Table Of ContentMon.Not.R.Astron.Soc.000,000–000(2011) Printed11January2012 (MNLATEXstylefilev2.2)
Magnetic Doppler imaging considering atmospheric structure
modifications due to local abundances: a luxury or a necessity?
1 2 3
2 O. Kochukhov , G. A. Wade and D. Shulyak
1 1DepartmentofPhysicsandAstronomy,UppsalaUniversityBox516,75120Uppsala,Sweden
0 2DepartmentofPhysics,RoyalMilitaryCollegeofCanada,Box17000,StnForces,Kingston,OntarioK7K7B4,Canada
2 3InstituteofAstrophysics,Georg-AugustUniversity,Friedrich-Hund-Platz1,D-37077Go¨ttingen,Germany
n
a
J Accepted2012January9.Received2012January9;inoriginalform2011December6
9
] ABSTRACT
R
Magnetic Doppler imaging is currently the most powerful method of interpreting high-
S
resolution spectropolarimetric observations of stars. This technique has provided the very
h. first maps of stellar magnetic field topologies reconstructed from timeseries of full Stokes
p vector spectra, revealing the presence of small-scale magnetic fields on the surfaces of Ap
- stars.ThesestudieswererecentlycriticisiedbyStiftetal.,whoclaimedthatmagneticinver-
o
sionsarenotrobustandareseriouslyunderminedbyneglectingafeedbackontheStokesline
r
t profilesfromthelocalatmosphericstructureintheregionsofenhancedmetalabundance.We
s
showthatStiftetal.misinterpretedpublishedmagneticDopplerimagingresultsandconsis-
a
[ tently neglectedsome of the most fundamentalprinciplesbehind magnetic mapping.Using
state-of-the-artopacitysamplingmodelatmosphereandpolarisedradiativetransfercodes,we
1
demonstratethatthevariationofatmosphericstructureacrossthesurfaceofastarwithchemi-
v
calspotsaffectsthelocalcontinuumintensitybutisnegligibleforthenormalisedlocalStokes
2
profilesexceptfortheraresituationofaverystronglineinanextremelyFe-richatmosphere.
0
Forthedisk-integratedspectraofanApstarwithextremeabundancevariations,wefindthat
9
1 theassumptionofameanmodelatmosphereleadstomoderateerrorsinStokesI butisneg-
. ligible for the circular and linear polarisationspectra. Employinga new magnetic inversion
1
code,whichincorporatesthehorizontalvariationofatmosphericstructureinducedbychem-
0
icalspots,wereconstructednewmapsofmagneticfieldandFeabundanceforthebrightAp
2
star α2CVn. The resulting distribution of chemical spots changes insignificantly compared
1
: tothepreviousmodellingbasedonasinglemodelatmosphere,whilethemagneticfieldge-
v ometrydoes notchange at all. This shows that the assertions by Stift et al. are exaggerated
i
X as a consequenceof unreasonableassumptions and extrapolations,as well as methodologi-
calflawsandinconsistenciesoftheiranalysis.Ourdiscussionprovesthatpublishedmagnetic
r
a inversions based on a mean stellar atmosphere are highly robust and reliable, and that the
presenceofsmall-scalemagneticfieldstructuresonthesurfacesofApstarsisindeedreal.In-
corporatinghorizontalvariationsofatmosphericstructureinDopplerimagingcanmarginally
improvereconstructionofabundancedistributionsforstarsshowingverylargeironoverabun-
dances.Butthiscostlytechniqueisunnecessaryformagneticmappingwith high-resolution
polarisationspectra.
Keywords: stars:chemicallypeculiar–stars: atmospheres– stars:magneticfields–stars:
spots–stars:individual:α2CVn
1 INTRODUCTION tantnewinsightsintotheprocessesoccurringinstellarouterlayers
and unique constraints on theoretical models (Donatietal. 2003,
2006a,b; Petitetal. 2008; Morinetal. 2008; Kochukhovetal.
Doppler Imaging (DI) of stellar surfaces represents one of the
2007;Kochukhov&Wade2010).
spectacularsuccessesofmodernobservationalstellarastrophysics.
Since the first applications of these tomographic modeling tech- Conventional DI interprets rotational modulation of the in-
niques to cool and hot stars in the 1970s and early 1980s tensity spectra of stars to reconstruct two dimensional maps of
(Khokhlova&Ryabchikova 1975; Goncharskiietal. 1977, 1982; photospheric temperature or abundance spots (inthecase of cool
Vogt&Penrod 1983), they have revealed the complex surface andhotstars,respectively).Basically,thestellarsurfaceisdivided
structuresof rotatingstarsinremarkable detail, providing impor- into a grid of pixels, each of which is assigned independent lo-
2 O. Kochukhov,G. A. WadeandD. Shulyak
calphysicalproperties(temperatureorabundance).Syntheticdisc- ing that DI maps based on single, mean stellar atmospheres are
integratedspectraarecomputed,takingintoaccountthelocaltem- highlyrobustandreliable.Ourpaperisorganisedasfollows.First,
perature/abundanceofeachpixel,aswellasitsrotationalDoppler inSection2webringtothereader’sattentionseveralkeyaspects
shift,andcompared toatimeseriesof observed spectrallinepro- ofDIandMDImethodologyoverlookedormisinterpretedbyS12.
files.Iterativeadjustmentofthelocalphysicalconditionsleadstoa Calculationofrealisticopacitysamplingmodelatmospheres,cor-
model(or“map”)thatiscapableofreproducingtheobservedline respondinglocalcontinuumintensities,andtheStokesIQUV pro-
profileshapesandmodulationindetail. filesisdescribedinSection3.UsingtheFeabundancedistribution
As the available observational data and computational re- andmagneticfieldtopologyreconstructed forα2CVnbyKW10,
sources have improved, DI models have become increasingly so- wecomputeinSection4.2thefullStokesprofilesofastrongFeII
phisticated, with better resolution and realism (e.g. inclusion of line, exploring the consequences of the lateral variation of atmo-
various physical effects such as differential rotation). A qual- sphericstructureontheintensityandpolarisationprofiles.Acom-
itative step forward was taken with the inclusion of magnetic plete MDI inversion based on a grid of model atmospheres with
fieldsbyPiskunov&Khokhlova(1983)forApstarsandbySemel different elemental abundances is presented for α2CVn in Sec-
(1989) for cool active stars. Zeeman or Magnetic Doppler Imag- tion4.3.Wesummariseanddiscussresultsofourinvestigationin
ing (ZDI or MDI; Brownetal. 1991; Piskunov&Kochukhov Section5.
2002; Kochukhov&Piskunov 2002) extends the basic princi-
ples of DI to allow mapping of stellar magnetic fields by inter-
preting timeseries of line profile Zeeman polarisation. Currently,
themostsophisticatedMDIcodes(Piskunov&Kochukhov 2002; 2 GENERALCONSIDERATIONS
Kochukhov&Piskunov 2002) are capable of simultaneous and
2.1 Robustnessandstabilityofmagneticinversions
self-consistentmappingofsurfaceabundancesortemperaturedis-
tributionsalongwiththevector magneticfieldfromtimeseriesof As with any complex data modelling technique, capabilities
lineprofilesintwo(IV)orfour(IQUV)Stokesparameters. and intrinsic limitations of MDI must be tested comprehen-
Recently, Stift,Leone&Cowley (2012) reported the results sively. The physical foundations and numerical methods im-
of experiments that seriously question the basic reliability of the plemented in the INVERS10 code were described in detail by
results of DI and MDI, particular as applied to chemically pecu- Piskunov&Kochukhov (2002, hereafter PK02). This paper pre-
liar Ap stars. As pointed out by Stift et al., DI techniques have sented a general discussion of the full Stokes vector inversion
revealed some surprising and exotic characteristics of the atmo- methodologyandcarriedoutacomprehensivetheoreticalandnu-
spheresoftheseobjects.Oneofthemostoutstanding isthelarge merical assessment of its key algorithms, such as polarised ra-
chemical abundance contrasts in atmospheres of Ap stars, which diative transfer and regularisation of the ill-posed stellar surface
according to DI analyses can be quite extreme (e.g. variations of mappinginverseproblems.TheforwardStokesparameterprofiles
several ordersof magnitudeacrossthestellarsurface). Inregions computed with INVERS10 were compared with the spectra pro-
of high enrichment, some metals are sometimes inferred to be ducedbytwootherindependentmagneticspectrumsynthesiscodes
only a factor of ∼30 less abundant compared to hydrogen (e.g., (Wadeetal. 2001). This study showed a very good agreement of
Kuschnigetal. 1998), which is challenging to explain theoreti- thelocalanddisk-integratedStokesprofilesproducedbyallthree
cally. The distributions of abundances can be extremely complex codes,providedtheyuseaconsistentdefinitionofStokesparame-
(Kochukhovetal.2004b),whilesometimestheyarerathersimple tersandthesameinputmodelatmosphereandatomicdata.
(e.g.,Lu¨ftingeretal.2010b).Thedistributionsmayshowstraight- A subsequent study by Kochukhov&Piskunov (2002, here-
forward relationships with the inferred magnetic field geome- afterKP02)presentedextensivenumericalexperimentsdesignedto
try(e.g.,Rice,Wehlau&Holmgren1997),butfrequentlytheydo evaluateperformanceofINVERS10.Thesetestsdemonstratedthat,
not.Recently,MDImappingusingfourStokesparameterdatasets givenhigh-resolutionIQUV observations,themagneticinversion
(Kochukhovetal.2004a;Kochukhov&Wade2010)hasrevealed code is capable of correctly reconstructing abundance and mag-
the unexpected presence of intense, small-scale structures in the neticfieldvectorsurfacedistributionssimultaneouslyandwithout
reconstructed magnetic field. The existence of these structures is anypriorassumptionsaboutthelarge-scalemagneticfieldgeome-
unexplained,andaddsaqualitativelynewelementtoourgrowing try.Atthesametime,itwasfoundthatanaccuratemappingofa
pictureofthemagneticstructureofApstars. globally-organisedmagneticfieldfromcircularpolarisationalone
TheessentialclaimofStiftetal.(2012,hereafterS12)isthat requires an additional multipolar constraint on the possible mag-
mapscomputedforApstarsusingDIandMDImethodsarefunda- neticfieldstructure.KP02showedthatthefullStokesvector-based
mentallyunreliableandthatthevarietyofunexplained properties simultaneousmagneticandabundanceinversionsaregenerallysta-
ofmapsdescribedaboveareartefactsofneglectingtheimpactof bleandrobust.Itwasfound thatthederived solutionsareunique
largeabundancesandabundancecontrastsonthelocalstellaratmo- and themagnetic mapping code cannot arrive toaspurious solu-
spheric structure(e.g., Khan&Shulyak 2007).In particular, Stift tionwhichwouldprovideagoodfittoobservations. Anotherim-
etal.statethatignoringtheseeffectsresultsinartificiallineprofile portantoutcomeoftheMDItestscarriedoutbyKP02isthatmag-
variability,whichtranslatesintospuriousabundanceandmagnetic neticinversionsextractinformationaboutmagneticfieldprimarily
structuresinDImaps.Ultimately,theyconclude thatcomplexDI fromtherotationalmodulationofStokesprofilesratherthanfrom
mapswithhigh-contrastabundancedistributionsareincorrect,and theDopplershifts,unlikefortheconventionalscalarDI,makingit
that the presence of small-scale structure and high-contrast mag- possibletoapplyMDItoveryslowlyrotatingstars.
neticspots,asinferredfromMDI,cannotatpresentbereliablyas- Thecomprehensive discussion oftheMDI methodology im-
certained,andarelikelyanartefactoftheinversionprocedure. plementedinINVERS10andextensivenumericaltestsofthiscode
In thispaper weconfront the assumptions and methodology presented by PK02 and KP02 are neglected by S12, who do not
of S12, demonstrating that their conclusions are largely founded mentioneitherofthesetwopapers.Thisoversightmakesmuchof
on fundamental errors and unrealistic assumptions, and conclud- thegeneralcriticismbyS12irrelevantbecausemanyoftheircon-
MagneticDopplerimagingconsideringmetallicity-dependentatmosphericstructure 3
cernsabout magneticinversionmethodology havebeenexplicitly cal harmonic expansion, was independently developed by Donati
addressedbyPK02andKP02. (2001) and successfully used for the Stokes IV ZDI of global
We note that the contentions of S12 about what DI inver- fields of Ap stars (Lu¨ftingeretal. 2010a), fully convective low-
sioncanorcannotdolackareferencetothebasicnumericaltests massstars(Donatietal.2006a;Morinetal.2008),solaranalogues
demonstrating robustness of their ownabundance inversion code. (Petitetal.2008),andlate-typeAp-stardescendants(Aurie`reetal.
Forinstance,comparingFig.7and8oftheirpaper,onecanseethat 2011). These developments were entirely neglected by S12, who
theauthorshavefailedtorecoverthetruedistributionofabundance subscribedtotheobsoleteviewofBrownetal.(1991)thatglobal
spotsevenwhentheyused aconstant magneticfieldandadopted magneticfieldscanbeonlymappedusingfullStokesdata.
thesamemeanatmosphericstructureforcalculationsoftheinput Discussing the IQUV magnetic mapping of α2CVn by
spectraandfortheinversion.Amajordiscrepancybetweenthein- KW10,S12insisted,citingFigs.6and8ofKW10,thatthesein-
put and reconstructed abundance maps revealed in such a simple versionsareunstablebecause“resultsdependtoostronglyonthe
testindicatesseriousproblemswiththeDIcodeusedbyS12.Ev- regularisationparameter”.Butthesefiguresshowexactlytheop-
idently,resultsbasedontheapplicationofthisuntestedinversion posite,demonstratingtheimpactofchangingtheregularisationpa-
codemustbeconsideredwithcaution. rameter (which controls the relative contributions of the penalty
functionandchi-squareofthefit)byastaggeringfactorof10.This
largeincreaseofregularisationwasrequiredtowipeoutthesmall-
2.2 RegularisationinmagneticDI
scalefieldstructures. Nevertheless, contrary totheclaim byS12,
Typically,agridofseveralhundredtoseveralthousandsurfaceel- thelarge-scalefieldtopology,characterisedbytheradialfieldcom-
ements is employed in DI. Because even a high-quality spectro- ponent, was very similar for the maps obtained with the optimal
scopicdatasetmayonlyrepresentafewhundredindependentcon- andartificiallylargeregularisation,sotheTikhonovregularisation
straintsonthestellarsurfacestructure,theconventional tempera- operatedasexpected.KW10havemadeitclearthatthetwovalues
tureorabundanceDopplerimagingproblemismathematicallyill- oftheregularisationparameterarebynomeansequivalentbecause
posed (Goncharskiietal. 1977; Vogt,Penrod&Hatzes 1987). As applyingthehigherregularisationvalueyieldsafarworsefittothe
aconsequence,alargefamilyofmaps–eachwiththesamelarge- observational data. S12 seem to have ignored these explanations
scale structure, but differing in smaller-scale details – is able to and the context of KW10 inversions with different regularisation
reproduce the observations. In fact, the surface grid is often suf- parameters,leadingtotheirmisinterpretationoftheMDIresults.
ficientlyfineastobeabletofittheobservational noise. Tosolve
theseproblemsapenalty(orregularisation)functionisincludedin
2.3 LineprofilemodellingandinterpretationofDImaps
the optimisation procedure, serving to limit the information con-
tent(maximumentropyregularisation)orhigh-frequencystructure The general goal of Doppler imaging – reconstructing a two-
(Tikhonovregularisation)ofthemap.Ithasbeendemonstratedthat dimensional stellar surface map – is always achieved through a
in conventional DI, for high resolution and signal-to-noise ratio modelling of the rotational modulation of spectral line profiles.
data sets, these two approaches lead to maps that are essentially However, itisimportant todiscriminatebetween thetasksof ob-
equivalent(Strassmeieretal.1991). tainingamapofacertainparametercharacterisingthestellarsur-
The situation is different for magnetic DI. Piskunov (2005) face and interpreting this map in terms of a physical quantity. In
showed that the underlying inverse problem based on full Stokes the early days of chemical spot mapping of Ap stars, DI stud-
observations iswell-posed and thushas a unique solution, and is ies reconstructed local equivalent widths instead of real elemen-
thereforenotinfluencedbytheselectionofaspecifictypeofreg- tal abundances (Goncharskiietal. 1983). The majority of current
ularisation.Inthiscasetheroleofregularisationislimitedtosup- DI and ZDI studies of cool active stars map brightness instead
pressingnumericalinstabilitiesinducedbyasparsewavelengthand of temperature, using a coarse temperature-independent analyti-
phasesampling typicalof realobservational data.Incontrast,the cal description of the local line profiles (e.g., Donatietal. 2003;
MDI problem based on only Stokes I and V is intrinsically ill- Marsdenetal. 2011). Modern abundance DI studies of Ap stars
posed, leadingtodifferent resultsdepending ontheappliedregu- (e.g.,Kochukhovetal.2004b;Rice,Holmgren&Bohlender2004;
larisationalgorithm.Forinstance, Brownetal.(1991)havefailed Lu¨ftingeretal.2010a)arenormallybasedonthetheoreticalspectra
torecoveradipolarmagneticfielddistributionfromStokesIV data calculatedfordifferentchemicalcompositionusingthesamemean
with their maximum entropy ZDI code but succeeded in recon- model atmosphere. The resulting maps are presented in the form
structing topologically more complex field geometries consisting of horizontal distributions of chemical abundance. But it is im-
ofisolatedspots. portanttorememberthateveninthiscase“elementalabundance”
Thiscounterintuitive situationwhen mapping simpler global is essentially a parameter controlling the local line strength and,
field topologies from Stokes IV spectra is seemingly more diffi- therefore, it is the former stellar surface property that is mapped
cult than recovering very complex fields of late-type stars (e.g., byaninversioncode.Interpretationofthelinestrengthparameter
Donatietal. 1999) has been addressed by KP02. They showed canbeviewedasaseparateanalysisstep,largelydecoupledfrom
that the maximum entropy constraint is inappropriate for global themappingitself.Assumptionsabouttheatmosphericstructureof
magnetic topologies since they do not correspond to a map with theregionscharacterisedbydifferentmetallicitymadeinthislat-
a lower information content compared to a distribution of iso- tersteparenotnecessarilyunderminingreconstructionofthetwo-
latedspotsonahomogeneous background. KP02achievedsome- dimensionalmapsandhavenopossibilityofchangingthetopology
what better results with IV imaging of dipolar fields using the ofthemainstellarsurfacefeaturesaslongastheresultinglocalpro-
Tikhonov regularisation and demonstrated that a fully satisfac- filescanmatchthoseproducedwithaphysicallycorrectandcom-
tory reconstruction of the global field topologies is possible with pletemodelbyintroducinganoffsetofchemicalabundancefrom
multipolarregularisation,whichpenalisesadeviationofthemag- itstruevalue.
netic geometry from a low-order multipolar solution. A simi- Ifamagneticfieldisreconstructedsimultaneouslywithchem-
lar magnetic inversion technique, employing a general spheri- icalmaps,thereisasimilardecouplingbetweenpolarisationprofile
4 O. Kochukhov,G. A. WadeandD. Shulyak
modellingandinterpretationoftheStokesI linestrength.Evenif QandUprofilesobservedforthe0.3–0.6phaseinterval.Belowwe
thelatterisinerrorbyasignificantamount,anydecentMDIcode showthatforStokesQUV thisdifferenceisactuallyconsiderably
would compensate for this problem by arriving to a different lo- moreimportantthantheimpactofusingindividuallocalmodelat-
calabundance,whichyieldsthelocalpolarisationprofilesclosely mospheres,asurgedbyS12.
matching the true ones for the same magnetic field strength and S12 objected the very possibility of resolving small-scale
orientation. One the other hand, an unaccounted variation of the surface features in relatively slowly-rotating stars like α2CVn
continuumbrightnessmayleadtoasystematicover orunderesti- (vesini=18.5 kms−1) on the grounds that the difference in
mationofthemagneticfieldstrengthinbright, respectivelydark, Doppler shifts between the leading and trailing edges of such
surfaceregions. spots is small in comparison to the instrumental resolution. Ac-
ThisrobustnessofDIwithrespecttoerrorsininterpretationof cording to them, the spatial resolution of the stellar DI images
thelocallineprofilesanddecouplingbetweenmagneticandabun- is entirely determined by the relation of vesini and instrumen-
dancemappingwasignoredinthediscussionbyS12.Throughout tal profile width. Any reader familiar with stellar surface map-
theirpapertheseauthorsincorrectlyimplythatanydifferencebe- ping will immediately recognise an obvious flow in this reason-
tween the local profiles corresponding to an approximate versus ing. S12consider thesurface resolution provided by asingle ob-
afullyrealistictreatmentofthelineformationwillautomatically servation,whileDImakesuseofmanytime-resolvedspectra(20
translatetomajorsurfacestructureartefacts,invalidatingmagnetic in the case of KW10 study of α2CVn), extracting information
mapping. This viewpoint is fundamentally flawed because it ne- not only from the Doppler shifts but also from rotational modu-
glects to take into account the intrinsic ability of an MDI code lation. The latter is more important for magnetic mapping since
tocompensate thelineintensityerrorsarising fromtheuse of an polarisationprofilescanexhibitahugerotationalvariabilityevenif
approximaterelationbetweenthelocallinestrengthandchemical vesiniisnegligible.NumericalexperimentsbyKP02havedemon-
composition. stratedthatthisallowsanapplicationofMDItoalltypesofearly-
type magnetic stars, regardless of their projected rotational ve-
locities. Indeed, magnetic DI maps have been obtained for many
2.4 Anevidenceforsmall-scalefieldsinApstars veryslowlyrotating(vesini.5kms−1)early-type(Donatietal.
Traditionally, analyses of magnetic field topologies of early-type 2006b; Lu¨ftingeretal. 2010b) and late-type (Petitetal. 2008;
starsreliedonfittinglow-ordermultipolarfieldmodelstothephase Morinetal.2008;Faresetal.2010)stars.Areadermayalsonote
curvesofintegralobservables,suchasthemeanlongitudinalfield that with systematic application of the reasoning of S12, one is
and themean fieldmodulus, inferredfromthelow- or moderate- bound toconclude that stellarsurface mapping iscompletely un-
resolution Stokes I and V observations (Landstreet&Mathys feasible with photometric data – a ludicrous assertion given the
2000; Bagnuloetal. 2002). Extension of these studies to broad- well-known success of many photometric star spot investigations
bandlinearpolarisationrevealedsignificantdeficienciesofthemul- (e.g.,Korhonen,Berdyugina&Tuominen2002;Lanzaetal.2009;
tipolarmodellingapproach, hintingatthepresence ofmorecom- Mosseretal. 2009; Lu¨ftingeretal.2010a). Forinstance, thehigh
plexmagnetictopologiesonthesurfacesofApstars(Leroyetal. photometricstabilityoftheCoRoTsatelliteallowedMosseretal.
1995; Leroy,Landolfi&Landidegl’Innocenti 1996). A direct (2009)todetectphotometricsignaturesofspotsassmallasafew
comparison of the multipolar field geometry predictions with the degreesacross,whereasaccordingtoS12suchastudyshouldnot
high-resolution MuSiCoS four Stokes parameter observations of haveprovidedanysurfacespatialresolutionatall.
Ap stars (Wadeetal. 2000; Bagnuloetal. 2001) demonstrated a Summarising, the general arguments by S12 against the de-
majorfailureofthesetopologicallysimplemodelsinreproducing tectabilityofthecomplexmagneticfieldsonApstarsaredemon-
theshapesandamplitudesoftheobservedStokesQandUprofiles. strablyincorrectsincetheyneglectsomeoftheverybasicDIprin-
Inversions based on these pioneering full Stokes vector ob- ciplesandignorealargebodyofcontraryevidenceintheliterature.
servations showed that small-scale field structures superimposed
ontheglobalbipolarbackground arerequiredtomatchthelinear
polarisation inside spectral line profiles (Kochukhovetal. 2004a;
KW10).TheKW10analysisofα2CVnprovidedthemostclear-cut 3 MODELATMOSPHERESANDLOCALPROFILES
exampleofthissituation. TheMDIbasedontheStokesI andV
3.1 AtmosphericmodellingwiththeLLmodelscode
subsetofthefourStokesparameterobservationsyieldedadipolar-
likefieldgeometry. Ontheotherhand, extendingtheinversionto Anymeaningfulanalysisoftheeffectsofnon-solarchemicalcom-
all four Stokes parameters revealed a pair of high-contrast mag- position on the atmospheric structure of Ap stars requires an ad-
neticspots superimposed onthe dipolar-likebackground. Asdis- vancedopacitysamplingmodelatmospherecode,whichallowsto
cussedabove, thisisdefinitelynotaregularisationartefact asthe carryoutaccuratecalculationsforarbitraryelementalabundances.
differencebetweenthelinearpolarisationprofilescorrespondingto Inthisstudyweuseversion8.8ofthewell-establishedmodelatmo-
the magnetic models withand without the small-scale features is spherecodeLLMODELS(Shulyaketal.2004).Thisprogramtreats
highlysignificant. atomiclineopacitybyadirect,line-by-linespectrumsynthesisand
TheclaimbyS12that“amere3outof20phasesareresponsi- isabletoaccountforindividualnon-solarchemicalabundancepat-
ble”forthiscomplexityofα2CVnmagneticmapsisfalse.KW10 terns (Khan&Shulyak 2007) and inhomogeneous vertical distri-
haveunambiguouslystatedinthetextoftheirpaperandclearlyil- butionofelements(Kochukhovetal.2009b;Shulyaketal.2009).
lustratedwithFig.5thatthediscrepancybetweenthe“simple”and The code also includes a provision for treating the effects of a
“complex”magneticfieldgeometriesislargeincomparisontothe strongmagneticfieldonthelineopacity,includingZeemansplit-
observational noise over at least half of the stellar rotational cy- tingandpolarisedradiativetransfer(Kochukhov,Khan&Shulyak
cle,correspondingtoapproximately10phases.Themodellacking 2005; Khan&Shulyak 2006), and a contribution of the Lorentz
small magnetic spots yields asubstantially higher chi-square and forcetotheequationofhydrostaticbalance(Shulyaketal.2007).
cannot evenqualitativelyreproduce thedouble-waveshapeofthe However,asdemonstratedbythepreviousstudies,forthemoderate
MagneticDopplerimagingconsideringmetallicity-dependentatmosphericstructure 5
Figure1.DistributionandvisibilityoftheFeabundancevaluesinthesur-
facemapderivedforα2CVnbyKW10.Upperpanel:Feabundanceforthe
visiblepartofthestellarsurface(latitude660◦).Theverticaldashedline Figure2.Temperatureasafunctionofthecontinuumopticaldepthatλ=
showstheweightedmeanabundance,logNFe/Ntot = −3.79,usedfor 5000 A˚ (upper panel) and Rosseland optical depth (lower panel) in the
thereferencemodelatmosphere.Thedashedlinesindicatefouradditional modelatmospheresofα2CVncomputedfordifferentFeabundances.
Fe abundances for which we also calculated model atmospheres. Lower
panel:Visibilityfunction(seeSection3.1)illustratingrelativecontributions
of the surface zones with different Fe abundances to the disk-integrated
spectraofα2CVn. mostly located on the stellar hemisphere directed away from the
observer. The lower panel of Fig. 1 quantifies this statement by
plottingavisibilityfunctionofdifferentFeabundances,definedas
2–4kGmagneticfieldofα2CVn,Zeemansplittingandpolarised theprojectedareasofthecorrespondingsurfaceelementssummed
radiativetransfercanbesafelyneglectedinthemodelatmosphere overall20rotationalphasesoftheα2CVnobservationsanalysed
calculations. by KW10. If we normalise the area under thisvisibility function
Using the final Fe abundance map derived for α2CVn by to1,therelativecontributionofzoneswithlogNFe/Ntot >−2is
KW10,wedeterminedameansurface-integratedFeabundanceof only1.1×10−2.Thus,contrarytotheclaimsbyS12,regionsofex-
logNFe/Ntot = −3.79usingaweightfunctionaccordingtothe tremeFeoverabundancecontributenegligiblytotheMDIanalysis
Eq.(64)ofPK02.ThefullamplitudeoftheFeabundancevariation ofα2CVn.
overthestellarsurfaceis4.9dex,butthelargestvaluesinthisrange Aimingtopresentabalancedanalysisoftheeffectsofmetal-
arerepresentedbyasmallnumberofsurfaceelementsandmakein- licity on the model atmosphere and spectra of α2CVn, we con-
significant contributiontothedisk-integrated Stokesprofilespec- sidered the Fe abundance range from logNFe/Ntot = −2.19 to
tra. Figure1 shows a distributionof the Feabundance values for logNFe/Ntot = −5.91,encompassing 90%ofthevisiblestellar
thevisiblepartofthestellarsurface,whichcorrespondstothelat- surface. We have covered this range with five model atmosphere
ituderangefrom−90◦ to+60◦ fortheinclinationanglei=120◦ calculations, differing only by the adopted Fe abundance. For
adoptedforα2CVn.MostofthediscussionbyS12isfocusedon simplicity,thesolarabundancesfromAsplund,Grevesse&Sauval
modelatmospherescomputedforextremeFeoverabundances,such (2005)areassumedforallotherchemicalelements.Forallmodel
aslogNFe/Ntot = −2.0,−1.75,and−1.5.However,thereader atmospherecalculationsreportedinthispaperweuseda72-layer
can readily see from Fig. 1 that these high abundances are quite depth grid with an equidistant spacing in the logτ5000 range be-
rare. In the actual Fe map published by KW10 only 2.5% of the tween −7 and +2. The frequency-dependent quantities were cal-
visiblesurfacehaslogNFe/Ntot >−2. culatedbetween50 A˚ and100000A˚ withaconstant wavelength
Infact,thezoneswithahighFeconcentration areevenless step of 0.1 A˚. The resulting sampling at ∼106 frequency points
important for the disk-integrated Stokes spectra because they are yieldsmorepreciselineopacitiesthantheATLAS12calculationsat
6 O. Kochukhov,G. A. WadeandD. Shulyak
120000frequenciesemployedinthestudybyS12.LLMODELSit- scopicandphotometricobservables (suchasenergydistributions,
erationswerecontinueduntilthetemperaturechangewaslessthan hydrogen lines, photometric parameters, metallic line spectra) in
1Katalldepths.Thisconditionalsosatisfiedthecriteriaoftheflux ordertoquantifytheimportanceofdetectedchangesinthecontext
constancyandradiativeequilibriumto0.1–0.3%.S12admittedthat ofparticularinvestigations.
their model calculations were plagued by convergence problems Let us note in passing that modification of the atmospheric
for high Fe abundances. We encountered no such problems with structure due to enhanced metal abundances is closely connected
LLMODELS. to a well-known effect of the redistribution of stellar flux from
Ourcomputationofthelineabsorptioncoefficientwasbased UV to longer wavelength regions (Leckrone 1973), making the
ontheatomicdatafor∼6.6×107linesextractedfromtheVALD Fe spot region darker in the UV and brighter in the optical
data base (Kupkaetal. 1999). In addition to the standard lists of wavelengths than the surrounding atmosphere (e.g., see Fig. 9
transitions between measured energy levels, this includes a new of Lu¨ftingeretal. 2010a). Contrary to the impression conveyed
massivepredictedlinelistcalculatedbyR.Kurucz1anddiscussed by S12, thisprominent influence of chemical composition on the
byGrupp,Kurucz&Tan(2009).Thesespectrallinedataareessen- structure of stellar surface layers is well-known to the Ap star
tialforaccurateopacitycalculationsinthecontextofApstaratmo- community and has been addressed in numerous applications of
sphericmodelling,especiallyforlargeFeoverabundancesexplored theLLMODELScodetoindividualstars(Kochukhovetal.2009b;
inthisstudy.Itisnotclearwhatpredictedlinelistwasemployed Shulyak,Kochukhov&Khan 2008; Shulyaketal. 2009, 2010b;
byS12. Lu¨ftingeretal.2010a)–asignificantandfertileresearchdirection,
The distribution of temperature with optical depth for five seeminglyunnoticedbyS12.
LLMODELSatmospheres computed for different Feabundance is
illustratedinFig.2.Lookingattheplotoftemperatureasafunction
3.2 Locallineprofilesandcontinuumintensities
ofτ5000,weseeaprominentbackwarming,whichleadstoasteeper
temperaturegradientandchangesthetemperatureatlogτ5000 =0 Toinvestigaterepercussionsofmetallicity-inducedchangesofat-
by500–1500K.Theseeffectsarelessextremeinthehigheratmo- mospheric structure on the local line shapes and continuum in-
sphericlayerscorrespondingtothetypicalline-formingregions. tensity, we computed the Stokes IQUV profiles of the FeII
Itisimportanttorealize,however,thatτ5000 reflectschanges 5018.44 A˚ line using polarised spectrum synthesis code SYN-
inthemodelstructureonlyatonewavelength.Thus,examiningthe MAST (Kochukhov,Makaganiuk&Piskunov 2010). Thisspectral
plotofT vs.τ5000,onecanonlysaythatforthemodelswithdif- line choice is determined by previous magnetic DI studies (K04,
ferentironabundancetheformationdepthofthecontinuumfluxat KW10), where this feature, along with FeII 4923.93 A˚, was one
λ = 5000A˚ correspondstosubstantiallydifferentlocaltempera- of the very few individual metal lines showing detectable lin-
ture.Butthispictureandtheresultingconclusions maychangeif ear polarisation signature in the MuSiCoS four Stokes param-
one chooses a monochromatic optical depth which is weakly af- eter spectra of Ap stars (Wadeetal. 2000). As emphasised by
fectedbytheFeabundance. KW10, FeII 4923 and 5018 A˚ generally represent a poor choice
Amorenaturalwaytocomparemodelatmospherestructures for Doppler imaging of chemical spots because these lines are
istoplotphysicalquantitiesagainstanopticaldepthcomputedwith saturated in a typical spectrum of a Fe-rich Ap star and thus
somemeanopacitycoefficient.Inparticular,anopticaldepthbased are weakly sensitive to the horizontal abundance variations. On
on the Rosseland opacity coefficient κR has a certain advantage: the other hand, strong lines are noticeably affected by vertical
spectralregionswiththesmallestopacitiestransfermostofthera- abundancestratification(Ryabchikova,Leone&Kochukhov2005;
diativeenergyandatthesametimecontributemosttoκR.Exam- Kochukhovetal. 2006), which is usually ignored in Doppler
ining the physical quantities of different models as a function of mapping.Non-magneticabundanceDIanalyses(Kochukhovetal.
τR allowsone totracechanges of themodel structure on anatu- 2004b; Lu¨ftingeretal. 2010a) as well as MDI studies based on
raldepthscalerepresentingfrequency-integratedpropertiesofthe theStokesIandV spectra(Kochukhovetal.2002;Lu¨ftingeretal.
radiationfield. 2010b)normallyavoidsuchstronglines,dealingwithweakorin-
ThelowerpanelinFig.2illustratesthedepthdependenceof termediatestrengthfeatures.Toassesshowthelatterlinesbehave
temperatureforthesamesetof LLMODELSatmospheresasmen- intheFe-richLLMODELSatmospheres,wehavecomputedasec-
tionedabove,butnowasafunctionofτR.Itisobviousthattheef- ondsetofStokesprofilesforafictitiousweakerironlinewiththe
fectsofdifferentironabundance introducemuchsmallerchanges same parameters as FeII 5018.44 A˚, but with loggf reduced by
comparedtothoseimpliedbytheT vs.τ5000plot.Inparticular,the 1dex.
temperaturevariationbetweendifferentmodelsisreducedtoonly Startingfromthenon-magneticcase,weshowinFig.3anor-
afewhundredK.Inthesub-photosphericlayersT(τR)isverysim- malised disk-centre Stokes I profiles of FeII 5018.44 A˚ and the
ilarforallmodelssimplybecauseofthefundamental propertyof weaker line for all five model atmospheres discussed above. One
τR:whenτR ≫ 1thetemperaturedistributionasymptoticallyap- set of line profiles was computed self-consistently with the Fe
proaches that of agrey model, which, asiswellknown, depends abundance adopted for the model atmosphere calculation, while
onlyonTeff andτR.Therefore,regionsofthesameτRexhibitthe another set representstheusual approach based onameanatmo-
sametemperature. sphericmodelwithlogNFe/Ntot =−3.79anddifferentFeabun-
Concluding, a huge variation in the model temperature dis- dancesinthespectrumsynthesis.Thesecalculationswererepeated
tributionsplottedagainst τ5000 shouldnot bemistakenforanex- for a 3.5 kG longitudinal field as well as for a4.0 kG transverse
haustiveandcompletecomparisonofdifferentmodels.Acomplete fieldinordertoassessanyimpactoftheatmosphericstructureon
analysisshouldalwaysincludenotonlycomparisonsofthedepth- the Stokes V and Q profiles, respectively. The V and Q spectra
dependent thermodynamic properties, but also different spectro- areillustratedinFig.3bandc.Allcalculationsincludedconvolu-
tionwithaGaussianinstrumentalprofilesappropriateforthefour
Stokes parameter MuSiCoS observations employed by K04 and
1 http://kurucz.harvard.edu/atoms.html KW10.
MagneticDopplerimagingconsideringmetallicity-dependentatmosphericstructure 7
Figure3.Comparisonofthenormaliseddisk-centreprofilesoftheFeII5018.44A˚ line(lowerprofilesineachpanel)computedself-consistentlyforthemodel
atmosphereswithdifferentironabundances(solidlines)withtheprofilescalculatedforthesameabundancesandasinglemeanatmosphere(dashedlines).
ThesamecomparisonispresentedforafictitiousweakerFeIIline(upperprofilesineachpanel).Thecorrespondingspectraareshiftedverticallyby0.2ofthe
continuum.Thethreerowsofplotsshowcalculationsfora)StokesIintheabsenceofmagneticfield,b)StokesV forthe3.5kGlineofsightmagneticfield,
c)StokesQforthe4.0kGtransversemagneticfield.
Figure4.Maximumrelativeprofiledifferenceofthenormaliseddisk-centre Figure5. SameasFig.4forthemaximumrelativeprofiledifferenceofthe
StokesIspectraoftheFeII5018.44A˚ linecalculatedself-consistentlywith normaliseddisk-centreStokesV inthe3.5kGlineofsightfield(circles)
amodelatmospherestructureandusingasinglemeanmodelatmosphere. andStokesQinthe4.0kGtransversefield(triangles).
Symbolsshowcalculationsforzeromagneticfield(circles),3.5kGlineof
sightfield(triangles),and4.0kGtransversefield(squares).
calculationsexceptforthestrongestline,forminginthemostFe-
richatmosphere.Eveninthatcase,thediscrepancyisonly4%of
Figures 3a–c leave no doubt that the difference between the thecontinuumforthelinewhichis83%deep.Ingeneral,themax-
normalisedlocallineprofilescomputedwithdifferentmodelatmo- imumrelativedifferencedoesnotexceedseveral%oftheStokesI
spheresandwithameanatmosphereanddifferentFeabundances profileamplitudeforFeII5018.44A˚ (Fig.4)andisentirelynegli-
isextremelysmall.Onecannotvisuallydistinguishthetwosetsof gibleforaweakerline.
8 O. Kochukhov,G. A. WadeandD. Shulyak
atmosphere).Asdiscussedabove,wedonotexpectthisregionto
contributesignificantlytothedisk-integratedlineprofiles.Thisis
directlyconfirmedbythespectrumsynthesiscalculationspresented
inthenextsection.
4 MAGNETICDIWITHSELF-CONSISTENTMODEL
ATMOSPHERESTRUCTURE
4.1 MagneticinversioncodeInvers13
Wehavecarriedoutaseriesofforwardlineprofilecalculationsto
assesstheimpactoftheanomalouslocalmodelatmospherestruc-
tureonthedisk-integratedStokesspectraofα2CVn.Fortheseex-
perimentsweusedournewmagneticDIcodeINVERS13.Thisef-
Figure6.Relative continuumintensity atλ=5018.44A˚ asafunctionof ficientandversatilecoderepresents afurtherdevelopment of our
theFeabundanceadoptedforthemodelatmospherecalculation. magnetic inversion tool INVERS10, which has been described in
detailbyPK02,evaluatedbyKochukhov&Piskunov(2002),and
wassubsequentlyappliedinseveralstudiesofanon-uniformdistri-
How does this difference compare with the sensitivity of butionofmagneticfieldsandchemicalabundancesonthesurfaces
StokesI toabundancevariation?ConsideringtheFeII5018.44A˚ ofApstars(K04,KW10,Lu¨ftingeretal.2010b;Kochukhovetal.
lineforthemodelwithlogNFe/Ntot =−2.19,wefoundthatthe 2011).
Fe abundance needs to be changed by just ≈0.10 dex, or 2% of INVERS13 and INVERS10 adopt similar regularisation and
thelogarithmicFeabundancerangeforα2CVn,toobtainthesame non-linearoptimisationstrategies,basedontheTikhonovregular-
maximumprofiledifferenceasseeninFig.4. isation and modified Levenberg-Marquardt method, respectively.
The metallicity-induced modification of the atmospheric However,insteadofusingasinglemodelatmospherestructureas
structureleadstoatinyvariationofthenormalisedStokesV andQ wasdoneinallpreviousDIstudiesofApstars,INVERS13iscapa-
profiles,illustratedinFig.3bandc,respectively.Intheworstcase bleofmappingthethreecomponentsofthemagneticfieldvector
scenario,thesechangesamounttoabout0.05–0.10ofthefullpo- andtreatingsimultaneouslyoneadditionalscalarsurfacemap(tem-
larisationprofileamplitude(seeFig.5).Generally,thepolarisation perature, abundance of a given element, magnetic field strength)
profiles become slightly broader, while their low-order moments fully self-consistently with the local model atmosphere structure.
remainunchanged. ThebasicmethodologyofapplyingINVERS13toApstarsisthus
The full Stokes profile calculations presented in this section verysimilartoaself-consistentmagneticandtemperaturemapping
fullysupporttheconclusionbyKW10that“thelocallineprofiles ofcoolactivestars,exceptthattheparameterdistinguishingdiffer-
are sensitive to model structure effects to a much smaller degree ent local model atmospheres is a chemical abundance instead of
thantochangesofabundanceormagneticfield”.Atthesametime, Teff.
enhancedmetallicityleavesamajorimprintonthecontinuumin- At the start of calculations, a grid of model atmo-
tensity, which is increased in the optical due to the flux redistri- spheres (either ATLAS, MARCS or LLMODELS) is pro-
bution from the short-wavelength regions. Figure 6 demonstrates cessed by INVERS13. The equations of chemical equilib-
that the Fe-rich zones are expected to be up to 40% brighter in rium (cool atmospheres) or ionisation balance (hot atmo-
thecontinuumatλ5018.44A˚ withrespecttothereferencemodel, spheres) are solved with the help of a robust molecular
whiletheregionswitharelativeFeunderabundanceturnouttobe equilibrium solver (Valenti,Piskunov&Johns-Krull 1998) im-
upto10%fainter.Theconsequencesofthesecontinuumintensity plemented in SME (Valenti&Piskunov 1996) and in the
changesforthedisk-integratedStokesIQUV profilesandforthe SYNTH3/SYNTHMAG/SYNMAST family of codes (Kochukhov
reconstructionofthesurfaceabundanceandmagneticfielddistri- 2007; Kochukhovetal. 2010). Then, INVERS13 calculates the
butionswillbeaddressedinthenextsection. depth-dependent line and continuum opacity coefficients for all
Thegeneralconclusionregardingtheinsignificanceofthedif- model atmospheres and all molecular and atomic spectral lines
ferential line-blanketing for the local spectral line shapes emerg- included in the input list. The atomic line parameters are taken
ingfromourcalculationsseemstobedrasticallydifferentfromthe from VALD (Kupkaetal. 1999), whereas the data on molec-
claimsmadebyS12inSection4oftheirpaper.Thisdiscrepancy ular transitions are taken from the MARCS opacity database
comes from the fact that S12 have neglected to consider the en- (Gustafssonetal.2008).
tireFeabundancerangeanddifferentlinestrengths,limitingtheir Thecoremoduleof INVERS13isthequadraticDELOalgo-
discussiontotheprofileofasingle,verystronglinecomputedfor rithm for highly accurate solution of the LTE polarised radiative
extreme Fe overabundances which occupy a tiny fraction of the transfer equation in an arbitrary stellar atmosphere permeated by
visiblesurfaceofα2CVn.Inaddition,theirlocal profileanalysis a depth-independent magnetic field with a given (local) strength
lackedacomparisonoftheintensitydifferenceswiththeactualpro- andorientation.PK02demonstratedthatthequadraticDELOfor-
fileamplitude,asinourFig.3,anddidnotatalladdresspolarisa- mal solver is superior in terms of the speed and accuracy to the
tionspectra.Asrevealedbyourmorecomprehensivecalculations, ZeemanFeautriermethodusedbyS12.Foreachelementinastel-
theatmospheric structurevariationsresult inasystematicchange lar surface grid optimally sampling different latitudes (see Fig.5
ofthecontinuumfluxesbuthasaminusculeinfluenceonthenor- inPK02),ourcodeevaluatesthelineabsorptionmatrixbyadding
malisedintensityandpolarisationlinesshapes,exceptforasmall opacitiesofallcontributingZeemancomponentsandcomputesthe
regionoftheparameterspace(thestrongestlineinthemostFe-rich local Stokes IQUV profiles and continuum intensity Ic by per-
MagneticDopplerimagingconsideringmetallicity-dependentatmosphericstructure 9
formingaquadraticinterpolationwithinthreesetsofmodelspec-
tra corresponding to the model atmospheres bracketing the value
of a scalar parameter. This also yields precise derivatives of the
StokesIQUV spectraandassociatedintensityIcwiththerespect
to ascalar parameter distinguishing different models in the input
model atmosphere grid. One-sided derivatives withrespect tothe
radial,meridional,andazimuthalmagneticfieldvectorcomponents
arecomputednumerically.ThefullcalculationofthelocalStokes
spectraandtheirderivativesthusconsistsof12separatepolarised
lineprofilecalculationsforeverysurfaceelementateveryrelevant
rotationalphase.
The local profiles are convolved with a Gaussian function
tomodelinstrumentalandradial-tangentialmacroturbulentbroad-
ening. Profilesare Doppler-shifted and summed with appropriate
weightsforeachrotationalphase.Asimilarphase-dependentdisk-
integrationiscarriedoutforthecontinuumintensity.Theresulting
StokesIQUV spectraarenormalisedbytherespectivecontinuum
fluxes.Thematrixofpartialderivativesofeveryobservedspectral
pixelwithrespecttoeverysurfaceparameterisestablishedtaking
intoaccount boththemagnetic and model structuresensitivityof
thelineprofilesandthemodelatmospheresensitivityofthecontin-
uumfluxes.Usingthesederivatives,thecodeestablishesthecurva-
turematrixrequiredbytheLevenberg-Marquardt algorithm,adds
regularisationtermsanditerativelyadjustssurfacemapstomatch
theobservations.
INVERS13 takes advantage of the standard MPI libraries
available at any modern multi-CPU computer to perform highly
efficientparallelradiativetransfercalculationswithanovelmaster-
slavetechniquediscussedbyPK02.Weusuallyrunthecodewith
onemasterprocesscollectingresultsoftheradiativetransfercalcu-
lationsby16–32slaveprocesses.Thecodewassuccessfullytested,
showingalinearscaling,withuptoseveralhundredCPUsinsimul-
taneousinversionsofoverathousand atomicandmolecularlines
spread over ten wavelength intervals. In practice, the number of
spectrallinesandwavelengthintervalsmodelledsimultaneouslyis
limitedonlybytheavailableRAM.Forthecalculationsdescribed
here, modelling of one or twoFeII spectral linesat 20 rotational
phases required less than one minute of computing time for the
forward spectrum synthesis and about threehours for a complete
inversiononamodest,10-year-old16-CPUcomputer.Thisexperi-
enceshowsthatcurrentlyavailablecomputingresourcesandstan-
dardscientificprogrammingtechniquesbasedonwell-established
FORTRANcodesenableafullyrealistictreatmentofthelateralvari-
ation of stellar atmosphere, without arecourse tothe next gener-
ation computers or exotic programming languages advocated by
S12.
INVERS13wasrigourouslytestedwithnumericalexperiments
forcool activestarsusingsimulatedcircularpolarisationand full
Stokes vector data (Kochukhov&Piskunov 2009). Thecode was
also applied to interpret circular polarisation observations of the
activeRSCVnstarIIPeg(Kochukhovetal.2009a).Acomprehen- Figure7.TheoreticalfourStokesparameterprofilesoftheFeII5018.44A˚
siveMDIstudyofthistargetiscurrentlyunderway. linecorrespondingtothefinalmagneticandabundancemapsderivedfor
α2CVnby KW10.This plot compares the self-consistent profiles (solid
line)withthestandardcalculationsusingameanmodelatmosphere(dashed
4.2 Disk-integratedIQUV profilesoftheFeII5018A˚ line line). For comparison, we also show profiles for the smoothed magnetic
fieldgeometrymodeldiscussedbyKW10(dottedline).Spectraforconsec-
We have used the forward polarised spectrum synthesis calcula- utiverotationalphasesareshiftedvertically.Rotationalphasesareindicated
tionswithINVERS10andINVERS13todirectlyanswertheques- ontheright.Thebarsatthelowerrightofeachcolumnshowthehorizon-
tion of how important is the local atmospheric structure for the talandvertical scale(0.5A˚ and1%oftheStokesI continuum intensity
KW10 analysis of the Fe abundance distribution and magnetic respectively).
field geometry of α2CVn. We adopted the magnetic and abun-
dancemapsderivedinthatstudyandemployed asingle ATLAS9
modelatmospherewithparametersTeff=11600K,logg=3.9and
10 O. Kochukhov,G. A.WadeandD. Shulyak
canbecompensatedbyamere∼0.1dexreductionoftheaverage
Feabundance,soitisnotimmediatelyobviousthataccountingfor
thelocalatmosphericstructureshouldleadtoanymajorchangeof
theabundanceinversionresultsandremoveextremelyhighabun-
dancesfromtheFemap.
Atthesametime,itisstrikingtoseehowlittleeffecttheuse
ofself-consistentmodellingoftheStokesprofileshasonthelinear
and circular polarisation spectra. The discrepancy in Stokes V is
only0.2–0.4%, whichexceedstheobservational uncertaintiesbut
isnegligiblecomparedtothemaximumcircularpolarisationsignal
of9%.Theimpactonthelinearpolarisationspectraisevensmaller.
ItisevidentfromFig.8thataccountingforthelocalatmospheric
structurechangesStokesQandU byabout0.1%,whichisbelow
thenoiselevelofobservationsforallbuttworotationalphases.
Itisenlighteningtocomparethedifferencebetweenthestan-
dard INVERS10 calculation and self-consistent INVERS13 model
withthediscrepancy betweentheformer spectraandthosecorre-
sponding to the smoothed version of the magnetic field topology
of α2CVn(thick solidline inFig. 8). Weremind thereader that
thisdiscrepancy,constitutingthemainevidenceforthesmall-scale
magneticfieldsinα2CVn,wasdismissedbyS12asinsignificant.
Figures7and8leavenodoubtthattheconclusionsofS12areer-
roneous. A signature of the small-scalefields yields a systematic
Figure 8. Phase dependence ofthe amplitude, observational uncertainty, modulation of the linear polarisation spectra, which is above the
andthedifferencebetweentheoreticalStokesparameterprofilesillustrated
observational noise for 15 out of 20 rotational phases in Stokes
inFig.7.ThisplotcomparestheamplitudeofStokesprofiles(blacksolid
Qandfor18phasesinU.Weseethattheactualsituationisjust
line)withthediscrepancybetweenthestandardandself-consistentcalcu-
the opposite of that claimed by S12 in Sections 5 and 7 of their
lation(dashedline),andobservationaluncertainty(dottedline).Forcom-
work:theinfluenceofstructuredfieldsishighlysignificantforthe
parison,wealsoshowthedifferencebetweenthestandardprofilesandthe
spectracorrespondingtothesmoothedmodelofthemagneticfieldtopology StokesVQUprofiles,whiletheassumptionofameanmodelatmo-
(redsolidline). sphereislargelyunimportantforV andcompletelynegligiblefor
QUeveninthispessimisticassessment(noabundanceadjustment,
astrongline).
[M/H]=+1.0forthefullStokesparametersynthesisoftheFeII TheresultsoftheforwardfullStokesprofilecalculationsun-
5018.44A˚ linewithINVERS10.Thesecalculationsareidenticalto equivocallydemonstratethatthespeculationsbyS12aboutthesig-
the final Stokes profilefitspresented by KW10. In particular, we nificanceofthelocalmodelatmospherestructureforthemodelling
considerthesamesetofrotationalphasesandtakeintoaccountthe ofpolarisationinα2CVnareincorrect.Atthispointthereaderis
spectralresolutionoftheMuSiCoSStokesvectordataavailablefor certainly entitled to ask what led S12 to these erroneous conclu-
α2CVn. sions.TheFedistributionconsideredbyS12inSection5oftheir
AnothersetoflineprofileswasproducedwiththeINVERS13 paperappearstobefarmoreextreme(log(NFe/Ntot) > −2over
codeusingthegridofLLMODELSatmospheresdiscussedinSec- 18%ofthevisiblestellarsurfacerightatthediskcentre)compared
tion 3.1. To avoid extrapolation in Fe abundance, we augmented totheonefound for α2CVnbyKW10(only 2.5%of thevisible
thisgridwithtwoextrememodelscomputedforlogNFe/Ntot = surfaceclosetothelimb).Theunspecified“eccentricdipole”with
−1.60 and −6.40. We checked that using a denser grid of 13 an unusually high contrast (fieldstrength variation by a factor of
modelatmospheresmakesnodifference.Theresultingnormalised 3.7insteadofafactorof2expectedforacentreddipole)seemsto
INVERS10andINVERS13Stokesparameterprofilesarecompared havebeenhand-pickedtomaximiseprofiledifferences.Curiously,
inFig.7.Inaddition,weplotasetofsyntheticINVERS10Stokes describingtheStokesparametercalculationsforthislargeFespot
IQUV parameters corresponding to the smoothed model of the and unusual magnetic geometry, S12 do not mention taking into
magnetic field geometry of α2CVn discussed by KW10. A de- accounttheinstrumentalprofileofthespectrograph,althoughthey
tailedassessment of thedifference betweenthe“standard”, “self- stressamoderateresolutionoftheMuSiCoSfourStokesparame-
consistent”and“smoothed”modelprofileswithrespecttotheob- ter data elsewhere in the paper. It is also very surprising that the
servational uncertainties is presented in Fig. 8. We note that for QandU profiles,forwhichtheeffectofself-consistentmodelling
polarisationthiscomparisonrepresentsaconservativeestimateof istheleastimportantbutwhichconverselyhaveplayedakeyrole
possible errors because we do not allow the code to adjust the inthediscoveryofsmall-scalemagneticfeaturesonthesurfaceof
StokesIprofileintensityasitwoulddoinarealmagneticinversion α2CVn,arecompletelyneglectedinthatpartoftheS12paper.
(seediscussioninSection2.3),therebyimprovingthefittoQUV. CalculationspresentedinSection7oftheS12paperaresim-
Whatcanwelearnfromthiscomparison?Clearly,theeffectof ilarly inadequate. In that part of their study the authors claimed
self-consistentcalculationsisnoticeableforStokesI.Theintensity to have assessed the feasibility of detecting a representative sig-
profile changes by 2–4% of the continuum, which is not negligi- nalofahigh-contrastmagneticspot.However,thescenarioconsid-
blecomparedtothecentrallineintensityof27–35%.Inagreement eredisnot applicable tothecaseof α2CVnbecause theadopted
withthepredictionmadeinSection3.2,theself-consistentinten- spot diameter is underestimated by a factor of two and the im-
sityprofilesaresystematicallydeeper duetoanincreasedcontin- posedcorrelationofthesmall-scaleabundanceandmagneticfield
uumbrightnessoftheFe-richsurfaceregions.However,thiseffect enhancementsiscontrarytowhatwasfoundforthatstar.Further-
