Recovery of the global magnetic field configuration of 78 Virginis from Stokes IQUV line profiles

0.89 MB·English

by  V. R. Khalack

Astronomy & Astrophysics manuscript no. 4326kw February 5, 2008
(DOI: will be inserted by hand later)
Recovery of the global magnetic field configuration of
78 Virginis from Stokes IQUV line profiles
6
0 V.R. Khalack1,2 and G.A. Wade3
0
2
1 D´epartement de physiqueet d’astronomie, Universit´ede Moncton, Moncton, N.-B., E1A 3E9, Canada
n
a 2 Main Astronomical Observatory,27 Zabolotnoho Str., 03680, Kyiv,Ukraine
J 3 Department of Physics, Royal Military College of Canada, PO Box 17000 stn ’FORCES’, Kingston, Ontario,

0 Canada K7K 4B4
3
Received date will be inserted by the editor Accepted date will be inserted by the editor

1
v
Abstract. The surface magnetic field configuration of the Ap star HD 118022 (78 Vir) has been reconstructed
7
in theframework of the magnetic charge distribution (MCD) method from the analysis of Stokes IQUV spectra
7
obtained using the MuSiCoS spectropolarimeter at Pic du Midi Observatory. Magnetically-sensitive Fe ii lines
6
were primarily employed in the analysis, supposing that iron is evenly distributed over the stellar surface. We
1
0 show that the Stokes IQUV profile shapes and variations of 78 Vir can be approximately fit assuming a global
6 magneticfieldconfigurationdescribedbyaslightlydecentered,inclinedmagneticdipoleofpolarsurfaceintensity
0 approximately3.3kG.Thederivedinclinationsofthestellarrotationalaxistothelineofsighti=24±5◦ aswell
/ as to the magnetic dipole axis β =124±5◦ are in good agreement with previous estimations by other authors,
h
whereas thesky-projectedposition angle⋆ ofthestellar rotation axisΩ∼110◦ is reportedhereforthefirsttime.
p
- In addition, several lines of Cr ii and Ti ii were studied, yielding evidence for non-uniform surface distributions
o of theseelements, and magnetic field results similar tothose derived from Fe.
r
t
s Key words. stars: chemically peculiar – stars: magnetic fields – line: polarisation – stars: individual: HD 118022,
a
78 Virginis
:
v
i
X
1. Introduction field configuration models. As argued by Borra (1980),
r
a the profiles were well reproduced with an inclined dipole
78 Virginis (HD 118022)is a bright Ap star, and the first
geometry that contains a moderate quadrupolar compo-
star other than the sun in which magnetic field was dis-
nent. Nevertheless, he also pointed out that a decentered
covered (Babcock 1947). The longitudinal magnetic field
(a=0.2) dipole model provides essentially the same fit
of 78 Vir, as first observed by Babcock, is variable, and
quality. Borra also concluded that the 78 Vir presents to
constantly negative. 78 Vir is also marginally variable
anobserveraremarkablyuniformmagneticfieldovermost
in broadband light, in radial velocity, and shows varia-
of its visible disk, at most phases. It is unfortunate that
tions of its line profiles (Preston 1969). As suggested by
due to the small inclination of the rotational axis (about
Preston (1969), the observed properties of 78 Vir can be ◦
25 )asignificantpartofthestellarsurfaceremainshidden
explained in the framework of the oblique rotator model
- a part which might have a less uniform field geometry.
(Stibbs 1950), in which the star rotates with a period of
approximately 3.7 days and has a rotation axis forming a
Theanalysisofbroadbandlinearpolarization(BBLP)
small angle with respect to the line of sight.
measurements of several Ap stars (including 78 Vir) by
The firstcomprehensivemodelling ofthe surfacemag-
Leroy et al. (1996) suggests that the magnetic fields
netic field structure of 78 Vir was performed by Borra
of many Ap stars exhibit departures from the standard
(1980). Borra obtained high-resolution measurements of
oblique rotator model assuming a pure dipole field ge-
circularpolarisationacrossthe Feii λ4520.2˚A line, which
ometry.Accordingtothoseauthors,significantdiscrepan-
he attempted to reproduce using nine different magnetic
cies between the BBLP observations and the “canonical
model”results(Landolfietal.1993)requiredtheassump-
Send offprint requests to: V. Khalack
e-mail: [email protected] tion of local departures from a dipolar field. In partic-
⋆ Ω increases clockwise from the axis to the North Celestial ular, Leroy et al. (1996) showed that the BBLP varia-
Pole and relates to the azimuth angle Θ specified by Landolfi tionsofmoststarscouldbereproducedassumingadipole
at al. (1993) as Ω=360◦−Θ. magnetic field geometrywith slightly expanded field lines
2 V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field
over some parts of the magnetic equator. For 78 Vir, the Table 1. Fundamental parameters of 78 Vir.
observed BBLP variations do not show especially strong
departures from the dipolar case, although according to Parameter Value Reference
Leroy et al. (1996) a modified dipolar model provides a Spectral type A1pCrSrEu Cowley et al. (1969)
Age 3.0+0.7×108y. this work
somewhat better fit to the BBLP curves for this star. −1.3
Distance 56.2±2.5 pcs Hipparcos (ESA 1997)
In order to better constrain the magnetic field con-
Period 3.d7220±0.d003 Preston (1969)
figuration of 78 Vir, in this paper we undertake a more
Teff 9200±290K Monier (1992)
detailed modelling of the surface magnetic field structure logg 4.50± 0.25 ibidem
based on the analysis of high resolutionline profiles in all Mv 1.18±0.10 this work
4 Stokes parameters. In Sec. 2 we discuss the properties L⋆ 27.3±2.5 L⊙ this work
of obtained Stokes IQUV spectra, and in Sec. 3 we sum- R⋆ 2.06±0.17R⊙ this work
marisethefundamentalcharacteristicsof78Vir.InSec.4 M⋆ 2.18±0.06M⊙ this work
we describe the features of the magnetic field modelling Vesini 12±1 km s−1 this work
framework,andderivetheglobalmagneticfieldconfigura- i 25◦±5◦ Leroy et al. (1996)
β 120◦±5◦ ibidem
tionofthestar,comparingobservedandcomputedStokes
profiles for various spectral features. In Sec. 5 we discuss
the derived globalmagnetic field characteristicsand their
agreement with earlier studies.
(ESA 1997), we find a somewhatlarger value for the stel-
lar radius R = 2.06±0.17 R⊙. Using the Hipparcos vi-
2. Observations sual magnitude and parallax we have found the absolute
visual magnitude of 78 Vir, M = 1.18 ± 0.10. Taking
v
Spectropolarimetric observations of 78 Vir were obtained
into account the bolometric correction (Flower 1996),
in 1997 February, 1998 February and 1999 January using
which corresponds to Monier’s (1992) 9200±290 K ef-
the2mT´elescopeBernardLyotatObservatoireduPicdu
fective temperature and the bolometric zeropoint correc-
Midi (Wade et al. 2000a). The MuSiCoS cross-dispersed ⊙
tion BC =-0.07, which is obtained assuming M =4.74
´echelle spectrograph (Baudrand & B¨ohm 1992) and ded- V bol
for the Sun (Bessell et al. 1998), we can find the ab-
icated polarimeter module (Donati et al. 1999) were em-
ployed for the observations. solute luminosity for 78 Vir L⋆ = (27.3 ± 2.5)L⊙. For
78 Vir Monier (1992) also estimates the integrated flux
The MuSiCoS spectrographis a table-top instrument, f =(2.7±0.27)×10−7erg sm−2s−1.Takingintoaccount
which allows the acquisition of a stellar spectrum in a ⋆
thedistancetothestar,thisprovidestheluminosityL =
givenpolarizationstate(StokesV,QorU)throughoutthe ⋆
(26.5±2.8)L⊙. Finally, employing the stellar radius and
spectralrangefrom4500to6600˚Awitharesolvingpower
the effective temperature (see Table 1) we obtain a third
ofabout35000,inasingle exposure.The spectrographis
fedbyadoubleopticalfibredirectly fromtheCassegrain- estimate of the absolute luminosity L⋆ = 27.4±4.5L⊙.
All of these values are in good mutual agreement. The
mounted polarimeter. The optical characteristics of the
polarization analyser, as well as the spectropolarimeter luminosity L⋆ = (27.3±2.5) L⊙ together with the effec-
tive temperature (see Table 1) allow us to find the stellar
observing procedures, are described in detail by Donati
etal.(1999).Observingdetails specific to the acquisition, mass M⋆ = 2.18±0.06M⊙ and age 3.0+−01..73 ×108 years
(see Fig. 1) by spline interpolation (Sandwell 1987) in
reduction and analysis of the 78 Vir spectra are provided
the model evolutionarytracks of Schaller et al. (1992) for
by Wade et al. (2000a).
metallicity Z=0.02.The derived age suggests that 78 Vir
The journal of spectropolarimetric observations is re-
has completed approximately 37% of its main sequence
portedbyWadeetal.(2000a).The52StokesV,QandU
life. The mass and radius provide logg = 4.16 ± 0.07.
spectra, obtained on 18 different nights, cover the whole
This value is marginally inconsistent with that of the
rotational period of 78 Vir approximately uniformly, and
Monier (1992), but itis ingoodagreementwiththe value
provide an averageS/N of about 370.
logg = 4.20 reported by King et al. (2003). A recent in-
vestigation of the Ursa Major stream characteristics per-
formedbyKingetal.(2003)derivedZ =0.016−0.02and
3. Fundamental parameters of 78 Vir
age(5±1)×108 yearsforthe stream.Theyfoundthat78
78 Vir was classified by Cowley et al. (1969) as Virisa”certainmember”ofthestreamfromphotometric
A1pCrSrEu. Adelman (1973a, 1973b) performed both a data,while the kinematic characteristicsofthe starargue
line identification and abundance analysis of this star. for its ”probable non-membership”. The derived age for
The distance to 78 Vir d = 48±15 pcs and its radius 78 Vir is marginally inconsistent with the UMa stream
R = 1.77±0.68R⊙ are derived by Monier (1992) using age. The published metallicity of 78 Vir is log(Fe/H)⋆−
theInfraredFluxMethod.Ontheotherhand,takinginto log(Fe/H)⊙ =−0.13 (Cayrel de Strobel et al. 1997) and
+1.58
account the mean angular diameter θ = 0.343 milliarcsec it is not clear if this star truly has a solar metallicity. For
a
obtained by Monier (1992) for this star and its distance this reason the derived uncertainties of the mass and age
d = 56.2± 2.5 pc derived from the Hipparcos parallax may be somewhat larger than indicated here.
V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field 3
Babcock (1947) observed that the longitudinal mag- 1111111.......8888888
netic field of 78 Vir was variable, but always negative.
ZZZZZZZAAAAAAAMMMMMMMSSSSSSS
Analysing Babcock’s data along with his own measure-
2222222.......5555555
ments, Preston (1969) determined the periodic character
1111111.......6666666
ofthemagneticfieldvariabilityandderivedtheephemeris:
JD (magnetic maximum)=2434816.9+3.d7220·E. (1) log L/Llog L/Llog L/Llog L/Llog L/Llog L/Llog L/Looooooo....... 1111111.......4444444 2222222.......11111118888888
Severalauthorshaveconfirmedthisperiodonthebasisof
furthermagneticfieldmeasurements(Wolff&Wolff1971; 2222222.......0000000
Wolff & Bonsack 1972; Wolff 1978; Borra 1980; Borra & 1111111.......2222222
Landstreet 1980; Landstreet 1982; Wade et al. 2000b),
while Landstreet (private communication) has estimated
1111111.......7777777
the accuracy of the period determination to be ±0.d003. 1111111
4444444.......1111111 4444444.......00000005555555 4444444 3333333.......99999995555555 3333333.......9999999 3333333.......88888885555555 3333333.......8888888
Preston (1969) also found that the crossover effect is llllllloooooooggggggg TTTTTTTeeeeeee fffffff fffffff [[[[[[[KKKKKKK]]]]]]]
very pronounced and is present throughout most of the
Fig.1. The position of 78 Vir on the H-R diagram
magnetic cycle. This feature is particularly remarkable
is compared with the theoretical evolutionary tracks of
at phase 0.85 (Wolff & Bonsack 1972, see also Wade et
Schaller et al. (1992), computed for solar metallicity and
al. 2000a). massesM = 1.7,2.0and 2.5M⊙. The age3.0×108 years
Applying the oblique rotator model with the dis-
andthe massM⋆ = 2.18M⊙ ofthe starareestimatedby
torted dipole approximation, Leroy (private communica-
interpolation of the theoretical evolutionary tracks. The
tion) found that they must adopt a period 3.7218 days
dashedlineshowstheisochrone,whichcorrespondstothe
in order to fit the observed BBLP variations. As dis- age of 78 Vir (3.0×108 y).
cussed in Sect. 1, their model, using local modifications
of the axisymmetric magnetic field imposed near the
magnetic equator, differs only mildly from a dipole con- 9200±290 K, a surface gravity logg = 4.5 and a pho-
figuration. The derived rotational axis inclination and tospheric metallicity [M/H] = 10. In addition, using a
dipole obliquity i = 25◦ and β = 120◦ provide a good model atmosphere with these values of logg and [M/H],
match to the longitudinal field and linear polarisation he has found that the effective temperature Teff =9300K
variations. Nevertheless, the period inferred is somewhat provides the best description of the spectral energy dis-
shorter than that obtained by other authors (Eq. 1), and tribution in the region from 1200˚A to 8000˚A at rota-
Wade et al. (2000b) reported that this period is only tional phase 0.0, while Teff =9200K seems to be the best
barelyconsistentwiththeavailablelongitudinalfieldmea- effective temperature, at phase 0.5. These temperatures
surements. are somewhat lower than most previous estimates, rang-
78Viralsoshowsphotometricvariabilityinthevisible ing from 9700K to 10700K (Mihalas & Henshaw 1966;
andtheinfraredwiththesameperiod(Eq.1).Combining Wolff 1967; Jugaku & Sargent 1968), probably because
the visual flux variability in the uvby bands, obtained by the models used by these authors were unblanketed and
Wolff & Wolff (1971), with their own new observations, calculated for solar abundances.
Catalano&Leone(1994)havetriedtoimprovetheperiod Thefundamentalparametersof78Viraresummarised
determination.Theiranalysisresultsinthenewephemeris in Table 1.
JD (ymin.)=2434816+(3.d722084±0.d000042)E, (2)
4. Modelling of the Stokes IQUV spectra
which was recently confirmed by Leone & According to the discussionof Sect.3, anATLAS9 model
Catanzaro (2001). These authors have shown with (Kurucz 1994) with parametersT =9250K,logg =4.5,
eff
the help of Hipparcos photometry that the light vari- [M/H]=0 and microturbulent velocity v =0 km s−1 suc-
t
ations of 78 Vir are not purely sinusoidal. Taking into cessfully approximates the stellar atmosphere of 78 Vir.
account all archival magnetic field observations as well Wedescribethestructureofthesurfacemagneticfield
as their own data, Leone & Catanzaro (2001) have of78Vir(assumedtobearigidlyrotatingandspherically
shown that the longitudinal field measurements are in symmetric star) using the magnetic charge distribution
phase only when the 3.722084 day period is adopted. (MCD) method (Gerth et al. 1997; Khalack et al. 2001).
Moreover, by adopting the 3.7218 day period suggested Originally, the MCD method considered a system of spa-
by Leroy (private communication), they find a 0.12-cycle tially separated point field sources with virtual magnetic
phase shift between the Hipparcos light curves and the charges in the stellar interior. In order to provide zero
other light curves. magneticfluxthroughthestellarsurfacethesumof“mag-
Monier (1992) has derived for 78 Vir from the sim- netic charges” should be kept to zero. These sources
ulation of the energy distribution in the spectral range produce a magnetic field whose potential at each point
from 1200˚A to 22000˚A an effective temperature T = on the stellar surface is specified by the superposition
eff
4 V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field
Table 2. Here the individual columns specify the ion, 4.1. Procedure
the wavelength, the quantum number J and the Land´e
From previous analyses of the Stokes IQUV spectra of
factor for lower and upper atomic levels, the observed
78 Vir, it has been found by Wade et al. (2000a) that
Stokes I profile depth and the line scaling factor (LSF)
the three strong Feii lines λ4923.93˚A, λ5018.44˚A and
(see Sect. 4.3), derived from the comparison of simulated
λ5169.03˚A show the strongest Zeeman signatures of any
equivalent width of the Stokes Q and U Zeeman profiles
lines in the optical spectra of CP stars. The real mag-
with the BBLP data (Leroy 1995). Asterisks marks the
netic field structure of 78 Vir is expected to differ only
LSF calculated from Stokes Q and U profiles simulated
marginallyfromacenteredmagneticdipole(basedonpre-
forafieldstructurethatisderivedfromonlyStokesI and
vious modeling efforts: Borra1980).The mostattention is
V profile variability (see Table. 3).
thereforepaidtothevariabilityoftheStokesI andV pro-
Ion λ,˚A Jlo glo Jup gup Iobs LSF files,whichcontainthemostinformationabouttheglobal
Feii 4620.52 3.5 1.21 3.5 1.40 0.33 0.035 surface magnetic field configuration. The Stokes I varia-
Feii 4635.32 2.5 1.20 3.5 1.13 0.32 0.036∗ tion provides information about the surface distribution
Fei 4635.85 1.0 1.49 2.0 1.89 of the chemical abundance, as well as the stellar radial
Feii 4923.93 2.5 2.00 1.5 2.40 0.58 0.079 velocity V and projected rotational velocity V sini. The
r e
Tii 5017.95 4.0 1.50 5.0 1.18 StokesV profilesaresensitivetotheglobalfieldconfigura-
Cri 5018.15 2.0 1.16 3.0 1.09 tion,inparticularthelower-ordermultipolarcomponents.
Feii 5018.44 2.5 2.00 2.5 1.87 0.60 0.102
The Stokes Q andU profilesalso providesome constraint
Crii 5018.84 3.5 1.24 3.5 1.42
on the global field morphology, but are most sensitive to
Feii 5100.61 4.5 1.54 4.5 1.34
Feii 5100.66 4.5 1.31 3.5 1.40 the smaller-scalestructure ofthe field. Unfortunately, the
Feii 5100.73 4.5 1.54 5.5 1.36 available Stokes Q and U profiles have low relative S/N
Feii 5100.85 1.5 0.80 1.5 0.72 0.47 0.083∗ ratio (Wade et al.2000a), and in this study they are used
Fei 5168.90 3.0 1.50 3.0 1.75 primarilytochecktheresultsoftheStokesI andV profile
Feii 5169.03 2.5 2.00 3.5 1.70 0.59 0.178 simulations and to determine the sky-projected position
Fei 5169.30 4.0 1.26 3.0 1.32 angle of the rotational axis, Ω. According to Khalack et
Feii 5197.48 2.5 1.20 1.5 0.72 al.(2001,2003)thisangle(0≤Ω<360◦)iscountedclock-
Feii 5197.58 2.5 0.57 1.5 0.44 0.39 0.073 wise from the rotation axis to the North Celestial Pole in
Feii 5362.74 3.5 1.15 4.5 1.24 the plane of the sky, and is related to the azimuth angle
Feii 5362.87 4.5 1.15 3.5 1.40 0.45 0.112∗ Θ specified by Landolfi et al. (1993) by Ω = 360◦ −Θ.
Cri 5362.96 3.0 1.33 3.0 1.06
The opposite counting of the position angle in the MCD
Feii 5362.97 3.5 1.44 4.5 1.34
Crii 5363.88 3.5 1.71 3.5 1.39 modeliscompensatedforbythenegativesignindefinition
Feii 6247.35 1.5 0.40 2.5 0.62 of the By-component of the local magnetic field. Hence a
Feii 6247.56 2.5 1.33 1.5 1.72 0.36 0.036∗ right-handed reference frame is applicable.
Feii 6432.68 2.5 2.00 2.5 1.65 0.24 0.026 Wehaveexaminedthespectraforadditionallineswith
Feii 6516.08 2.5 2.00 3.5 1.58 0.25 0.032 prominent polarisation signatures, in order to compile a
list of lines that are especially appropriate for this task.
To perform the line identification we use the VALD-2 re-
sources(Kupkaetal.1999;Ryabchikovaetal.1999).The
of potentials of individual sources (Gerth et al. 1997; adopted list contains Feii, Fei, Crii, Cri, Tiii, Tii and
Gerth & Glagolevskij 2001, 2004). Since the number of Mgii lines, but for the present study we will concentrate
sourcesisusuallymorethanone,weactuallyoperatewith primarily on the strongest Feii lines. The main reason
a systemof severalmagnetic dipoles located in the stellar for this is that Fe is presumably almost uniformly dis-
interior.Inordertominimizethenumberoffreemodelpa- tributed overthe stellar surface(see Sect. 4.2) andin this
rameters,themostconvenientwayistoconsiderasystem way we exclude from the fitting procedure a model of the
of two sources, which mathematically formulate a mag- surface abundance distribution. Table 2 presents the final
netic dipole (Khalack et al. 2001). When the dipole is listofFeiilinesusedinthesimulationprocedure,together
centered on the stellar centre and the dipole size is much with lines of some other elements that are responsible for
smaller than the stellar radius (Khalack 2002), the MCD blends.
modeltransformstotheconventionalmodelofasymmet- Some of the Feii lines listed in Table 2 have no
ricinclinedmagneticrotator(Stibbs1950).Otherwise,we detectable features in the Stokes Q and U spectra.
dealwithamorecomplexmagneticfieldconfiguration.The Nevertheless, they are comparatively strong lines (in the
mathematical verification of the MCD model as well as Stokes I spectra) with clear variability in the Stokes V
the procedure of specification of the angle β between the spectra, and are analysed without the linear polarization
rotational and magnetic dipole axes, the coordinates and data in order to check our final results.
the field strengths of the positive and negative magnetic All line profiles are simulated using the Zeeman2
poles on the basis of the derivedfree model parameters is polarised spectrum synthesis code (Landstreet 1988;
described in detail by Khalack et al. (2003). Wade et al. 2001). The code has been modified to in-
V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field 5
clude a magnetic field described within the framework of spectra are calculated with approximately the same re-
the MCD method (Khalack et al. 2003), and to allow for solving power as the observed spectra, this does not pro-
an automatic minimization of the model parameters us- vide a direct coincidence of wavelengths in the simulated
ing the downhill simplex method (Press et al. 1992). The and observed spectra. Therefore, during the χ2 function
relatively poor efficiency of the downhill simplex method, evaluation the simulated spectral intensity at the exact
requiringalargenumberoffunctionevaluations,isawell- observed wavelength is calculated using a linear interpo-
known problem. Repeating the minimization routine 3÷4 lation.
times in the vicinity of a supposed minimum in the pa- The scale of variability and the measurement errors
rameter space allows us to check if the method converges are different for each of the four Stokes IQUV spectra
to a global minimum. In our case, this technique requires (Wade et al. 2000a). In order to balance (from a statisti-
acomparativelylongcomputationaltime due to the large cal point of view) the information flow from each of the
amountofanalysedobservationaldataandthelargenum- four Stokes spectra, the respective χ2 contributions are
ber of free parameters. weighted.
The free model parameters employed in the line pro- The weights are derived from the comparison of the
file simulation can be divided into two groups: those that best fit values of χ2,χ2,χ2 with χ2 (when the ordinary
V Q U I
describe the magnetic field structure, and those that de- χ2 function is minimized) and provide approximately the
scribe the stellar geometry and atmospheric characteris- sameweightedvalueofthese functionsforthemajorityof
tics. The first group includes the parameters Qr, a1 and analysedlines(seeTable3).Theweightedχ2w functionsis
a2 (which specify the modulus of “magnetic charges”and specified in the following way:
their distance from the center of the star, expressed in
the units of stellar radius), and the parameters λ1,δ1 χ2 = 1[χ2+4χ2 +6(χ2 +χ2)], (4)
and λ ,δ (which specify the spherical coordinates of the w 4 I V Q U
2 2
“charges” in the rotational reference frame of the star). or
Meanwhile, the second group includes the parameters Ω
(thepositionangle),i(therotationalaxisinclinationwith χ2 = 1[χ2+4χ2], (5)
respect to the line of sight), V (the heliocentric radial w 2 I V
r
velocity), V sini (the projected rotational velocity), and
e ifweworkonlywiththeStokesI andV spectra.Thevalue
log(N /N )(theabundance(s)oftheelement(s)forming
x tot oftheχ2 functionisreducedthroughouttheminimization
thelinetobemodeled).InsteadofV siniwecanobviously w
e procedure,varyingthevaluesofallfreemodelparameters.
use only V as a free parameter, but V sini is preferable
e e Whenitreachesitsglobalminimum,wecanalsocalculate
here, because it allows us to compare our estimate of this
the ordinary χ2 function:
parameter directly with the results of other authors. The
minimum number of free model parameters is 11 for the 1
χ2 = [χ2+χ2 +χ2 +χ2]. (6)
MCD model if we do not take into account the linear po- 4 I V Q U
larization data, and 12 if we do. Sometimes, to obtain a
good fit to the line profiles we need to include in the sim- We begin by simulating the strong Feii λ4923.927 ˚A,
ulation lines of other chemical elements (see Table 2). In λ5018.44 and λ5169.03 lines, for which the minimization
this casethe numberoffreemodelparametersgrowswith process has been repeated for 13÷15 times starting from
the number of chemicalelements under investigation,but different locations in the free model parameter space in
usually does not exceed 15 parameters. order to avoid local minima. In the case of 78 Vir, the
Initially we specify arbitrarily the values of the free downhill simplex method converges to four global min-
modelparametersintherangeoftheirpossiblevaluesand ima, which are caused by the decentered dipole model
simulate the Stokes IQUV spectra at the phases of the symmetry in the MCD method. Two global minima cor-
◦
observations.Wethencomparethesimulatedprofileswith respond to the parameter sets i,β,Ω and i,β,180 + Ω
theobservedspectra,andcalculatethe reducedχ2,which which differ only by the angle Ω and provide exactly the
weadoptasameasureofthefitquality.Theexpressionfor same value of the χ2-function. Models with the oppo-
w
theχ2 function,whichreflectstheagreementbetween(for site direction of stellar rotation (180◦ −i,180◦−β,Ω or
example) the simulated Stokes I (I (ϕ)) and observed 180◦−i,180◦−β,180◦+Ω)do notresultin minima.The
Stokes I (Iobs) spectra is given by: λj i otherconfigurations(i,180◦−β,Ωandi,180◦−β,180◦+Ω)
i,λj
correspondto models in whichthe positive magnetic pole
χ2 = 1 NI 1 Ni Iio,bλsj −Iλj(ϕi) 2, (3) fbalceestoththeeobcasesrevoerf.7T8hVesier,cwonhfiergeurwaetioanlwsaayrse sneoettahpepnliecga--
I NI Xi=1 Ni Xj=1 σ[Iio,bλsj] ! ative magnetic pole, while the positive pole is partially
visible just for phases close to ϕ = 0.0 (Babcock 1947;
where σ[Iobs] corresponds to the measurement errors, ϕ Borra1980).Theothertwoglobalminimahaveparameter
i,λj i
is the rotationalphaseofthe observation,N specifies the setswhichdifferfromthe previousonesbythe locationof
I
number of spectra, while N is the number of pixels in the “magnetic charges”.The two “magnetic charges”can
i
each analysedline profile. In fact, although the simulated be located in two different stellar hemispheres (separated
6 V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field
Table 3. Results of Feii lines simulation for T = 9250K, logg = 4.5 and v =0 km s−1. The first column indicates
eff t
the free parameters, while the other columns specify the parameter values for the given line profile. The last column
providesthe averagedestimationerrorsforeachparameter.The first6 rowsshow the fit qualityfor the eachanalysed
Stokesprofile (Eq.3), weightedandunweighted(Eqs.4-6)χ2-function.The following12 rowsshow the bestfit values
ofthe freemodelparameters,while the final9 rowsprovidethe characteristicsofthe magnetic dipole poles,whichare
derived from the free parameters.
Line, ˚A 4620 4635 4923 5018 5100 5169 5197 5362 6247 6432 6516 σer
χ2 2.92 10.91 8.96 6.97 7.40 13.10 16.06 5.54 2.27 8.83 5.94
I
4χ2 6.14 7.86 9.14 8.69 6.97 9.09 8.71 6.64 5.00 12.24 7.39
V
6χ2 7.75 - 7.73 5.76 - 8.23 6.35 - - 12.15 9.30
Q
6χ2 7.42 - 10.04 6.82 - 10.79 6.80 - - 9.99 6.19
U
χ2 6.06 9.39 8.97 7.06 7.19 10.30 9.45 6.09 3.64 10.81 7.20
w
χ2 1.75 6.44 3.55 2.81 4.57 4.64 5.11 3.60 1.76 3.90 2.59
Qr,kG 180 254 303 202 230 221 151 192 398 173 289 50
a1,10−3 8.4 6.7 4.3 6.7 5.3 7.3 12.6 9.7 5.4 8.9 6.4 1.2
λ1 13◦ 24◦ 34◦ 41◦ 33◦ 15◦ -4◦ 19◦ 69◦ 23◦ 20◦ 5◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
δ1 -45 -46 -52 -51 -60 -30 -40 -44 -39 -45 -39 8
a2,10−3 3.1 3.7 3.5 5.2 4.3 2.5 1.6 3.7 5.1 3.8 2.7 0.3
λ2 118◦ 125◦ 144◦ 135◦ 145◦ 85◦ 111◦ 101◦ 113◦ 96◦ 111◦ 10◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
δ2 -16 -9 -9 -14 -12 -11 -11 -26 -11 -27 -11 10
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Ω 109 - 109 109 - 109 102 - - 164 128 17
i 22◦ 24◦ 22◦ 23◦ 25◦ 21◦ 29◦ 27◦ 26◦ 29◦ 24◦ 5◦
log(Fe/Ntot) -3.36 -2.82 -3.26 -3.24 -3.20 -3.12 -3.25 -2.86 -3.17 -2.97 -3.50 0.22
Vr -8.78 -8.42 -7.82 -7.51 -8.69 -7.23 -7.93 -8.25 -7.99 -6.57 -6.74 1.18
Vesini 10.9 11.1 11.9 12.0 11.5 12.6 11.9 13.0 11.9 13.7 11.5 0.9
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
β 125 124 122 120 123 118 129 126 123 126 122 5
a0,10−3 4.5 3.9 2.9 6.0 3.4 4.3 6.5 5.8 4.8 5.6 3.6 1.6
a,10−3 4.5 3.8 2.6 0.7 3.4 3.4 6.2 4.4 2.1 3.8 3.4 1.6
Bp, kG 3.25 3.95 3.47 3.18 3.13 3.02 4.00 3.45 3.42 2.69 3.92 0.7
λp -9◦ -10◦ -6◦ -7◦ -10◦ -8◦ -11◦ -8◦ -11◦ -8◦ -8◦ 5◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
δp -35 -33 -30 -30 -33 -28 -37 -36 -34 -36 -32 12
Bn, kG -3.18 -3.82 -3.46 -3.16 -3.12 -2.95 -3.85 -3.36 -3.41 -2.62 -3.86 0.7
λn 170◦ 170◦ 174◦ 173◦ 170◦ 172◦ 168◦ 171◦ 169◦ 171◦ 172◦ 5◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
δn 35 33 30 30 33 28 37 35 33 36 32 12
bytheequatorialplane),orinasinglehemisphere.Ifthey this procedure, and taking into account the uncertainties
are not significantly shifted from the stellar centre, they of the observationaldata and the obtained minimal value
canformthesamedipoleaxisandwillproducealmostthe of χ2-function, we can estimate the errors of the best-fit
samesurfacemagneticfieldconfiguration.The differences parameters.
betweenthetwomodelscanberevealedbythesensitivity
of χ2 function on the parameter values, but the minima
w 4.2. Results
will still exist. Preference is given to the deepest global
minimum,obtainedforthemodelwiththetwo“magnetic
Our initial assumption that iron is uniformly distributed
charges” located in the same stellar hemisphere. For the
over the surface of 78 Vir appears to be valid, given that
other lines the minimization process was performed only the analysed Feii lines reveal no significant variability of
3÷4times(althoughifwetestedthecontributionofblends the StokesI profileswith rotationalphase.EachFeii line
to the analysed profiles, we ran the minimization routine
(or group of lines) shown in Table 1 was analyzed inde-
severalmoretimes).Supposingthatallanalysedlinescon-
pendently of the others,using a stellar atmosphere model
tain the signatures of the same magnetic field structure, with T = 9250K, logg = 4.5, v =0 km s−1. For each
eff t
we chose the initial locations in parameter space not far
best-fit simulation the derived free model parameters are
fromtheparametersetobtainedfromthestronglineanal- givenin Table 3.During the simulationofsome Feiilines
ysis.
the Stokes Q and U profiles were not taken into account,
In order to evaluate the fit errors (and therefore the and consequently the angle Ω is not determined for those
uncertaintiesonthederivedfreeparameters),wecalculate lines.
deviations of the simulated profiles produced as a result The other data in Table 3 are derived from the free
of small variations of each of the free parameters, thus model parameters (Khalack et al. 2003). Here β defines
introducing a small shift along one axis in the χ2 hyper- the angle between the magnetic dipole axis and the stel-
spacefromthepointofthefunctionminimumvalue.Using lar rotation axis (the angle exists if these two axes cross
V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field 7
a) I/Ic b) V/Ic c) Q/Ic d) U/Ic
3333333333333333333333333333333333333333333333333.................................................5555555555555555555555555555555555555555555555555 0000000000000000000000000000000000000000000000000.................................................8888888888888888888888888888888888888888888888888
TiI CrI FeII CrII TiI CrI FeII CrII TiI CrI FeII CrII TiI CrI FeII CrII
| | | | .9740 | | | | 0000000000000000000000000000000000000000000000..............................................3333333333333333333333333333333333333333333333 | | | | | | | |
0000000000000000000000000000000000000000000000000.................................................7777777777777777777777777777777777777777777777777
....8888999955558888
3333333333333333333333333333333333333333333333333 .9740 .9793 .9851
.......8888888444444488888884444444 0000000000000000000000000000000000000000000000..............................................2222222222222222222222222222222222222222222222
....8888999955558888 ....9999000044441111 ....9999111133332222
..........7777777777555555555588888888885555555555
0000000000000000000000000000000000000000000000000.................................................6666666666666666666666666666666666666666666666666
.......8888888444444488888884444444 .......8888888555555533333337777777 .......8888888555555588888886666666
.............6666666666666777777777777711111111111118888888888888
2222222222222222222222222222222222222222222222222.................................................5555555555555555555555555555555555555555555555555 ................6666666666666666444444444444444422222222222222228888888888888888 ..........7777777777555555555588888888885555555555 ..........7777777777666666666688888888881111111111 ..........7777777777777777777777777777776666666666
...................5555555555555555555111111111111111111166666666666666666661111111111111111111 0000000000000000000000000000000000000000000000000.................................................5555555555555555555555555555555555555555555555555 .............6666666666666777777777777711111111111118888888888888 0000000000000000000000000000000000000000000000..............................................2222222222222222222222222222222222222222222222 .............6666666666666777777777777799999999999990000000000000 .............6666666666666888888888888877777777777774444444444444
......................4444444444444444444444333333333333333333333355555555555555555555554444444444444444444444 ................6666666666666666444444444444444422222222222222228888888888888888 ................6666666666666666555555555555555500000000000000002222222222222222 ................6666666666666666555555555555555599999999999999999999999999999999
2222222222222222222222222222222222222222222222222 .........................4444444444444444444444444000000000000000000000000033333333333333333333333337777777777777777777777777 0000000000000000000000000000000000000000000000000.................................................4444444444444444444444444444444444444444444444444 ...................5555555555555555555111111111111111111166666666666666666661111111111111111111 ...................5555555555555555555222222222222222222255555555555555555550000000000000000000 ...................5555555555555555555333333333333333333333333333333333333339999999999999999999
............................3333333333333333333333333333555555555555555555555555555588888888888888888888888888880000000000000000000000000000
......................4444444444444444444444333333333333333333333355555555555555555555554444444444444444444444
......................4444444444444444444444444444444444444444444400000000000000000000004444444444444444444444 ......................4444444444444444444444444444444444444444444455555555555555555555551111111111111111111111
...............................3333333333333333333333333333333111111111111111111111111111111155555555555555555555555555555557777777777777777777777777777777 0000000000000000000000000000000000000000000000..............................................1111111111111111111111111111111111111111111111
.........................4444444444444444444444444000000000000000000000000033333333333333333333333337777777777777777777777777
..................................2222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777 0000000000000000000000000000000000000000000000000.................................................3333333333333333333333333333333333333333333333333 .........................4444444444444444444444444111111111111111111111111100000000000000000000000006666666666666666666666666 .........................4444444444444444444444444111111111111111111111111199999999999999999999999993333333333333333333333333
1111111111111111111111111111111111111111111111111.................................................5555555555555555555555555555555555555555555555555 ............................3333333333333333333333333333555555555555555555555555555588888888888888888888888888880000000000000000000000000000
.....................................2222222222222222222222222222222222222333333333333333333333333333333333333366666666666666666666666666666666666661111111111111111111111111111111111111 0000000000000000000000000000000000000000000000..............................................1111111111111111111111111111111111111111111111 ............................3333333333333333333333333333666666666666666666666666666666666666666666666666666666669999999999999999999999999999 ............................3333333333333333333333333333777777777777777777777777777766666666666666666666666666663333333333333333333333333333
...............................3333333333333333333333333333333111111111111111111111111111111155555555555555555555555555555557777777777777777777777777777777
........................................1111111111111111111111111111111111111111666666666666666666666666666666666666666655555555555555555555555555555555555555550000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000.................................................2222222222222222222222222222222222222222222222222 ...............................2222222222222222222222222222222999999999999999999999999999999988888888888888888888888888888882222222222222222222222222222222 ...............................3333333333333333333333333333333000000000000000000000000000000033333333333333333333333333333339999999999999999999999999999999
...........................................1111111111111111111111111111111111111111111000000000000000000000000000000000000000000044444444444444444444444444444444444444444445555555555555555555555555555555555555555555 ..................................2222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777
..................................2222222222222222222222222222222222444444444444444444444444444444444455555555555555555555555555555555550000000000000000000000000000000000 ..................................2222222222222222222222222222222222555555555555555555555555555555555544444444444444444444444444444444440000000000000000000000000000000000
1111111111111111111111111111111111111111111111111 ..............................................0000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555 .....................................2222222222222222222222222222222222222333333333333333333333333333333333333366666666666666666666666666666666666661111111111111111111111111111111111111
.....................................1111111111111111111111111111111111111777777777777777777777777777777777777722222222222222222222222222222222222222222222222222222222222222222222222222 .....................................1111111111111111111111111111111111111888888888888888888888888888888888888811111111111111111111111111111111111113333333333333333333333333333333333333
0000000000000000000000000000000000000000000000000.................................................1111111111111111111111111111111111111111111111111
........................................1111111111111111111111111111111111111111666666666666666666666666666666666666666655555555555555555555555555555555555555550000000000000000000000000000000000000000
........................................1111111111111111111111111111111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000009999999999999999999999999999999999999999 ........................................1111111111111111111111111111111111111111111111111111111111111111111111111111111177777777777777777777777777777777777777774444444444444444444444444444444444444444
...........................................1111111111111111111111111111111111111111111000000000000000000000000000000000000000000044444444444444444444444444444444444444444445555555555555555555555555555555555555555555
0000000000000000000000000000000000000000000000000.................................................5555555555555555555555555555555555555555555555555 0000000000000000000000000000000000000000000000 ...........................................0000000000000000000000000000000000000000000444444444444444444444444444444444444444444466666666666666666666666666666666666666666669999999999999999999999999999999999999999999 0000000000000000000000000000000000000000000000 ...........................................0000000000000000000000000000000000000000000555555555555555555555555555555555555555555522222222222222222222222222222222222222222223333333333333333333333333333333333333333333
0000000000000000000000000000000000000000000000000 ..............................................0000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555
5555555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111118888888888888888888888888888888888888888888888888 5555555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111119999999999999999999999999999999999999999999999999 5555555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111118888888888888888888888888888888888888888888888888 5555555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111119999999999999999999999999999999999999999999999999 5555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111118888888888888888888888888888888888888888888888 5555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111119999999999999999999999999999999999999999999999 5555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111118888888888888888888888888888888888888888888888 5555555555555555555555555555555555555555555555000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111119999999999999999999999999999999999999999999999
Fig.2. Strong lines. Observed (solid lines) and simulated (dashed lines) Stokes I (a), V (b), Q (c) and U (d) line
profiles for Feii λ5018.44˚A line. The vertical bars show the size of the respective observational errors. The observed
and simulated profiles are shifted by 0.15, 0.045, 0.02 and on 0.015 along the vertical axes for Stokes I, V, Q and U
profiles respectively.
each other). The distance of the magnetic dipole center one set of Stokes I profiles are presented in the figure, all
from the center of the star and one-half of the magnetic were taken into account during the calculation of the χ2-
I
dipole size are represented by variables a and a, respec- function (Eq. 3). The best-fit values of the Stokes IVQU
0
tively. The variables (B ,λ ,δ ) and (B ,λ ,δ ) specify χ2-functions for the Feii λ4923.927 line are given in the
p p p n n n
the location of the positive and negative magnetic poles third column of Table 3.
at the stellar surface and the respective strength of the The secondline profileis composedmainly ofthe Feii
magnetic field. λ5018.44˚A line,butisalsocontaminatedbycontributions
from the Tii λ5017.95, Cri λ5018.15 and Crii λ5018.84
lines. It seems that the Crii λ5018.84˚A is responsible for
4.2.1. The strong lines
the blend in the red wing of the Stokes I and V profiles
The Feii λ4923.927, λ5018.44 and λ5169.03 lines are the (Fig. 2). Supposing a uniform Cr distribution, this blend
most prominent Fe lines in the spectrum of 78 Vir, and is well fit for a Cr abundance of logCr/Ntot=-3.38 dex.
showtheclearestStokesQandU signatures.Thefirstpro- Thebestfitχ2 valuesfortheStokesIVQU profilesofFeii
file is formed essentially by the single line Feii λ4923.927 λ5018.44 are given in the fourth column of Table 3.
(seeTable2)andappearstocontainnoimportantblends. In the case of Feii λ5169.033, the profile is blended
The agreement between the observed and simulated data by the weaker Fei λ5168.898 and λ5169.296 lines (see
for Feii λ4923.927 is similar to that shown in Fig. 2 (for Table 2). According to Kochukhov et al. (2004) loggf=-
Fe ii λ5018). Independent Stokes I spectra (each corre- 0.786 given in the GRIFON list for the Fei λ5169.296˚A
sponding toa slightlydifferentphase)wereobtainedwith line is too high to match the solar spectrum, and they
eachoftheStokesV,QandU spectra,andalthoughonly recommend to use a decreased loggf=-2.15. This value
8 V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field
a) I/Ic b) V/Ic c) Q/Ic d) U/Ic
3333333333333333333333333333333333333333333333333.................................................5555555555555555555555555555555555555555555555555
FeII FeII FeII FeII
| 0000000000000000000000000000000000000000000000000.................................................7777777777777777777777777777777777777777777777777 | 0000000000000000000000000000000000000000000000..............................................3333333333333333333333333333333333333333333333 | |
.9740
.9740
.9793 .9851
....8888999955558888
....8888999955558888 0000000000000000000000000000000000000000000000..............................................2222222222222222222222222222222222222222222222
3333333333333333333333333333333333333333333333333 ....9999000044441111 ....9999111133332222
.......8888888444444488888884444444 0000000000000000000000000000000000000000000000000.................................................6666666666666666666666666666666666666666666666666
.......8888888444444488888884444444
.......8888888555555533333337777777 .......8888888555555588888886666666
..........7777777777555555555588888888885555555555
..........7777777777555555555588888888885555555555
.............6666666666666777777777777711111111111118888888888888 ..........7777777777666666666688888888881111111111 ..........7777777777777777777777777777776666666666
0000000000000000000000000000000000000000000000000.................................................5555555555555555555555555555555555555555555555555 .............6666666666666777777777777711111111111118888888888888
2222222222222222222222222222222222222222222222222.................................................5555555555555555555555555555555555555555555555555 ................6666666666666666444444444444444422222222222222228888888888888888 0000000000000000000000000000000000000000000000..............................................2222222222222222222222222222222222222222222222 .............6666666666666777777777777799999999999990000000000000 .............6666666666666888888888888877777777777774444444444444
................6666666666666666444444444444444422222222222222228888888888888888
...................5555555555555555555111111111111111111166666666666666666661111111111111111111 ................6666666666666666555555555555555500000000000000002222222222222222 ................6666666666666666555555555555555599999999999999999999999999999999
0000000000000000000000000000000000000000000000000.................................................4444444444444444444444444444444444444444444444444 ...................5555555555555555555111111111111111111166666666666666666661111111111111111111
......................4444444444444444444444333333333333333333333355555555555555555555554444444444444444444444 ...................5555555555555555555222222222222222222255555555555555555550000000000000000000 ...................5555555555555555555333333333333333333333333333333333333339999999999999999999
......................4444444444444444444444333333333333333333333355555555555555555555554444444444444444444444
.........................4444444444444444444444444000000000000000000000000033333333333333333333333337777777777777777777777777
2222222222222222222222222222222222222222222222222 ......................4444444444444444444444444444444444444444444400000000000000000000004444444444444444444444 ......................4444444444444444444444444444444444444444444455555555555555555555551111111111111111111111
0000000000000000000000000000000000000000000000..............................................1111111111111111111111111111111111111111111111
.........................4444444444444444444444444000000000000000000000000033333333333333333333333337777777777777777777777777
............................3333333333333333333333333333555555555555555555555555555588888888888888888888888888880000000000000000000000000000 0000000000000000000000000000000000000000000000000.................................................3333333333333333333333333333333333333333333333333
.........................4444444444444444444444444111111111111111111111111100000000000000000000000006666666666666666666666666 .........................4444444444444444444444444111111111111111111111111199999999999999999999999993333333333333333333333333
...............................3333333333333333333333333333333111111111111111111111111111111155555555555555555555555555555557777777777777777777777777777777 ............................3333333333333333333333333333555555555555555555555555555588888888888888888888888888880000000000000000000000000000
0000000000000000000000000000000000000000000000..............................................1111111111111111111111111111111111111111111111 ............................3333333333333333333333333333666666666666666666666666666666666666666666666666666666669999999999999999999999999999 ............................3333333333333333333333333333777777777777777777777777777766666666666666666666666666663333333333333333333333333333
..................................2222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777 ...............................3333333333333333333333333333333111111111111111111111111111111155555555555555555555555555555557777777777777777777777777777777
1111111111111111111111111111111111111111111111111.................................................5555555555555555555555555555555555555555555555555 0000000000000000000000000000000000000000000000000.................................................2222222222222222222222222222222222222222222222222 ...............................2222222222222222222222222222222999999999999999999999999999999988888888888888888888888888888882222222222222222222222222222222 ...............................3333333333333333333333333333333000000000000000000000000000000033333333333333333333333333333339999999999999999999999999999999
.....................................2222222222222222222222222222222222222333333333333333333333333333333333333366666666666666666666666666666666666661111111111111111111111111111111111111 ..................................2222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777
..................................2222222222222222222222222222222222444444444444444444444444444444444455555555555555555555555555555555550000000000000000000000000000000000 ..................................2222222222222222222222222222222222555555555555555555555555555555555544444444444444444444444444444444440000000000000000000000000000000000
........................................1111111111111111111111111111111111111111666666666666666666666666666666666666666655555555555555555555555555555555555555550000000000000000000000000000000000000000 .....................................2222222222222222222222222222222222222333333333333333333333333333333333333366666666666666666666666666666666666661111111111111111111111111111111111111
...........................................1111111111111111111111111111111111111111111000000000000000000000000000000000000000000044444444444444444444444444444444444444444445555555555555555555555555555555555555555555 0000000000000000000000000000000000000000000000000.................................................1111111111111111111111111111111111111111111111111 .....................................1111111111111111111111111111111111111777777777777777777777777777777777777722222222222222222222222222222222222222222222222222222222222222222222222222 .....................................1111111111111111111111111111111111111888888888888888888888888888888888888811111111111111111111111111111111111113333333333333333333333333333333333333
........................................1111111111111111111111111111111111111111666666666666666666666666666666666666666655555555555555555555555555555555555555550000000000000000000000000000000000000000
1111111111111111111111111111111111111111111111111 ..............................................0000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555 ........................................1111111111111111111111111111111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000009999999999999999999999999999999999999999 ........................................1111111111111111111111111111111111111111111111111111111111111111111111111111111177777777777777777777777777777777777777774444444444444444444444444444444444444444
...........................................1111111111111111111111111111111111111111111000000000000000000000000000000000000000000044444444444444444444444444444444444444444445555555555555555555555555555555555555555555
0000000000000000000000000000000000000000000000 ...........................................0000000000000000000000000000000000000000000444444444444444444444444444444444444444444466666666666666666666666666666666666666666669999999999999999999999999999999999999999999 0000000000000000000000000000000000000000000000 ...........................................0000000000000000000000000000000000000000000555555555555555555555555555555555555555555522222222222222222222222222222222222222222223333333333333333333333333333333333333333333
0000000000000000000000000000000000000000000000000 ..............................................0000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555
6666666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666666 6666666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666666.................................................5555555555555555555555555555555555555555555555555 6666666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666666 6666666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666666.................................................5555555555555555555555555555555555555555555555555 6666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666 6666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666..............................................5555555555555555555555555555555555555555555555 6666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666 6666666666666666666666666666666666666666666666555555555555555555555555555555555555555555555511111111111111111111111111111111111111111111116666666666666666666666666666666666666666666666..............................................5555555555555555555555555555555555555555555555
Fig.4. The same as at the Fig. 2, but for Feii λ6516.08˚A.
stellarrotationalphaseforthisparticularline isverysim-
111111111 ilar to that shown in Fig. 2 for Feii λ5018.44. The sixth
columnofTable3presentsthebestfitparametersandthe
000000000.........555555555
sssssssss model characteristics for Feii λ5169.033.
m/m/m/m/m/m/m/m/m/
∆∆∆∆∆∆∆∆∆Vr, kVr, kVr, kVr, kVr, kVr, kVr, kVr, kVr, k 000000000 StokTehseI caonndcoVrdparnocfieleovfartiahteionosbsaesrvaedfunacntidonsoimf ustlaeltleadr
---------000000000.........555555555 rotation is extremely good for these lines. On the other
hand, the fit to the Stokes Q and U profiles is only ap-
---------111111111
---------000000000.........222222222555555555 000000000 000000000.........222222222555555555 000000000.........555555555 000000000.........777777777555555555 111111111 111111111.........222222222555555555 proximate.Thesimulatedprofilesshowgenerallythesame
PPPPPPPPPhhhhhhhhhaaaaaaaaassssssssseeeeeeeee intensity, qualitative structure and variability as the ob-
Fig.3. The difference in radial velocity between the servations.However,evenwiththeserelativelynoisydata,
observed and calculated Stokes I profiles for the Feii it is clear that the model does not reproducethe observa-
lines λ4923.927˚A (crosses),λ5018.44˚A (open circles)and tions within the errors at some phases (for example, see
λ5169.033˚A (open squares). The mean radial velocities phases 0.1174,0.5339 and 0.9851 in Fig. 2).
V =-7.82 km s−1, -7.51 km s−1, -7.23 km s−1 obtained In order to verify the assumption of a uniform surface
r
from the simulation of Feii λ4923.927˚A, λ5018.44˚A, distribution of iron in the atmosphere of 78 Vir, we have
λ5169.033˚A lines respectively (see Table 3), have been calculatedthedifferenceinradialvelocitybetweentheob-
taken into account. served and simulated Stokes I profiles for the strong Feii
lines λ4923.927,λ5018.44 and λ5169.033˚A (see Fig. 3).
The mean(averagedforallthe availableobservational
is employed here for the respective profile simulation and phases) radial velocities obtained from the simulation are
provides much better agreement with the observed data. taken into account in the calculation of the velocities of
The behaviour of the best fit Stokes IVQU profiles with therespectivesimulatedprofiles.Fig.3showsthatthede-
V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field 9
rived differences in Vr almost coincide for all three lines. a) I/Ic b) V/Ic
A moderate disagreementmay exist only for the data ob-
tainedinthe vicinityofrotationalphaseϕ=0.5,whenthe 0000000000000000000000000000000000000000000000000000....................................................8888888888888888888888888888888888888888888888888888
negative magnetic pole is most visible. The variation of 3333333333333333333333333333333333333333333333333333....................................................5555555555555555555555555555555555555555555555555555 FeII FeII FeII FeII
radial velocity, which shows a reasonably coherent varia- | | .9740 | |
.9740
tion with phase from about -1 to +1 km/s, may reflect a ....8888999955558888 0000000000000000000000000000000000000000000000000000....................................................7777777777777777777777777777777777777777777777777777
mildly non-uniform distribution of Fe, unmodelled struc- ....8888999955558888
.......8888888444444488888884444444
ture in the magnetic field, or other unaccounted-forphys- 3333333333333333333333333333333333333333333333333333
..........7777777777555555555588888888885555555555 .......8888888444444488888884444444
ical processes in the stellar atmosphere.
0000000000000000000000000000000000000000000000000000....................................................6666666666666666666666666666666666666666666666666666
.............6666666666666777777777777711111111111118888888888888 ..........7777777777555555555588888888885555555555
4.2.2. The moderate strength lines ................6666666666666666444444444444444422222222222222228888888888888888 .............6666666666666777777777777711111111111118888888888888
2222222222222222222222222222222222222222222222222222....................................................5555555555555555555555555555555555555555555555555555 ...................5555555555555555555777777777777777777755555555555555555557777777777777777777
From the list of the Feii lines selected for analysis 0000000000000000000000000000000000000000000000000000....................................................5555555555555555555555555555555555555555555555555555 ................6666666666666666444444444444444422222222222222228888888888888888
......................5555555555555555555555111111111111111111111166666666666666666666661111111111111111111111
(see Table 2) the λ4620.52, λ5197.58, λ6432.68 and ...................5555555555555555555777777777777777777755555555555555555557777777777777777777
λ6516.08 lines are weaker than the λ4923.927, λ5018.44 .........................4444444444444444444444444333333333333333333333333355555555555555555555555554444444444444444444444444
and λ5169.03 lines. Due to the weaker polarised signal, a 0000000000000000000000000000000000000000000000000000....................................................4444444444444444444444444444444444444444444444444444 ......................5555555555555555555555111111111111111111111166666666666666666666661111111111111111111111
............................4444444444444444444444444444000000000000000000000000000033333333333333333333333333337777777777777777777777777777
2222222222222222222222222222222222222222222222222222
few spectra with comparatively large observationalerrors .........................4444444444444444444444444333333333333333333333333355555555555555555555555554444444444444444444444444
...............................3333333333333333333333333333333555555555555555555555555555555588888888888888888888888888888880000000000000000000000000000000
have been excluded from the simulation (phases 0.5757,
0.5847 and 0.5914). ..................................3333333333333333333333333333333333111111111111111111111111111111111155555555555555555555555555555555557777777777777777777777777777777777 0000000000000000000000000000000000000000000000000000....................................................3333333333333333333333333333333333333333333333333333 ............................4444444444444444444444444444000000000000000000000000000033333333333333333333333333337777777777777777777777777777
ThelineFeiiλ4620.521isprimarilyresponsibleforthe .....................................2222222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777777 ...............................3333333333333333333333333333333555555555555555555555555555555588888888888888888888888888888880000000000000000000000000000000
formation of the observed profile. No blends were taken 1111111111111111111111111111111111111111111111111111....................................................5555555555555555555555555555555555555555555555555555
into account during the simulation in this case. The Feii ........................................2222222222222222222222222222222222222222333333333333333333333333333333333333333366666666666666666666666666666666666666661111111111111111111111111111111111111111 ..................................3333333333333333333333333333333333111111111111111111111111111111111155555555555555555555555555555555557777777777777777777777777777777777
0000000000000000000000000000000000000000000000000000....................................................2222222222222222222222222222222222222222222222222222
λ5197.577line is blended by the weak Feii λ5197.48line, ...........................................1111111111111111111111111111111111111111111666666666666666666666666666666666666666666655555555555555555555555555555555555555555550000000000000000000000000000000000000000000 .....................................2222222222222222222222222222222222222888888888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777777
butprovidesthemaincontributiontotheobservedprofile. ..............................................1111111111111111111111111111111111111111111111000000000000000000000000000000000000000000000044444444444444444444444444444444444444444444445555555555555555555555555555555555555555555555
........................................2222222222222222222222222222222222222222333333333333333333333333333333333333333366666666666666666666666666666666666666661111111111111111111111111111111111111111
The behaviour of the best fit Stokes IVQU profiles with 1111111111111111111111111111111111111111111111111111 .................................................0000000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555555
stellar rotational phase for these lines is very similar to 0000000000000000000000000000000000000000000000000000....................................................1111111111111111111111111111111111111111111111111111 ...........................................1111111111111111111111111111111111111111111666666666666666666666666666666666666666666655555555555555555555555555555555555555555550000000000000000000000000000000000000000000
that shown in Fig. 4 for Feii λ4620.521˚A. The first and
..............................................1111111111111111111111111111111111111111111111000000000000000000000000000000000000000000000044444444444444444444444444444444444444444444445555555555555555555555555555555555555555555555
seventhcolumnsofTable3presentthebestfitparameters
and the model characteristics for Feii lines λ4620.521˚A 0000000000000000000000000000000000000000000000000000 .................................................0000000000000000000000000000000000000000000000000444444444444444444444444444444444444444444444444411111111111111111111111111111111111111111111111115555555555555555555555555555555555555555555555555
and λ5197.577˚A,respectively. 0000000000000000000000000000000000000000000000000000....................................................5555555555555555555555555555555555555555555555555555
The Feiiλ6432.68˚A line is alsoessentiallyunblended. 5555555555555555555555555555555555555555555555555555111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000....................................................5555555555555555555555555555555555555555555555555555 5555555555555555555555555555555555555555555555555555111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000001111111111111111111111111111111111111111111111111111 5555555555555555555555555555555555555555555555555555111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000....................................................5555555555555555555555555555555555555555555555555555 5555555555555555555555555555555555555555555555555555111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000001111111111111111111111111111111111111111111111111111
The agreement between the observed and simulated
Fig.5.Observed(solidlines)andsimulated(dashedlines)
Stokes IVQU data for this particular line is very simi-
lar to that presented by Fig. 4. The Feii λ6516.08˚A line Stokes I (a) and V (b) profiles for Feii λ5100.607˚A,
λ5100.664˚A,λ5100.727˚A and λ5100.852˚A lines. The ver-
is the primary contributor to the formation of the cor-
tical bars show the size of respective observationalerrors.
responding observed Stokes IVQU profiles. The agree-
The observed and simulated profiles are shifted on 0.15,
ment of the best-fit simulation with the observed data
forFeiiλ6516.08˚A line is shownatFig.4.The tenth and and on 0.045 along the vertical axes for Stokes I and V
profiles correspondingly.
eleventh columns at the Tables 3 contain the best fit pa-
rametersandmodelcharacteristicsfortheFeiiλ6432.68˚A
andλ6516.08˚A lines,respectively.Thebestfitofthesim-
observedandsimulateddataforthisparticularlineissim-
ulatedStokesIVQU profilesfortheFeiiλ6516.08˚A lineis ilarto thatshowninFig.5.The superpositionofthe Feii
statisticallybetterthanthatobtainedforFeiiλ6432.68˚A.
λ5100.607, λ5100.664, λ5100.727 and λ5100.852 lines is
responsible for the formation of Stokes I and V profiles
4.2.3. Separate analysis of the Stokes I and V spectra at ∼ 5100.7 ˚A (see Table 2). Fig. 5 shows the simulated
Stokes I and V profiles, which fit well the observed pro-
The other Feii lines λ4635.316˚A, the λ5100˚A-group, files (see also the respective χ2 value in the fifth column
λ5362.87˚A and λ6247.56˚A show no variability in the of Table 3).
Stokes Q and U spectra and hence only the Stokes I and In the region of the Feii λ5362.87 line the observed
V spectra are taken into account during the simulation. Stokes I and V profiles are formed mainly by this line
These lines are selected for the analysis in order to check and by a weak contribution from the Feii λ5362.74,
theresultingmagneticfieldstructureof78Vironthebasis λ5362.98,Criλ5362.87andCriiλ5363.88lines.TheFeii
of a more thorough line list. λ6247.557 line also provides the main contribution to the
The Feii λ4635.316 line is the main contributor to observed Stokes I and V profiles and is blended by the
the formationofthe correspondingobservedStokesI and Feii λ6247.35 line. The best fit simulated data show al-
V profiles, although it is blended by the comparatively mostthesamefitqualityasisshowninFig.5.Simulation
weak Fei λ4635.846˚A line. The agreement between the of the Stokes I and V profiles in the regions of these
10 V.R. Khalack and G.A. Wade:Recovery of the78 Vir global magnetic field
lines results in almostthe same configurationof the mag- 000000000.........111111111
netic field structure as obtained from the analysis of the
stronger Feii lines, for which all four Stokes parameters 000000000.........000000000555555555
were taken into account during the simulation.
%)%)%)%)%)%)%)%)%)
Q/I (Q/I (Q/I (Q/I (Q/I (Q/I (Q/I (Q/I (Q/I (ccccccccc 000000000
000000000000000 ---------000000000.........000000000555555555
---------000000000.........111111111
---------000000000.........222222222555555555 000000000 000000000.........222222222555555555 000000000.........555555555 000000000.........777777777555555555 111111111 111111111.........222222222555555555
---------------555555555555555000000000000000000000000000000
PPPPPPPPPhhhhhhhhhaaaaaaaaassssssssseeeeeeeee
Bl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, GBl, G 000000000.........111111111
---------------111111111111111000000000000000000000000000000000000000000000 000000000.........000000000555555555
%)%)%)%)%)%)%)%)%)
U/I (U/I (U/I (U/I (U/I (U/I (U/I (U/I (U/I (ccccccccc 000000000
---------------111111111111111555555555555555000000000000000000000000000000
---------------000000000000000...............222222222222222555555555555555 000000000000000 000000000000000...............222222222222222555555555555555 000000000000000...............555555555555555 000000000000000...............777777777777777555555555555555 111111111111111 111111111111111...............222222222222222555555555555555
PPPPPPPPPPPPPPPhhhhhhhhhhhhhhhaaaaaaaaaaaaaaassssssssssssssseeeeeeeeeeeeeee ---------000000000.........000000000555555555
Fig.6. Longitudinal magnetic field data with phase ob-
---------000000000.........111111111
tained from analysis of Feii λ6432.68 (solid line), Feii ---------000000000.........222222222555555555 000000000 000000000.........222222222555555555 000000000.........555555555 000000000.........777777777555555555 111111111 111111111.........222222222555555555
λ5018.44 (dashed line), Feii λ6516.08 (dotted line) and PPPPPPPPPhhhhhhhhhaaaaaaaaassssssssseeeeeeeee
Feii λ5197.58 (dash-dotted line). The open squares with Fig.7. Comparison of scaled net linear polarization data
1σ error bars correspond to the LSD data published by obtained from the best fit simulations of Feii λ4923.927
Wade et al. (2000b). (solid line) and Fe ii λ6516 (dashed line), and from LSD
profiles(Wadeetal.2000b)(opensquares)for78Virwith
respective 1σ error bars.
4.3. Integral magnetic field characteristics
field configuration is capable of reproducing them. Using
In order to check the agreement of the derived magnetic the same stellar atmosphere model, the longitudinal field
fieldmodelwithotheravailablemagneticfielddatafor78 calculated from the model fits to the strongest Feii line
Vir, we calculate the intensity-weighted, averaged (over λ5018.44˚A and to the weak Feii line λ6432.68˚A show a
the visible stellar disk)longitudinalmagneticfield Bl and good agreement with the LSD Bl data (see Fig. 6). The
the normalized equivalent widths of the Stokes Q and U longitudinal fields calculated from the simulation results
profilesforalltheanalysedphases.Thenormalizationpro- for the other Feii lines appear to be shifted downward in
cedure is performed in accordance with the method de- longitudinalfieldintensityrelativetotheLSDB variation
l
scribed by Wade et al. (2000b) over the passband of the (see Fig. 6).
analysed line profiles. The calculated Stokes Q and U equivalent widths for
All the available longitudinal magnetic field mea- all the analysed line profiles are multiplied by the respec-
surements for 78 Vir are plotted in Fig. 6 of tive line scaling factor in order to fit them to the LSD
Leone & Catanzaro (2001). This figure shows the Stokes Q and U data. For each analysed line a scaling
good agreement of the Bl measurements obtained factor is determined with the help of the Least Squares
by Wade et al. (2000b) using the Least-Squares method. The respective results are given in Table 2,
Deconvolution (LSD) technique (Donati et al. 1997) with taking into account that the LSD equivalent widths of
themajorityofotherobservationaldata.Therefore,inthis Stokes Q and U profiles have a line scaling factor of 0.1
paper we just compare our results with the LSD longitu- (Wadeetal.2000b)fromcomparisonwiththeBBLPdata
dinal field data. (Leroy 1995). Fig. 7 presents the Stokes Q and U equiv-
As demonstrated by Wade et al. (2000b), the 78 Vir alent widths variability with phase,derivedfrom the best
Stokes Q and U LSD equivalent widths are proportional fitsimulationsofFeiiλ4923.927(withΩ=109◦)andFeii
to the BBLPmeasuredatthat phase(Leroy 1995)with a λ6516.08 (with Ω=128◦). They are scaled by a factor of
line scaling factor. This factor might be different for the 0.08 (for λ4923) and 0.03 (for λ6516) and plotted over
analysedlines (seethe lastcolumninTable 2),butisthe the Stokes Q and U equivalent widths derived from LSD
same for the both Stokes Q and U equivalent widths for profiles. It is remarkable that the simulated polarimetric
a particular spectral line. data for Feii λ4923.927˚A and λ5018.44˚A lines have al-
Neither the B measurements nor the BBLP data are most the same scaling factor as the LSD Stokes Q and
l
included directly into the model minimization procedure. U equivalent widths. Theoretically, the line scaling fac-
However,giventhatthemodelsuccessfullyreproducesthe tor depends onthe Land´efactorandonthe degreeof line
Stokes profiles with which these quantities are fundamen- saturation(Wadeetal.2000a).Table2showsthatingen-
tally related, we should expect that the model magnetic eral the lines with a high mean Land´e factor have also a