i n a t i o n , t h e o b l i q u i t y , a n d t h e m e a n

f i e l d i n t e n s i t y t h a t a r e c o m p a r a b l e t o t h o s e r e p o r t e d b y o t h e r a u t h o r s ( e . g . L e r o y e t a l . , 1 9 9 6 ; L a n d s t r e e t & M a t h y s , 2 0 0 0 ; B a g n u l o e t a l . , 2 0 0 0 ) . W h i l e a n o n u n i f o r m d i s t r i b u t i o n o f a w a s a l s o c o n s i d e r e d i n t h i s m o d e l i n g , n o e v i d e n c e f o r l i n e p r o f i l e v a r i a b i l i t y d u e t o s u c h n o n u n i f o r m i t i e s w a s f o u n d ( t h e r e s u l t a n t u n i f o r m C a a b u n d a n c e w a s = - 5 . 3 ) . T h i s i s n o t C r B r e s u l t s i n e s s e n s u r p r i s i n g : t h e g e o m e t r y o f t i a l l y t h e s a m e h e m i s p h e r e o f t h e s t a r b e i n g v i s i b l e t h r o u g h o u t a n e n t i r e r o t a t i o n . T h i s g e o m e t r y i s i l l u s t r a t e d i n F i g . 1 2 . T h e r e f o r e e v e n i f s u r f a c e a b u n d a n c e n o n u n i f o r m i t i e s d o e x i s t t h e y w i l l p r o d u c e l i t t l e o r n o l i n e p r o f i l e v a r i a b i l i t y . A n o t h e r p o s s i b i l i t y i s t h a t a n o n u n i f o r m C a d i s t r i b u t i o n m i g h t s a m p l e t h e m a g n e t i c e q u a t o r ( s a y ) , r e s u l t i n g i n S t o k e s p r o f i l e s w i t h a d i f f e r e n t g e n e r a l m o r p h o l o g y t h a n w o u l d b e o b t a i n e d f o r a u n i f o r m d i s t r i b u t i o n . A g a i n , n o e v i d e n c e f o r s u c h a s c e n a r i o w a s o b s e r v e d . C r B i s w e l l k n o w n t o h a v e a n o n - a x i s y m m e t r i c m a g n e t i c f i e l d , a n d s o a n a x i s y m m e t r i c m o d e l s u c h a s t h a t d e s c r i b e d a b o v e w i l l o n l y v e r y a p p r o x i m a t e l y d e s c r i b e t h e s u r f a c e c o n f i g u r a t i o n . B a g n u l o & W a d e ( i n p r o g r e s s ) a r e s t u d y i n g h o w t h e f i t t o t h e S t o k e s p r o f i l e s c a n b e i m p r o v e d e m p l o y i n g a n o n - a x i s y m m e t r i c m u l t i p o l a r m o d e l c o n s i s t i n g o f a m a g n e t i c d i p o l e + g e n e r a l q u a d r u p o l e . P r e l i m i n a r y f i t s b y l e a s t - s q u a r e s t o a fe w ( ~ 1 0 ) l i n e s s i m u l t a n e o u s l y a r e e n c o u r a g i n g -- t h e a g r e e m e n t b e t w e e n t h e o b s e r v e d a n d c a l c u l a t e d S t o k e s p r o f i l e s i s s u b s t a n t i a l l y b e t t e r t h a n t h a t a c h i e v e d u s i n g a n a x i s y m m e t r i c m o d e l .

7 . O u t l o o k
T h e r e c u r r e n t l y e x i s t s a r e a s o n a b l y l a r g e c o l l e c t i o n o f f o u r S t o k e s p a r a m e t e r d a t a s u i t a b l e f o r m o d e l i n g . B a s e d o n T a b l e 2 a n d t h e f i g u r e s p r e s e n t e d b y W a d e e t a l . ( 1 9 9 9 a ) , t h e r e a r e 4 s t a r s ( 5 3 C a m , 7 8 V i r , C r B , C V n ) f o r w h i c h c o m p l e t e S t o k e s d a t a s e t s

e x i s t fo r w h i c h t h e o b s e r v a t i o n s a r e o f s u f f i c i e n t l y q u a l i t y t h a t m o d e l i n g o f i n d i v i d u a l s p e c t r a l l i n e s i s p o s s i b l e . T h e r e a r e f u r t h e r m o r e t h r e e a d d i t i o n a l o b j e c t s ( 4 9 C a m , H D 3 2 6 3 3 , H D 7 1 8 6 6 ) f o r w h i c h t h e o b s e r v a t i o n s a r e p r o b a b l y o f i n s u f f i c i e n t q u a l i t y f o r m o d e l i n g o f i n d i v i d u a l s p e c t r a l l i n e s , b u t f o r w h i c h L S D p r o f i l e s e x i s t . T h e e x i s t e n c e o f t h e s e o b s e r v a t i o n s a p p e a r s t o b e m o t i v a t i n g a n u m b e r o f i n v e s t i g a t i o n s a i m e d a t t h e i r i n t e r p r e t a t i o n , a n d t h e f u ll e x p l o i t a t i o n o f t h e s e d a t a w i l l l i k e l y t a k e a n u m b e r o f y e a r s t o c o m p l e t e . M e a n w h i l e , t h e d e v e l o p m e n t o f n e w highr e s o l u t i o n s p e c t r o p o l a r i m e t e r s ( f o r e x a m p l e t h e E S P a D O n S s p e c t r o p o l a r i m e t e r f o r t h e C a n a d a - F r a n c e H a w a i i T e l e s c o p e ) c o n t i n u e s . T h i s i s k e y , a s t h e b r i g h t e s t , s t r o n g e s t f i e l d , s h a r p e s t - l i n e d n o r t h e r n A p s t a r s h a v e a l r e a d y b e e n o b s e r v e d u s i n g t h e M u S i C o S s p e c t r o p o l a r i m e t e r , l e a v i n g o n l y t h e m o r e c h a l l e n g i n g o b j e c t s , m a n y o f w h i c h f a l l o u t s i d e t h e c a p a b i l -


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i t i e s o f t h i s i n s t r u m e n t . N e w , m o r e s e n s i t i v e i n s t r u m e n t s a r e r e q u i r e d t o c a r r y o n . I n a d d i t i o n , w h i l e M u S i C o S h a s t h e a d v a n t a g e o f w i d e s p e c t r a l c o v e r a g e , i t c a n a c h i e v e o n l y i n t e r m e d i a t e ( R = 3 . 5 x 1 0 ) s p e c t r a l r e s o l u t i o n . S u c h a r e s o l u t i o n c o r r e s p o n d s t o a b o u t 8 .5 k m s
-1 4

, a l l o w i n g l e s s t h a n 3 r e s o l u t i o n e l e -

m e n t s a c r o s s t h e r o t a t i o n a l l y - b r o a d e n e d l i n e p r o f i l e o f 4 9 C a m ( t h e m o s t r a p i d l y r o t a t i n g A p s t a r fo r w h i c h 4 S t o k e s p a r a m e t e r r e s u l t s h a v e b e e n o b t a i n e d t o d a t e ) . S p e c t r o p o l a r i m e t e r s w i t h s u b s t a n t i a l l y h i g h e r r e s o l u t i o n ( 1 - 2 x 1 0 ) w o u l d a l l o w t h e s t u d y o f S t o k e s p r o f i l e s a n d t h e i m a g i n g o f s t e l l a r s u r f a c e s w i t h m u c h g r e a t e r d e t a i l .
5

R e f e r e n c e s
B a b e l J . , M i c h a u d G . , 1 9 9 1 , A s t r o p h y s . J . , 3 6 6 , 560

B a g n u l o S ., L a n d o l f i M . , 1 9 9 9 , A s t r o n . A s t r o p h y s . , 4 3 6 , 1 58 B a g n u l o S ., M o n i n D . , L e o n e F . , S t i f t M . J . , 2 0 0 0 , i n t h i s b o o k B a g n u l o S ., M a t h y s G . , L a n d o l f i M . , Land i D e g l ' I n n o c e n t i M . , 2 0 0 0 , A s t r o n . A s t r o p h y s . , i n p r e s s B a u d r a n d J . , B o h m . , 1 9 9 2 , A s t r o n . A s t r o p h y s . , 2 5 9 , 711 D o n a t i J . - F . , C a t a l a C , W a d e G . A . , G a l l o u G . , D e l a i g u e G . , R a b o u R , 1 9 9 9 , A s t r o n . A s t r o p h y s . S u p p l . S e r . , 1 3 4 , 1 49 D o n a t i J . F . , S e m e l M ., C a r t e r B . D . , R e e s D . E . , C a m e r o n A . C . , 1 9 9 7 , M o n . N o t . R . A s t r o n . S o c , 2 9 1 , 658 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 M . , L a n d i D e g l ' I n n o c e n t i E . , 1 9 8 9 , A s t r o n . A s t r o p h y s . , 2 1 6 , 1 1 3 L a n d s t r e e t J . D . , 1 9 8 2 , A s t r o p h y s . J . , 2 5 8 , 6 3 9


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