Table Of ContentAstronomy&Astrophysicsmanuscriptno.13860 (cid:13)c ESO2010
January29,2010
Magnetic Doppler imaging of α2 Canum Venaticorum in
⋆
all four Stokes parameters
Unveiling the hidden complexity of stellar magnetic fields
O.Kochukhov1 andG.A.Wade2
1 DepartmentofPhysicsandAstronomy,UppsalaUniversity,Box516,UppsalaSE-75120,Sweden
2 DepartmentofPhysics,RoyalMilitaryCollegeofCanada,Box17000,Kingston,Ontario,K7K4B4Canada
0
1 Received14December2009/Accepted28January2010
0
2 ABSTRACT
n
Context.Strongorganizedmagneticfieldshavebeenstudiedintheuppermainsequencechemicallypeculiarstarsformorethanhalf
a
acentury.However, onlyrecentlyhaveobservational methodsandnumerical techniquesbecomesufficientlymaturetoallowusto
J
recordandinterprethigh-resolutionfourStokesparameterspectra,leadingtothefirstassumption-freemagneticfieldmodelsofthese
9 stars.
2 Aims. Here we present a detailed magnetic Doppler imaging analysis of the spectropolarimetric observations of the prototypical
magneticApstarα2CVn.Thisisthesecondstarforwhichthemagneticfieldtopologyandhorizontalchemicalabundanceinhomo-
] geneitieshavebeeninferreddirectlyfromphase-resolved observations of lineprofilesinallfour Stokesparameters, freefromthe
R
traditionalassumptionofalow-ordermultipolarfieldgeometry.
S Methods. WeinterprettherotationalmodulationofthecircularandlinearpolarizationprofilesofthestrongFeiiandCriilinesinthe
. spectraofα2CVnrecordedwiththeMuSiCoSspectropolarimeter.Thesurfaceabundancedistributionsofthetwochemicalelements
h
andafullvectormapofthestellarmagneticfieldarereconstructedinaself-consistentinversionusingourstate-of-the-artmagnetic
p
DopplerimagingcodeInvers10.
-
o Results. Wesucceededinreproducingmostofthedetailsoftheavailablespectropolarimetricobservationsofα2CVnwithamag-
r neticmapwhichcombinesaglobaldipolar-likefieldtopologywithlocalizedspotsofhigherfieldintensity.Wedemonstratethatthese
t small-scalemagneticstructuresareinevitablyrequiredtofitthelinearpolarizationspectra;however,theirpresencecannotbeinferred
s
a fromtheStokesIandVobservationsalone.Wealsofoundhigh-contrastsurfacedistributionsofFeandCr,withbothelementsshow-
[ ingabundanceminimaintheregionofweakerandtopologicallysimplermagneticfield.
Conclusions. OurmagneticDoppler imaginganalysisofα2CVnandpreviousresultsfor53Camsupport theviewthattheupper
1
mainsequence starscanharbour fairlycomplex surfacemagnetic fieldswhichresembleoblique dipolesonly atthelargestspatial
v
scales. Spectrainall four Stokes parametersareabsolutely essential tounveil andmeaningfully characterize thisfieldcomplexity
5
inApstars.Wethereforesuggest thatunderstanding magnetismofstarsinotherpartsof theH-Rdiagramissimilarlyincomplete
2
withoutinvestigationoftheirlinearpolarizationspectra.
0
0 Keywords.polarization–stars:atmospheres–stars:chemicallypeculiar–stars:magneticfields–stars:individual:α2CVn
.
2
0
0 1. Introduction and weak stellar winds) within the atmosphere. The character-
1 istic consequence of this interaction is the presence of strong
Magneticfieldsplayafundamentalroleinthephysicsoftheat-
: chemicalabundancenon-uniformitiesinphotosphericlayers.
v mospheresofasignificantfractionofstarsontheH-Rdiagram.
i Theweightofopinionholdsthatthe magneticfieldsof up-
X Themagneticfieldsofintermediate-massuppermainsequence permainsequencestarsareprimarilyfossilfields–theslowly-
stars(theApstars)haveverydifferentcharacteristics,andprob-
r decayingremnantsof magnetic field accumulatedor generated
a ably a different origin, than those of late- type stars like the
duringthecomplexprocessofstarformation.Inthisframework,
sun(e.g.Mestel2003).InApstars,thelarge-scalesurfacemag-
the global and local propertiesof the magnetic field (the mean
neticfieldisobservedtobestaticontimescalesofatleastmany
intensity,obliquityrelativetothestellarrotationaxis,andlarge-
decades,andappearstobe“frozen”intoarigidlyrotatingatmo-
scaletopology;butalsoitssmaller-scalestructure)maywellpro-
sphere.Themagneticfieldisgloballyorganized,permeatingthe
videuniqueinformationaboutinvisibleinteriorprocesseswhich
entire stellar surface,with a relativelyhighfield strength(typi-
have occurred or are occurring within the star: differential ro-
callyofafewhundredsuptoafewtensofthousandsofgauss).
tation, meridional circulation currents, global- and local-scale
Magneticfieldappearstobepresentin onlyasmallfractionof
dynamoaction,etc.Thereforeaknowledgeofthedetailedmag-
intermediate-massmainsequencestars,anditspresencestrongly
neticfieldstructureofApstars–onbothlargeandsmallscales–
influencesenergyandmasstransport(e.g.,diffusion,convection
cancontributesignificantlytounderstandingthephysicsofsuch
stars.
Sendoffprintrequeststo:O.Kochukhov,
e-mail:[email protected] The characteristics of magnetic fields in upper main se-
⋆ Based on data obtained using the Te´lescope Bernard Lyot at quence stars are inferred from the influence of the Zeeman ef-
ObservatoireduPicduMidi fect on their spectra. Even a relatively weak magnetic field
2 O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters
(a few tens of gauss) will clearly imprint its presence on the the analysis of α2CVn, to examine to what extent the smooth
local emergent spectrum of a star, by splitting and polariz- map derived by Kochukhovetal. (2002) is attributable to their
ing line profiles. The large majority of magnetic field data lack of linear polarizationdata. This is the topic of the present
in the literature measure the longitudinal Zeeman effect via paper. In Sect. 2 we describe the spectropolarimetric observa-
induced circular polarization within photospheric absorption tions fromwhich the MDI mapsare derived.In Sect. 3 we de-
lines, as quantified by the mean longitudinal magnetic field. termine the physical parameters of the star, with the particular
While such measurements represent a powerful tool for de- goal of constraining the orientation of the stellar rotation axis
tecting organized magnetic fields, they are relatively insensi- withrespecttotheline-of-sight.InSect.4wereviewbasicprin-
tive to the topology of the magnetic field, typically constrain- ciples of MDI, and describe the spectral lines used, their asso-
ing the strength and orientation of the field’s dipole compo- ciated atomic data, and the determinationof globalparameters
nent. If line circular polarization (the V Stokes parameter) is tobeusedinthemappingprocedure.InSect.5wedescribethe
measuredusinghighspectralresolvingpower,additionaldetail resultsofthemapping,payingparticularattentiontothesignif-
canberecoveredbyexploitingtherotationalDopplereffectvia icance and robustness of small-scale structures detected in the
Magnetic Doppler Imaging (MDI; see Piskunov&Kochukhov fieldmodulusmap.
2002;Kochukhov&Piskunov2002).Thiswasdemonstratedby
Kochukhovetal.(2002),whomappedthemagneticfieldofthe
2. Spectropolarimetricobservations
Ap star α2CVn using a timeseries of high-resolution Stokes I
and V line profiles, interpreted using MDI. To converge to a The main set of spectropolarimetric observations of α2CVn
unique solution, the map of α2CVn was forced to resemble, was obtained in 1997–1999 using the (now decommissioned)
tothegreatestextentpossiblegiventheconstraintsimposedby MuSiCoSspectropolarimeteratthe2-mBernardLyotTelescope
the data, a non-axisymmetricmultipolar configuration.The re- (TBL)atPicduMidiobservatory.
sultingmapshowedadominantdipolecomponent,withasmall The MuSiCoS instrument consists of a table-top cross-
quadrupolarcontribution.Overall,themapwasquitesmooth,in dispersed e´chelle spectrograph (Baudrand&Bohm 1992), fed
agreementwithexpectations. by optical fibres directly from a Cassegrain-mounted polariza-
Additional magnetic field structural detail is accessible tion analysismodule.Thisinstrumentallowsthe acquisitionof
throughthespectrallinelinearpolarization(QandU Stokespa- a stellar spectrumin a givenpolarizationstate (StokesV, Q or
rameters) inducedby the transverseZeeman effect. As demon- U)throughoutthespectralrange450to660nminasingleexpo-
stratedbyLeroy(1995),Zeemanlinearpolarization,evenwhen sure.Theresolvingpowerisλ/∆λ≃ 35000.Theopticalcharac-
measuredphotometricallyusingbroadbandpasses,providesin- teristicsofthe spectropolarimeterandcorrespondingobserving
formation about the smaller-scale structure of the magnetic proceduresaredescribedindetailbyDonatietal.(1999).
fieldthatisnotavailablefromcircularpolarization.Wadeetal. When MuSiCoS is used for polarization observations,
(2000b)extendedtheseobservationalinvestigationstotheirnat- starlight enters the polarimeter at the Cassegrain focus. The
ural culmination by obtaining high-precision measurements of beam then may optionally pass through a rotatable λ/4 re-
Zeemanpolarizationinbothcircularandlinearpolarization(i.e. tarder (in the case of Stokes V observations) or not (in the
in all four Stokes I, V, Q and U parameters) for a sample of case of Stokes Q or U observations). The beam then inter-
about 15 Ap stars, with a resolving power sufficient to distin- sectsaSavart-typebeamsplitterwhichseparatesthestellarlight
guishthevariationofthepolarizationacrossindividualspectral intotwobeamswhicharerespectivelypolarizedalongandper-
lines.Foraboutadozenofthesestars,multipleStokesIQUVse- pendiculartotheinstrumentalreferenceazimuth.Theanalyzed
quenceswereacquired,allowingacharacterisationofthelongi- beamsaretheninjectedintothedouble50µmfibre,whichtrans-
tudinalandtransversecomponentsofthemagneticfieldondif- ports the light to the spectrograph. Spectra in both orthogonal
ferent regions of the stellar surface. For the A2p star 53 Cam, polarizationsaretherebyrecordedsimultaneouslyonthethinned
Wadeetal.(2000b)andBagnuloetal.(2001)demonstratedthat 1024×1024pixelsiteCCDdetector.
the existing low-order multipolar models of the magnetic field Asinglepolarimetricobservation,yieldinganintensityspec-
wereunabletoreproducetheobservedStokesQandU spectra, trumandoneotherStokesparameter,consistsofasequenceof4
althoughtheyacceptablymatchedtheStokesVspectra.Theyin- subexposures,betweenwhichtheretarder(forcircularpolariza-
terpretedtheirresultsasindicatingthatthemagnetictopologyof tionStokesV)orthepolarimetricmoduleitself(forlinearpolar-
53Camwassignificantlymorecomplexthanthatenvisionedby izationsStokes Q andU)isrotatedby±90◦ followingthepro-
themodels. cedure suggested by Semeletal. (1993). This has the effect of
Totestthisproposal,Kochukhovetal.(2004b)appliedMDI exchangingthebeamswithinthewholeinstrumentandswitch-
to the 53 Cam data of Wadeetal. (2000b) and Bagnuloetal. ingthepositionsofthetwoorthogonallypolarizedspectraonthe
(2001) – the first attemptto reconstructa stellar magneticfield CCD.Thisobservingproceduresuppressesallfirst-orderspuri-
usingmeasurementsofspectrallinesinthefourStokesparam- ouspolarizationsignaturesdowntowellbelowthenoiselevel.
eters. Kochukhovet al. were able to reproducethe variable in- Spectra obtained at the TBL using the MuSiCoS spectro-
tensityandmorphologyofthepolarizedlineprofilesof3strong graphandpolarimeterwerereducedusingtheESpRITreduction
Fe ii lines throughout the star’s rotational cycle. In agreement package(Donatietal.1997).TheESpRIT-reducedspectrawere
with the prediction of Wadeetal. (2000b) and Bagnuloetal. thenpost-processedtoimprovethequalityofthecontinuumnor-
(2001), the derived field topology was surprisingly complex. malization. A full description of the acquisition and reduction
In particular, while the radial field exhibited an approximately procedureof the MuSiCoS observationsofα2CVn is provided
dipolar behaviour, the tangential field and the field modulus byWadeetal.(2000b).
showedstructureonmuchsmallerscales. A complete list of the spectropolarimetric observations of
Inthecontextoftheiranalysisof53Cam,Kochukhovetal. α2CVnanalyzedinourstudyispresentedinTable1.Thistable
demonstratedthattheconstraintonthetransversefieldprovided providesinformationontheUTdateofobservation,listsStokes
by linear polarization measurements was essential to detecting parametersobtained,theirJuliandates,meanrotationphasesand
the fine structure of the field. It was thereforenaturalto revisit thesignal-to-noiseratio.TheStokesV spectrumrecordedon25
O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters 3
February 1997 was treated separately from the linear polariza- (Maury&Pickering1897) andisaprototypeofspectrumvari-
tion observations on that night due to a gap in observing time able stars. It was classified as A0pSiEuHg by Cowleyetal.
exceeding1%oftherotationperiod. (1969) and was targeted by many notable magnetic and spec-
AcloseexaminationoftheStokesQandUprofilesobtained trum variability studies (Babcock&Burd 1952; Pyper 1969;
on20February1997(ϕ = 0.391,see Wadeetal.2000b),com- Cohen1970;Borra&Landstreet1977),includingearlyattempts
paringwiththeobservationsclosestinphase,suggeststhattheir to reconstruct maps of abundance spots and magnetic field
signs are inverted. These particular linear polarization spectra with the Doppler Imaging technique (Goncharskiietal. 1983;
were obtainedwith an interruptionin the observingprocedure, Khokhlova&Pavlova1984;Glagolevskiietal.1985).
which possibly led to a sign error. We have obtained an ad- InthepreviousDIstudyofα2CVn(Kochukhovetal.2002)
ditional observation of α2CVn on 26 July 2004, at the rota- wehavedeterminedtheeffectivetemperatureandsurfacegravity
tion phase ϕ = 0.413. The Stokes Q and U profiles from this usingtheoreticalmodelatmospherefittotheopticalspectropho-
phase are inverted in comparisonto the Feb 20, 1997 observa- tometry and hydrogen Balmer line profiles. We found that the
tion and agree with the other spectra obtained in 1997–1999. ATLAS9modelscomputedwith10timessolarmetalabundance
Consequently, we decided to disregard the linear polarization provideasatisfactoryfittoobservationsforT =11600±150K
eff
spectra from 20 February 1997, using only the Stokes V data andlogg=3.9±0.1.SubsequentlyLipski&Ste¸pien´(2008)used
fromthatnight. the same model atmosphere code to fit the optical spectral en-
Thus,inthemagneticDIanalysispresentedbelowwehave ergy distribution simultaneously with the UV fluxes recorded
modeled20separatephasesofpartialorcompleteStokesparam- by the IUE satellite. They showed that the UV data require
eterobservationsofα2CVn.Fortherotationphaseswhere2or T ≈10750K,whichisabout1000KcoolerthanT inferred
eff eff
3intensityprofileswereavailablewitheachoftheotherStokes from the optical spectrophotometry. The final T =11250 ±
eff
parameters,wehaveconstructedanaverageStokesI spectrum. 500 K recommended by Lipski&Ste¸pien´ (2008) is formally
All spectra were then interpolated onto a common wavelength notdifferentfromthetemperatureadoptedbyKochukhovetal.
gridasrequiredforcomparisonwiththeoreticalcalculations. (2002),buthasamuchlargeruncertaintyreflectingthedifficulty
Allobservationsofα2CVn arephasedusingtheephemeris offittingthestellarspectruminabroadwavelengthregion.
ofFarnsworth(1932):
It is evident that further progress in deriving a realistic
model structure of the atmosphere of α2CVn can be achieved
JD(Eu )=2419869.720+5d.46939×E, (1)
max bycomputingmodelswithindividualisedchemicalcomposition
andmagneticfield(Shulyaketal.2004;Kochukhovetal.2005).
which gives the time of the maximum intensity of the spectral
linesofEuiiandroughlycorrespondstothenegativeextremum Nevertheless,forthepurposeofourinvestigation,andtoretain
the possibility of a direct comparison with the results of the
of the longitudinal magnetic field phase curve (Wadeetal.
previous DI study, it is sufficient to use the model atmosphere
2000a).
parametersofKochukhovetal.(2002),whichrepresentreason-
ablywellthemeanatmosphericpropertiesofthestar.Usingfor-
3. Physicalpropertiesofα2CVn wardspectrumsynthesiscalculationswehaveverifiedthatmod-
ificationofT by±500Kproducesonlyamarginalchangeof
eff
OurmodelingoftheStokesIQUVspectraofα2CVnandthein- the line intensities, comparable or smaller than the uncertainty
terpretationoftheirrotationalvariabilityintermsofthesurface duetoimperfectatomicdata.Thus,thetemperatureuncertainty
mapsofbothmagneticfieldandchemicalelementsrequiresde- isnotacriticalfactorinouranalysisofα2CVn.
termining certain basic physical and geometrical properties of
In addition to the uncertainties mentioned above, the as-
thetargetstar:
sumption of a constant model structure is not fully correct for
spottedApstarssincethelineandcontinuousopacityisdiffer-
i Atmospheric parameters – T , logg and average chemical
eff entinsideandoutsidespots.However,thiscommonassumption
composition–areneededasinputsforamodelatmosphere
is justified forDopplerimagingstudiessimilar to oursbecause
code, which provides us with the vertical temperature-
thelocallineprofilesaresensitivetomodelstructureeffectstoa
density stratification used for the numerical calculation of
muchsmallerdegreethantochangesofabundanceormagnetic
thelocalStokesprofilesandcontinuumintensities;
field.
ii RotationperiodP andprojectedrotationalvelocityv sini
rot e
are needed to establish the mutual phasing of observations Taking into account the distance to α2CVn, d=35.2 ±
andtoevaluatetherotationalDopplershiftacrossthestellar 1.1 pc which follows from the revised Hipparcos parallax
surface; (vanLeeuwen2007),andadoptingV = 2.89±0.04fromPyper
iii Two angles, i and Θ, establish the orientation of the (1969),wefindtheabsolutevisualmagnitudeMV=0.16±0.08.
stellar axis of rotation with respect to the observer For the temperature Teff=11600 ± 500 K and the bolomet-
(Piskunov&Kochukhov2002). riccorrectionBC=−0.42measuredbyLipski&Ste¸pien´ (2008)
we obtain L=101 ± 12 L and R=2.49 ± 0.26 R assuming
⊙ ⊙
Some of these parameters can be adjusted by optimiz- M⊙ =4.75fortheSun(Bessell2000)andusinganuncertainty
bol
ing the fit to the observations with multiple inversions of 0.1 for the bolometric correction of α2CVn. Then, employ-
(Kochukhov&Piskunov2002).Otherparametersproducelittle ingthisstellarluminosityandtemperaturewecandeterminethe
impactontheStokesspectraandDopplerimagesandhencere- evolutionary state of α2CVn by comparing its observed posi-
quireapriorestimation. tion in the H-R diagram with the predictions of the theoretical
evolutionarytracks by Schalleretal. (1992) and Schaereretal.
(1993). Assuming an overall metallicity Z = 0.018, we derive
3.1.Atmosphericparameters
astellarmass M=2.97±0.07 M andanageof1.65+0.6×108
⊙ −0.7
α2CVn (12 CVn A, HD112413, HR4915, HIP63125) years.Theseparametersyieldthesurfacegravitylogg=4.12±
was the first star given the “A-type peculiar” classification 0.09,whichcanbebroughtinagreementwiththespectroscopic
4 O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters
Table1.FourStokesparameterspectropolarimetricobservationsofα2CVnusedforMagneticDopplerImaging.
UTDate Stokes JD(2450000+) S/N
Parameters Q U V ϕ δϕ range
18Feb1997 Q U V 498.511 498.531 498.489 0.038 0.004 420–550
19Feb1997 Q U V 499.532 499.555 499.511 0.225 0.004 650–690
20Feb1997a V 500.422 0.387 550
21Feb1997 Q U V 501.490 501.504 501.470 0.582 0.003 270–320
22Feb1997 Q V 502.549 502.519 0.773 0.003 320–490
23Feb1997 Q U V 503.497 503.516 503.474 0.949 0.004 240–280
25Feb1997b V 505.581 0.331 470
25Feb1997 Q U 505.695 505.715 0.353 0.002 410–410
07Feb1998 Q U V 852.562 852.584 852.539 0.771 0.004 740–890
09Feb1998 Q U V 854.561 854.584 854.539 0.137 0.004 820–1020
11Feb1998 Q U V 856.545 856.567 856.524 0.499 0.004 910–1020
12Feb1998 Q U V 857.675 857.694 857.657 0.706 0.003 820–880
13Feb1998 Q U V 858.555 858.574 858.537 0.867 0.003 710–810
15Feb1998 Q U V 860.605 860.626 860.586 0.242 0.004 500–720
16Feb1998 Q U V 861.606 861.627 861.584 0.424 0.004 560–750
17Feb1998 Q U V 862.533 862.554 862.515 0.594 0.004 560–560
19Feb1998 Q U V 864.603 864.638 864.576 0.973 0.006 430–590
15Jan1999 Q V 1194.621 1194.600 0.310 0.002 400–450
19Jan1999 Q U V 1198.652 1198.672 1198.633 0.049 0.004 770–840
26Jul2004 Q U V 3213.362 3213.381 3213.400 0.413 0.004 440–480
ColumnsgivetheUTdateforthebeginningofthenightwhenobservationwasobtained,theStokesparametersobserved(inadditiontoStokes
I),theJulianDatesofmid-exposureforeachobservedStokesparameter,themeanphaseϕforallobservedStokesparameters,themaximum
differenceδϕbetweenϕandthephasesofindividualStokesparameterobservations,therangeofthesignal-to-noiseratioper2.6km/sspectral
pixelfortwoorthreeexposures.
aTheStokesQandUspectraaredisregarded.Seetextfordetailedexplanation.
bTheStokesVobservationisconsideredindividuallyduetoalarger-than-averagephasedifferencewithrespecttothelinearpolarizationspectra
obtainedonthatnight.
Table2.Fundamentalparametersofα2CVn. surementsobtainedoverthe time spanof 25 years(Wadeetal.
2000a).Thus,therotationalperiodofFarnsworth(1932)isade-
Parameter Value Ref. quateforouranalysis.Wedonotattempttoimproveitsinceno
T 11600±500K (1),(2) recentextensivephotometricorspectroscopictime-seriesobser-
eff
logg 3.9±0.1 (1) vationsareavailableforα2CVn.
Distance 35.2±1.1pc (3)
Theprojectedrotationalvelocityisoneofthekeyinputpa-
M 0.16±0.08 (4)
V rametersforDIinversionsandneedstobedeterminedwitharel-
BC −0.42±0.1 (2)
ativelyhighprecision(Kochukhov&Piskunov2002).Itisbest
L 101±12L (4)
⊙
R 2.49±0.26R (4) foundbyoptimizingthefittolineprofilesobservedathighspec-
⊙
M 2.97±0.07M (4) tral resolution. As described in Sect. 4.3, the intensity profiles
⊙
Age 1.65+0.6×108yr (4) ofthestrongFeiiandCrii linesinthe spectrumofα2CVn are
−0.7
v sini 18.4±0.5kms−1 (4) reproducedwithv sini=18.4±0.5kms−1.Thisvalueissome-
e e
P 5d.46939 (5) what larger than 17.4± 0.5 kms−1 found by Kochukhovetal.
rot
i 120±5◦ (4) (2002)usingCriilinesofdifferentstrength.Althoughformally
Θ 115±5◦ (4) notsignificant,thisdiscrepancycouldreflectthedifferenceinin-
strumentalprofileor,possibly,beanartifactofverticalchemical
References.(1)Kochukhovetal.(2002);(2)Lipski&Ste¸pien´ (2008);
stratificationneglectedinourspectrumsynthesis.
(3)vanLeeuwen(2007);(4)thiswork;(5)Farnsworth(1932).
Theorientationofthestellarrotationalaxisischaracterized
by the two angles i and Θ. The inclination, 0◦ ≤ i ≤ 180◦, is
estimateifthelatteriscorrectedforthestrongHedeficiencyof the angle between the rotational axis and the observer’s line-
theatmosphereofα2CVn(Kochukhovetal.2002). of-sight. The values i > 90◦ correspond to the situation when
The fundamental parameters of α2CVn determined in our we see a clockwise rotation of the star from the visible ro-
study,aswellasthoseadoptedfrompreviousinvestigations,are tation pole. The azimuth angle, 0◦ ≤ Θ ≤ 360◦ determines
summarizedinTable2. the sky-projectedpositionangleof therotationalaxis. Thisan-
gle is counted counterclockwise from the North Celestial Pole
(Landolfietal.1993).Theazimuthangleisrelevantandthedis-
3.2.Rotationalparameters tinctionbetweeniand180◦−icanbemadeonlywheninterpret-
The widely-used rotation period of α2CVn, 5d.46939, was de- ing linear polarizationobservations.Note that since the Stokes
Q and U parametersdependon trigonometricfunctionsof 2Θ,
termined by Farnsworth (1932) based on a series of intensity
measurementsoftheblueEuiilinestakenatseveralobservato- thevaluesofΘandΘ+180◦cannotbediscriminated.
riesduringtheyears1912–1932.Usingthisrotationalperiodone NumericaltestsofthefourStokesparameterinversionpro-
cansuccessfullyphasetogetherlongitudinalmagneticfieldmea- cedure(Kochukhov&Piskunov2002)indicatethatthetiltand,
O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters 5
Fig.1.Photopolarimetricmeasurementsofthelongitudinalmag-
netic field in α2CVn (Borra&Landstreet 1977; Landstreet
1982, upper panel) and the total net linear polarizationderived
from the LSD Stokes Q and U measurements of Wadeetal.
(2000a)(lowerpanel)comparedwiththepredictionsofthedipo-
larmodel(solidcurves)withparametersB = 4.6kG,i = 119◦
p
and β = 78◦. The model with the same B and i′ = 180◦ −i,
p
β′ =180◦−βyieldsindistinguishablecurves.
Fig.2. Comparisonof the net linear polarization(symbols)ob-
especially, the azimuth angle of the rotational axis can be re- tainedfromtheLSDStokesQandU profilesofα2CVn(upper
liably determined by minimizing the χ2 of the fit to observa- andmiddlepanels,respectively)withthedipolarmodelpredic-
tionsininversionswithdifferentiandΘ.However,ourmagnetic tions(lines).Thesolidlinecorrespondstothemagneticfieldge-
DI technique is fairly costly in computing time, which makes ometrygivenbyB =4.6kG,i=119◦,β=78◦,Θ=110◦.The
p
exploring all possible rotational axis orientations impractical. dashed line shows predictions of the best-fitting model for the
Insteadwestartbyfindinga plausiblerangeforeachangleus- sameB ,180◦−i,180◦−βandΘ=119◦.Thelowerpanelshows
p
ingasimpledipolarfieldmodelandthenrefinetheangleswith thetotalchi-squareofthefittonetlinearpolarizationcurvesas
aseriesofMDIreconstructions. a function of the azimuth angle Θ for the dipolar model with
The inclination angle is inferred from the simultaneous i = 119◦, β = 78◦ (solid line) and for the modelwith comple-
fit of the dipolar field model to the Balmer line photopo- mentaryangles180◦−iand180◦−β(dashedline).
larimetric longitudinalfield measurements(Borra&Landstreet
1977; Landstreet 1982) and the total linear polarization, P ≡
L
(Q/I)2+(U/I)2, obtained from the net linear polarization ducesthemainfeaturesofthehB iandP phasedependenceif
mpeasurements of Wadeetal. (2000a). We employ longitudinal weadoptparametersB =4.6±0.z5kG,β=L78±7◦,i=119±12◦
p
field measurements obtained from H lines to avoid the strong (or 180◦ − β=102◦ and 180◦ − i=61◦). The inclination angle
modulationduetotoabundancenonuniformitiesthatisevident obtainedinthiswayisconsistentwithi=53±8◦thatcanbede-
in longitudinal field measurements derived from, e.g., lines of rivedfromP ,Randv siniofα2CVnusingtheusualoblique
rot e
Fe or Cr (Wadeetal. 2000a). The dipole modelparametersin- rotatorrelation.
cludethepolarfieldstrengthB ,magneticobliquityβandangles In the second step we find the values of Θ which provide
p
iandΘ.ThetheoreticalP iscomputedfollowingLandolfietal. thebestmatchtotheobservedrotationalvariationofthe Qand
L
(1993). Using the total linear polarizationinstead of individual U net linear polarizationfor the previously determineddipolar
net Q andU measurementsallowsusto avoidsolvingforΘ at modelparameters.Fig.2demonstratesthataglobalχ2minimum
thisstageoftheanalysis. isfoundforΘ≈110◦andthatβ=78◦,i=119◦provideabetter
Thecomparisonofthemagneticobservableswiththedipo- matchtotheobservationsthan180◦−βand180◦−i.Thesame
lar modelpredictionsis presentedin Fig.1. Thescatter around conclusion was reached by Wadeetal. (2000a). Thus, prelimi-
thebest-fitcurveisconsiderablebutthemodelevidentlyrepro- narymodelingsuggestsi≈120◦andΘ≈110◦.Thefinalvalues
6 O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters
oftheseparametersareestablishedinSect.4.3withthehelpof We apply regularization to ensure stability of the complex
multiple MDI inversions which take into account chemical in- optimizationprocessofmagneticDIandtoobtainauniqueso-
homogeneitiesandincorporatedeviationsof the magneticfield lution independent of the initial guess and of the surface dis-
topologyfromasimpledipole. cretization.ThegeneralformoftheregularizationfunctionalR
implementedinInvers10is
4. MagneticDopplerimaging R = Λ1XX(Bi−Bj)2+Λ2X(Bi−Bmi ult)2 (5)
i j i
4.1.Magneticinversiontechnique
AresdoeltuatiiloenddciirsccuulsasrioannodflMinaegarneptioclaDriozpaptiloenrIsmpeacgtirnagiussipnrgovhiidgehd- + Λ3Xi Xj (ε1i −ε1j)2+Xi Xj (ε2i −ε2j)2+....
byPiskunov&Kochukhov(2002).InthatpublicationtheMDI The first and third terms representthe Tikhonov regularization
code Invers10 was introduced, and key numerical techniques
functions for magnetic field and abundance distributions, re-
employed in magnetic inversions were discussed. Here we spectively. Λ and Λ are correspondingregularization param-
1 3
brieflyrecountthebasicprinciplesofourMDImethodology,re- eters, selected in such a way that D/R≈2–10 at the conver-
ferringthe readerto Piskunov&Kochukhov(2002) forfurther
gence. Application of the Tikhonov regularization leads to lo-
detailsandforanin-depthdescriptionoftechnicalissues. cal smoothingof the solutionby minimizingthe differencebe-
The aim of the magnetic inversion is to derive the surface tween thevalueof parametersin the gridpointi andallneigh-
distributionsofmagneticfieldandchemicalabundancesbymin-
bouringpoints j.AppropriatechoiceoftheTikhonovregulariza-
imizingthetotaldiscrepancyfunction tion ensures that the magnetic and abundance distributions re-
constructedbyourcoderepresentthesimplestpossiblesolution
Ψ=D+R, (2)
which is still compatible with observational data. At the same
where D characterizes the discrepancy between the observed time, the Tikhonov regularization does not impose an a priori
andcomputedphase-resolvedspectraandRistheregularization mathematical model for the global magnetic field geometry or
functional.ForDIwithpolarizationdata abundancedistribution.
A comprehensive series of numerical experiments with
D= ω Fcomp(B,ε1,ε2,...)−Fobs 2/σ2 , (3) Invers10waspresentedbyKochukhov&Piskunov(2002).This
X kh kϕλ kϕλi kϕλ paper explored the performance of the magnetic DI method
kϕλ
for differentcombinationsof stellar parameters, magnetic field
where F are the computed and observed Stokes parameter strengths and geometries by reconstructing surface maps from
kϕλ
profiles for rotation phase ϕ and wavelength point λ. The in- simulatedStokesparameterobservations.Thesenumericaltests
dex k represents summation over the available Stokes param- convincingly demonstrate that a successful inversion, reveal-
eters (all four in the present study of α2CVn), while weights ingmagneticfield structuresatboththelargeandsmallscales,
ω are introduced to ensure approximately the same contribu- is possible when the code is applied to the full Stokes vec-
k
tionofeachStokesparametertoD.Inpracticethisisachieved tor data set in conjunction with Tikhonov regularization. In
by choosing ω proportionalto the phase-averaged amplitudes fact, Piskunov (2005) showed that, from the basic mathemati-
k
of the observed Stokes parameters. For our α2CVn data set calstandpoint,theunderlyinginverseproblemofmagneticmap-
ω :ω :ω :ω =1:45:45:6. ping with all four Stokes parameters is well-posed and has a
I Q U V
TheoreticalStokesIQUV spectraforagivenrotationphase unique solution. In this case regularization is only needed to
dependonthesurfacetopologyofthemagneticfield Bandthe suppressinstabilities producedby the noise in real data and its
geometryoftheabundancedistributionsε1,ε2,etc.Normalized sparsephaseandwavelengthcoverage.However,duetothelack
StokesprofilesareobtainedbysummingtheDoppler-shiftedlo- of high-qualityStokes IQUV stellar spectra, the only previous
calStokesspectraX(k) overadiscretesurfacegridanddividing magneticDIinversionbasedonpolarizationobservationsinall
i
bythedisk-integratedintensityintheunpolarizedcontinuum four Stokes parameters was our study of the Ap star 53Cam
(Kochukhovetal.2004a).
Nϕ Nϕ While Stokes IQUV datasets suitable for Magnetic DI are
Fkcϕoλmp =XSi(ϕ)X(ik)(B,ε1,ε2,...,∆λD)/XSi(ϕ)Iic. (4) relatively rare, comparable Stokes IV datasets are more com-
i=1 i=1 mon. Magnetic DI with only circular polarization observations
hasthereforebeenmorewidelyapplied.However,suchdatasets
Here,theindexirunsoverasetofN surfacezonesvisibletothe
ϕ result in maps which are less informative and robust com-
observerattherotationalphaseϕandS(ϕ)istheprojectedareaof
i pared to those computed from full Stokes vector data. The IV
surfaceelementi.Forthepresentstudyofα2CVnweuseagrid mapping problem is intrinsically ill-conditioned and the out-
with695surfacezones,whichisfullysufficientgiventhemod- come of such inversions depends sensitively on the choice
eratevesiniofthestarandthelimitedspectralresolutionofthe of regularization function and the initial guess. Our numeri-
observationaldata.ThelocalStokesparameterprofilesX(k) are cal experiments with Invers10 (Piskunov&Kochukhov 2002;
i
computedforagivenmodelatmosphereandlocalvaluesofthe Kochukhov&Piskunov 2002) showed that a successful re-
abundancesandmagneticfieldbysolvingnumericallythepolar- construction of the global field topology of Ap stars using
izedradiativetransferequationinitsgeneralform,asdescribed only Stokes IV data requires imposing a prior assumption that
byPiskunov&Kochukhov(2002).Thus,theoreticalspectraare strongly restricts the range of possible derived magnetic field
computed by Invers10 fully self-consistently with the current topologies. In our DI code this assumption is implemented in
magneticandabundancemapsandtreatingthelineformationin theformofso-calledmultipolarregularization(thesecondterm
themagnetizedstellaratmospherewithoutanyofthesimplifying in Eq. (5)), which directs the solution toward a general, non-
assumptionsthatarecommonlyused,e.g.aMilne-Eddingtonat- axisymmetric second-order multipolar expansion of the mag-
mosphereorfixedlocalGaussianlineprofiles. neticfieldstructure.ThemultipolarmodelfieldBmult,calculated
O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters 7
Table3.Atomicparametersofspectrallinesemployedinmag-
neticinversions.
Ion λ(Å) E (ev) loggf J J g g
lo lo up lo up
Crii 4824.127a 3.871 −0.970 4.5 4.5 1.34 1.34
Feii 4824.198 10.288 −1.215 4.5 4.5 1.45 1.22
Feii 4824.838 8.145 −1.893 4.5 3.5 1.25 1.24
Feii 4923.927 2.891 −1.320 2.5 1.5 2.00 2.40
Feii 5018.440a 2.891 −1.271b 2.5 2.5 2.00 1.87
Feii 5019.462 5.569 −2.697 3.5 4.5 1.14 1.10
Thecolumnsspecifytheion,thecentralwavelengthλ,excitation
potentialoftheloweratomiclevelE ,quantumnumbersJandLande´
lo
factorsgforboththelowerandupperlevels.
aDominantblendcomponents.
bOscillatorstrengthisadjustedintheMDIinversion.
by our code at each iteration, is equivalent to the dipole plus
Fig.3.TherelativestandarddeviationofthefittoCriiandFeii
quadrupoleparameterizationofBagnuloetal.(1996).StokesIV
time series of the Ap stars α2CVn (Kochukhovetal. 2002), Stokes I profiles is plotted as a function of vesini (symbols).
Thesolidcurveshowsthequadraticfitinthe17.5–19.5kms−1
HD24712 (Lu¨ftingeretal. 2010) and HD72106 (Folsometal.
2008) were interpreted using Invers10 in the multipolar regu- region.Theverticaldashedlinecorrespondstotheoptimalpro-
jectedrotationalvelocityv sini=18.4±0.5kms−1.
larizationmode.Allthesestudiesinferredbasicallydipolarfield e
geometriesforthetargetstars.
The availability of MuSiCoS four Stokes parameter time-
resolved spectra for α2CVn provides us with the capability to line shows less prominent Stokes Q and U profiles because it
performmagneticinversionconstrainedonlybylocal,Tikhonov is somewhatweaker than the Feii lines, and exhibits a smaller
regularization(Λ ,0,Λ =0,seeSect.5.1).Atthesametime, Zeeman effect (g¯ = 1.3). Nevertheless, simultaneous recovery
1 2
itisalsoveryinstructivetocomparetheseultimateMDIresults ofthesurfacemagneticfielddistributionfromthelinesofmore
with the maps obtained disregarding linear polarization obser- than one chemical element always improves the robustness of
vations and globally constraining the magnetic inversion with MDI(Kochukhov&Piskunov2002).
themultipolarregularization.InSect.5.2wepresentresultsfor Thecompletelinelistadoptedinthemagneticinversionsof
magnetic DI in the circular polarization mode, with Λ = 0, α2CVn is presented in Table 3. Parameters of the two strong
1
Λ ,0. Feii lines, one Crii line and three weaker Feii blends are ex-
2
tractedfromtheVALDdatabase(Kupkaetal.1999).Theoscil-
lator strength of the lines from Feii multiplet 42, to which the
4.2.Thechoiceofspectrallines
4923.93and5018.44Ålinesbelong,arenotoriousforthelarge
Theamplitudeoflinearpolarizationsignaturesinmetallinesof scatter of loggf recommended in different literature sources.
eventhemoststrongly-magneticApstarsistypicallyatthelevel Eventherelativeloggf valuesoftheselinesarenotknownwith
ofonly10−3 (Wadeetal.2000b).Consequently,wearelimited the precision necessary for detailed spectrum synthesis mod-
tostudyingtheStokes QandU profilesofafewdeepest,mag- eling, promptingKochukhovetal. (2004a) to performseparate
netically sensitive individual lines in the stellar spectra. Using MDIinversionsforeachoftheselinesintheirstudyof53Cam.
suchfeaturesisusuallynotoptimalforabundancemappingbe- To overcome this problem here we modified Invers10, adding
cause the lines are saturated over a significant part of the stel- the oscillator strength of the Feii 5018.44 Å line to the list of
larsurfaceandthustheirvariationissubduedincomparisonto free parameters determined by the code. Table 3 lists the final
weaker lines of the same ions. For this reason, and also due to loggf for this lines, which is 0.051 dex lower than the value
therelativelylowresolvingpoweroftheMuSiCoSobservations, recommendedbyVALD.
wedonotexpecttoachievethesamequalityofabundanceinver- Additionalline data requiredforcalculationof the Zeeman
sionsas demonstratedbyKochukhovetal. (2002) in the previ- splitting includes Lande´ factors of the upper and lower atomic
ousDIanalysisofα2CVn.However,thisreducedsensitivityof levelsandcorresponding J quantumnumbers.These quantities
thestronglinestoabundanceinhomogeneitiesmakethemmore arealsospecifiedinTable3,withLande´factorsoriginatingfrom
suitableforourprimarygoalofthemappingmagneticfieldus- thetheoreticalcomputationsbyKurucz(1993)andextractedus-
ingfourStokesparameterprofiles. ingtheVALDinterface.
We found three lines in the spectrum of α2CVn appropri-
ate for magnetic DI. The two Feii features at λ 4923.93 and
4.3.Optimizationofv sini,iandΘ
5018.44 Å were previously used by Kochukhovetal. (2004a) e
andKhalack&Wade(2006)inthestudiesofotherApstarsob- Magnetic inversion is sensitive to the adopted projected ro-
served with the MuSiCoS spectropolarimeterand proved to be tational velocity and the three-dimensional orientation of the
effective for the reconstruction of the magnetic field topology. stellar rotational axis. Numerical experiments presented by
TheirmeanLande´factors,g¯ =1.7–1.9,areamongthelargestfor Kochukhov&Piskunov (2002) showed that solutions with in-
metallinesof suchintensity.Both linesshow a clearand com- correct values of these parameters tend to have a higher χ2 of
plexsignalintheStokesQandU spectra,detectedathighsig- the fit to the observedStokes profiles. For v sini this degrada-
e
nificancelevelsforthemajorityofrotationphases(see Fig.5). tionofthefitqualityismostclearlyseeninStokesI.Ameasure
WealsousetheCrii4824.13Ålineformagneticmapping.This ofthefinaldiscrepancybetweentheobservedandcomputedin-
8 O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters
Especially impressive is the ability of the model to reproduce
thesystematicdetailsofthelinearpolarizationprofiles,suchas
thepeculiar”w”-shapeofStokesUaroundphase0.4.Theagree-
mentbetweentheobservationsandsyntheticStokesV spectrais
excellent.
Thesurfacemagneticfielddistributionofα2CVnrecovered
byInvers10fromthephase-resolvedStokesprofilesoftheCrii
and Feii lines is presented in Fig. 6. This plot shows spherical
projectionsofthemagneticmap,displayingseparatelythefield
strengthandthefieldorientationdistributions.Inthisandsimilar
plotsthe star is shownat 5 equidistantrotationalphases, at the
aspectanglecorrespondingtoi= 120◦andΘ = 0◦.Theoverall
structure of the magnetic field in α2CVn is dipolar-like in the
sense that approximately half of the stellar surface is covered
withtheoutward-directedradialfieldwhileanotherhalfexhibits
inward-directedfield.Thisglobalmagneticfieldtopologyagrees
with the orientation of the dipolar axis suggested in previous
magneticanalysesofα2CVnwheremultipolarmodelswerefit-
Fig.4.Therelativetotaldiscrepancyfunctionisplottedagainst
tedtointegralmagneticobservables(Borra&Landstreet1977;
theazimuthangleΘadoptedfortheinversion.Differentsymbols
Gerthetal.1999)ortoStokesIVlineprofiles(Kochukhovetal.
showresultsobtainedforthreevaluesoftheinclinationanglei.
2002).However,thefieldstrengthdistributionillustratedinthe
FromthesecurvesweinfertheoptimalvaluesΘ=115±5◦and
upper panel of Fig. 6 reveals that the derived magnetic field
i=120±5◦.
topologyofα2CVnresemblesthatofadipoleonthelargestspa-
tial scale. The outcome of our four Stokes parameter inversion
tensityprofilesoftheCriiandFeiilinesisillustratedinFig.3. suggeststhatthefieldis,infact,farmorecomplex.Inparticular,
Examiningtheinversionresultsfor7trialvaluesofv siniinthe there isa definite asymmetryin the field strengthand structure
e
rangefrom17to20kms−1wefindaminimumforv sini=18– between the negativemagnetic pole (ϕ ≈ 0.0) and the positive
e
19 kms−1. A parabolicfit to these results yields a minimumat one(ϕ≈ 0.5).Thefieldisclearlystrongerinthelattercaseand
v sini=18.4±0.5kms−1.Weadoptthisrotationalvelocityfor its structure is dominated by the high-contrast magnetic spots
e
allsubsequentinversionspresentedinourpaper. wherethefieldstrengthreaches4.5kGlocally.Thisisapproxi-
Wefollowedasimilarstrategytooptimizethechoiceofthe mately1kGhigherthanthelocalfieldattheoppositesideofthe
inclinationandazimuthangles.Themodelingofthemeanlongi- star.
tudinalfieldandnetlinearpolarizationsuggeststhattheseangles Therectangularprojectionofthemagneticfielddistribution
arecloseto120◦and110◦,respectively.Todeterminethevalues reconstructedforα2CVnispresentedinFig.7forallthreecom-
of i and Θ we carried out inversions for the ranges i = 110– ponentsofthefieldvector.Thisfigureagaingivesanimpression
130◦ and Θ = 90–140◦ with a 10◦ step for both angles. The of the global dipolar topology locally distorted by small-scale
normalized total discrepancy function obtained by Invers10 at features. The radial field componentis most similar to the sur-
the convergenceis illustrated in Fig. 4 for the entire gridof 18 facemagneticstructureexpectedforanobliquedipole,whilethe
inversions.Itisevidentthati=120◦givesasmallerdiscrepancy meridionalandazimuthalcomponentsdeviatesignificantlyfrom
thaneitheri = 110◦ or130◦,whiletheoptimalΘisconstrained asimple,low-ordermultipolarshape.
tothe110–120◦intervalindependentofinclination.Finally,we Is it possible that the complexity seen in the MDI maps
establishi=120±5◦andΘ=115±5◦andadoptthisorientation of α2CVn is an artifact produced by the noise in observations
ofthestellarrotationaxisforallmagneticinversionspresented combined with an insufficiently strong regularization adopted
below. in the magnetic mapping? We have thoroughly examined this
possibilitybyconductinginversionswithdifferentvaluesofthe
Tikhonovregularizationparameterformagneticfield,Λ .These
5. Results 1
testsconfirmthatourchoiceofregularizationiscorrectbecause
5.1.FullStokesvectorinversion anysubstantialincreaseofthesmoothingofthemagneticmaps
by the Tikhonov regularization noticeably worsens the agree-
We reconstruct the magnetic field topology and surface dis- mentbetweentheoreticalspectraandthelinearpolarizationob-
tributions of Cr and Fe in a simultaneous inversion, using 20 servations. For example, with the Tikhonov regularization in-
phasesof observationsin all four Stokesparameters.The best- creasedbyafactorof10thecodeconvergestoamarkedlysim-
fitting theoretical profiles obtained by Invers10 at the conver- pler field topology, with the field strength not exceeding 3 kG
gence are shown in Fig. 5 with the thick solid line. Evidently, (Fig.8).TheresultingfitqualityisunchangedforStokes I and
the observed line profile variability in the intensity and polar- is only slightly worse for Stokes V relative to the profiles cor-
ization is satisfactorily reproduced.The only minor systematic respondingtothecomplexfieldmodel.However,thesmoothed
discrepancy is seen in the profile of the Feii 5018.44 Å line, field structurefails to reproducethe Stokes Q andU spectraat
which is overestimated in the theoretical spectra correspond- phasesϕ=0.353–0.594,whenthemostcomplexpartofthesur-
ing to the rotation phases around negative magnetic extremum facemagneticfielddistributioncrossesthelineofsight.Forall
(ϕ=0.949–0.049).Thelinearpolarizationprofilesarewell-fitted these phases the observed linear polarization profiles exhibit a
byInvers10.Occasionaldiscrepanciesbetweentheobservations double-wavestructurewithtwominimaandtwomaximaacross
andthemodelspectra(e.g.,StokesQprofileof Feii5018.44Å eachspectralline(e.g.the”w”-shapedprofilesofStokesU).On
forϕ=0.225)arealwayslimitedtoonlyoneoutthethreelines the other hand, these features are well-reproducedby the best-
studied and thus is likely due to observational uncertainties. fittingcomplexmagneticfielddistribution.In contrast,theoret-
O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters 9
icalprofilescorrespondingto the smoothedmagneticmap lack ousresults,thisfigurealsoshowsthetheoreticalprofilesforour
the double-wave structure. The resulting degradation of the fit optimalfourStokesparameterinversion(sameasinFig.5).At
qualityfortheQandU spectraisreadilyseenintheincreaseof the resolution of the MuSiCoS spectra we see no difference in
χ2by30–60%. the intensity and circular polarization between the two sets of
Thisanalysisshowsthatthesmall-scalefeaturesinourmag- syntheticprofiles.YetthepictureisverydifferentforStokes Q
neticmapofα2CVnaredirectlyconnectedtotheparticularmor- andU.Themodeltopologyinferredfromthecircularpolariza-
phology of the linear polarization signatures, observed consis- tion alone evidently yields systematically higher amplitude of
tently in all spectral lines studied.Thus, we can assert that our the linear polarization profiles and cannot match the complex
finalmagneticfieldtopologyofα2CVn(Fig.6)istrulythesim- shape of the observed Q and U signal around rotational phase
plestpossiblefieldstructureconsistentwiththeavailableStokes ϕ≈0.5.Thisdegradationofthefitqualityisreadilyseenwitha
parameterobservations.Therefore,theinferredcomplexityand factorof2–3increaseoftheχ2forbothQandUspectra.Thus,a
substantiallocaldeviationofthestellarmagneticgeometryfrom successfuldescriptionofthephasevariationoftheintensityand
the low-ordermultipolarfield arereal. It isremarkablethat we circular polarization spectra gives no guarantee that the result-
areabletodetectandcharacterizethesecomplexmagneticstruc- ing modelof the stellar surface magneticfield is also adequate
turesusingStokesIQUV data,butnotusingStokesIV data. fortheStokesQandU data.
ThesurfaceabundancemapsofCrandFereconstructedby The magnetic map derived in the Stokes IV inversion is
Invers10 simultaneously with the magnetic field topology are shown in Fig. 11. This field structure is very close to a dipo-
presented in Fig. 9. We infer a high-contrast distribution for lar topologywith a mildnon-axisymmetricquadrupolarcontri-
bothelements,withtheabundancevaryingby3–4dexandpro- bution (B = 3.7 kG, B = 1.0 kG). The global field compo-
p q
nouncedabundanceminimaatphaseϕ=0.Theseabundanceim- nent and distribution of the radial field is similar to those seen
agesarequalitativelysimilartotheiron-peakelementmapsre- in Fig. 6 but there is no evidence of the small-scale magnetic
constructedbyKochukhovetal.(2002)buthavealowersurface structure which is needed to fit the linear polarization profiles.
resolutionforthereasonsdiscussedinSect.4.2.Inaddition,the WeconcludethattheStokesIV analysisofthestellarmagnetic
mapobtainedfromthetwostrongFeiilinesexhibitsnoticeably topologiesisfundamentallylimitedwith respecttothescale of
smallerabundancevaluesintheregionaroundthenegativemag- magneticstructureswhichcanberesolvedwiththismethod.In
netic pole (logNFe/Ntot <−6 compared to logNFe/Ntot ≥−4.5 the particularcase of α2CVn itcan be used to assess the over-
obtained by Kochukhovetal. 2002). This can possibly be as- all, globaldipolar-likefield topologybut, comparedto the four
cribed to the effects of vertical chemical stratification, which Stokes parameter inversion, it cannot providedetailed maps of
we do not take into account in our analysis. A vertical abun- the distributions of the magnetic field strength and orientation
dance distribution with a transition from a high abundance in acrossthestellarsurface.
the deeper atmosphere to a lower element concentration in
the higher layers is often observed in cooler Ap stars (e.g.,
Kochukhovetal.2006,2009).Ifsuchstratificationispresentin 5.3.Multipolarexpansionofmagneticmaps
somepartsofthesurfaceofα2CVn,itwouldleadtosubstantial
weakeningofthecoresofintrinsicallystrongFeiilinesrelative Sphericalharmonicexpansionprovidesaconvenientmethodfor
tointrinsicallyweaklines,yieldingasmallerabundanceifsuch quantitativeassessmentanddetailedcharacterizationofthemag-
lines are analyzed neglecting chemical stratification. However, netic field maps obtained in the MDI inversions.It also allows
these effects generally do not significantly influence polariza- ustoobjectivelycomparethemagneticfieldtopologyofα2CVn
tionprofilesandthusadetailedtreatmentofverticalabundance with the field geometryof other Ap stars, in particular 53Cam
inhomogeneitiesisbeyondthescopeofourstudy. which was previously studied in all four Stokes parameters by
Kochukhovetal.(2004a)usingthesameinversiontechnique.
Hereweapplythesphericalharmonicexpansionmethodin-
5.2.StokesIV inversionwithmultipolarregularization
troducedbyPiskunov&Kochukhov(2002)andemployedinthe
MagneticinversioninallfourStokesparameters,similartoour analysisof53CambyKochukhovetal.(2004a).Thethreevec-
analysis of α2CVn, remains very uncommon in the research tor components of the magnetic field distribution are approxi-
fieldofstellarmagnetismduetotheconsiderabledifficultiesas- matedwithasuperpositionofrealsphericalharmonicseries,in-
sociatedwiththeacquisitionofthenecessaryobservationaldata. cludingbothpoloidalandtoroidalexpansionterms.Intheanal-
Instead,magneticDIandZeemanDopplerImaging(ZDI)stud- ysisofα2CVnwetruncatethemultipolarexpansionatℓ = 10.
ies employing time-resolved Stokes IV spectra are much more This givesus 240 poloidaland toroidalmultipolarcoefficients,
widelyappliedtostudyfieldtopologiesofmagneticApstarsand whicharedeterminedbysolvingalinearleast-squaresproblem.
activelate-typestars.Inthissectionweassesshowmuchofthe Differentlatitudes on the stellar surface are treated with differ-
complexityofthemagneticfieldofα2CVn,evidentfromthefull entweightsto accountforthe variationof the field reconstruc-
Stokesvectorreconstruction,canberetrievedintheanalysisof tionqualityandtoexcludetheinvisiblepartofthestellarsurface
anincomplete,circularpolarization-onlydataset.Asdescribed (hiddentousduetothestellargeometry).
in Sect. 4.1, in this experiment we use Invers10 in Stokes IV The results of the multipolar expansion are presented in
inversionmodewiththemultipolarregularization.Wecarryout Fig.12forthemagneticfieldtopologyofα2CVnobtainedfrom
magneticfieldreconstructionforthesamethreeCrandFespec- all four Stokes parameters (Fig. 6), from the Stokes IV spec-
tralfeatures,althoughinprinciplethenumberofusablespectral tramodeledusingmultipolarregularization(Fig.11)andforthe
linesismuchlargerifwearenotconcernedwiththeanalysisof surfacemagneticfieldof53Cam.ForthelatterstartheMDImap
theStokesQandU spectra. wasproducedwiththesamemodifiedInvers10codeasapplied
The fit to the I and V spectra of α2CVn and predictionfor inthepresentstudyofα2CVn,usingtheStokesIQUV profiles
thelinearpolarizationprofilesobtainedbyourcodeatthecon- ofthethreeFeiilinesstudiedbyKochukhovetal.(2004a).The
vergenceofthe IV mappingproblemisillustratedwiththethin updated magnetic map of 53Cam does not differ appreciably
line in Fig. 10. To enable a direct comparison with the previ- fromtheaveragefieldtopologyinferredbythoseauthors.
10 O.KochukhovandG.A.Wade:MagneticDopplerimagingofα2CanumVenaticoruminallfourStokesparameters
Multipolar expansion coefficients, (Cm)2+(C−m)2, are conducting inversions with different intensities of the regular-
q ℓ ℓ
izationparameterforthemagneticfield.Weconfirmedthatour
plotted in Fig. 12 as a function of ℓ and m, separately for the
choice of regularization is correct because any substantial in-
poloidal and toroidal expansion terms. The greyscale for each
creaseofthesmoothingofthemagneticmapsnoticeablywors-
magnetic map is renormalized so that the largest coefficient is
ened the agreement between the observed and computed lin-
black and the smallest is white. Fig. 12 suggests that the field
ear polarization observations. We thereby confirmed that the
geometryofα2CVnisdominatedbyℓ = 1modes.Atthesame
smaller-scalestructuresinourmaparereal,andrequiredinor-
time, the contributionof the higher-ℓ modesand the amplitude
der to reproducethe observations.We also computedmagnetic
of the toroidal field components is also non-negligible for the
fieldmapsusingjusttheStokesIVspectra.Thesemapswereno-
MDImapobtainedfromallfourStokesparameters(leftcolumn
ticeably less structured than those derivedfrom the full Stokes
in Fig. 12). For this surface magnetic field distribution we see
IQUV dataset, confirmingthatthelinearpolarizationobserva-
contributionsofmodeswithℓupto5–6.Ontheotherhand,the
tionsareessentialtodetectingthiscomplex,smaller-scalecom-
IVmappingresults(middlecolumninFig.12)suggestaconsid-
ponentofthemagneticfield.
erablysimplerfield,showingveryclearlythereductionofthein-
Asphericalharmonicdecompositionwasappliedtoquanti-
formationcontentwhenlinearpolarizationspectraareexcluded
tativelystudythederivedmagneticfieldtopology.Thisanalysis
from the Doppler Imaging analysis. In this case the dominant
suggeststhatthefieldgeometryofα2CVnisdominatedbyℓ=1
dipolarmodeisdistortedonlybyamarginalcontributionofthe
modes.Atthesametime,thecontributionofthehigher-ℓmodes
non-axisymmetricℓ=2andaxisymmetricℓ=3–8components,
andtheamplitudeofthetoroidalfield componentsisalso non-
whilethetoroidalfieldispracticallyabsent.
negligible,withcontributionsofmodeswithℓupto5–6.
Despite a consistent analysis using the same inversion
methodologyand MuSiCoS spectropolarimetricdata of similar Wealsore-visitedtheanalysisof53Camperformedin2004,
obtaininganewfieldmapofthisstarusingthecurrentversionof
qualityandphasecoverage,thelevelofcomplexityturnsoutto
themappingsoftware.Despitethisconsistentanalysisusingthe
be dramatically differentfor the surface magnetic field topolo-
sameinversionmethodologyandspectropolarimetricdataofthe
gies of α2CVn and 53Cam. For α2CVn the ℓ = 1 mode is
sameorigin,andofsimilarqualityandphasecoverage,thelevel
more importantrelative to other componentsand the contribu-
ofmagneticfieldcomplexityisfoundtobedramaticallydiffer-
tion of the toroidal field is small. For 53Cam the power is not
entforα2CVnand53Cam.Forα2CVntheℓ=1modeismore
concentrated in the dipolar componentbut spread out over the
important relative to other componentsand the contribution of
entireℓ =1–10range,withthebroadmaximumatℓ=4–7,and
thetoroidalfieldissmall.For53Camthepowerisnotconcen-
thetoroidalcomponentsarenoticeablystrongerthaninα2CVn.
tratedinthedipolarcomponentbutisdistributedovertheentire
Therefore,weconcludethatthesurfacemagneticfieldcomplex-
ℓ=1–10range,withabroadmaximumatℓ=4–7,andtoroidal
ityofthetwoApstarsstudiedusinghigh-resolutionfourStokes
componentsthatnoticeablystrongerthanthoseofα2CVn.Also,
parameterobservationsisintrinsicallydifferent.
theclearhemisphericasymmetryofthequalitativefieldproper-
tiesofα2CVn–withsmall-scalestructuresconfinedessentially
tothestronger,positivepole,theoriginofwhichisnotknown–
6. Discussion
isnotobviouslyreflectedinthefield of53Cam. Therefore,we
In this paper we have described what is only the second self- concludethat the surface magnetic field complexityof the two
consistentanalysisofhigh-resolutionStokesIQUV lineprofiles Apstarsstudiedusinghigh-resolutionfourStokesparameterob-
of a magnetic Ap star. For the prototypical spectrum variable servationsisintrinsicallydifferent.Thereasonforthisdifference
α2CVn,wehaveemployedphase-resolvedpolarizationspectra iscurrentlyamystery–similarmapsofalargersampleofstars
to derive a detailed map of the surface magnetic field intensity willbeneededbeforeaclearerunderstandingwillbepossible.
andorientationusingthe MagneticDopplerImagingapproach. The surface abundance maps of Cr and Fe show high-
Inaddition,wehavederivedsimilarlydetailedmapsofthesur- contrastdistributionsofbothelements,varyinginabundanceby
facedistributionsoftheabundancesofFeandCr. 3–4 dex and with pronouncedabundance minima at a location
We find that the overall structure of the magnetic field in corresponding roughly to the negative magnetic pole. We also
α2CVnisdipole-like,withapproximatelyhalfofthestellarsur- observed that the map obtained from the two strong Feii lines
face covered with the outward-directed radial field while the exhibitsnoticeablysmallerabundancesintheregionaroundthe
other exhibits inward-directed field. This is in agreement with negative magnetic pole than those obtained for this star using
the field geometry suggested in previous magnetic analyses of weakerFelinesbyKochukhovetal.(2002).Wediscussedhow
α2CVn obtained from fitting of multipolar models. However, thiscouldbebeascribedtotheeffectsofverticalchemicalstrati-
thefieldstrengthdistributionwederiverevealsthatthefieldis, fication,whichwouldleadtosubstantialweakeningofthecores
in fact, far more complex than the simple geometry suggested of intrinsically strong Feii lines relative to intrinsically weak
by earlier models. In particular, there is a definite asymmetry lines, yielding a smaller abundance if such lines are analyzed
in the field strength and structure between the negative mag- neglectingchemicalstratification.
netic pole and the positive pole.The field is clearlystrongerat Theinvestigationsofthemagneticfieldsofbothα2CVn(in
thepositivepoleanditsstructureisdominatedbyhigh-contrast thispaper)and53Cam(byKochukhovetal.2004b)leadtothe
magneticspots where the field strengthreaches4.5 kG locally. viewofahiddencomplexityofthemagneticfieldsofApstars–
Interestingly, this is approximately 1 kG higher than the local acomplexitywhichisonlyrevealedwiththehelpoflinearpolar-
field atthe oppositeside of the star. These spotsare analogous izationmeasurements.Despite thesuccesses achieved,the data
to those detected in the A4p star 53Cam by Kochukhovetal. onwhichthesestudieshavebeenbasedisfundamentallylimited.
(2004b). We investigated the possibility that the defining char- The resolving power of the MuSiCoS spectropolarimeter was
acteristicofthesemaps–thehigh-contraststructureofthefield rather low – just 35000– and the throughputof the instrument
strengthatsmallerspatialscales–mayresultfrominsufficiently below 1%. These characteristics led to polarization spectra in
smoothingas a consequenceof underestimationof the regular- which Zeemansignatureswere often undetectable(particularly
ization intensity. We thoroughly examined this possibility by inlinearpolarization),oronlydetectableatrelativelylowsignif-
