Astron. Astrophys. 358, 929–942 (2000) ASTRONOMYAND

ASTROPHYSICS

Modelling of magnetic fields of CP stars

III. The combined interpretation of five different magnetic observables:theory, and application to β Coronae Borealis

S. Bagnulo1, M. Landolfi 2, G. Mathys3, and M. Landi Degl’Innocenti4

1 Universiẗat Wien, Institut f̈ur Astronomie, T̈urkenschanzstrasse 17, 1180 Wien, Austria (bagnulo@astro.univie.ac.at)2 Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy (landolfi@arcetri.astro.it)3 European Southern Observatory, Casilla 19001, Santiago 19, Chile (gmathys@eso.org)4 C.N.R., Gruppo Nazionale di Astronomia, Unità di Ricerca di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy (mlandi@arcetri.astro.it)

Received 6 October 1999 / Accepted 2 February 2000

Abstract. The most common techniques for detecting mag-netic fields of chemically peculiar (CP) stars are based on thedetermination of some quantities related to the magnetic mor-phology, which are obtained through the analysis of StokesI andV profiles, and on observations of frequency integrated StokesQ andU . In previous papers of this series we set up a formal-ism aimed at describing the magnetic field of CP stars in termsof a superposition of a dipole and a quadrupole field, and weproposed a method to recover the magnetic morphology fromthe combined analysis of those observable quantities which cansimply be expressed by means of analytical formulas, namely:the mean longitudinal field, the crossover, and the mean squarefield. Here we extend this method in order to include thosequantities for which numerical integrations are required, i.e.,the mean field modulus and the broadband linear polarisation.Estimates of stellar radius and projected equatorial velocity arealso explicitly taken into account. We present an application tothe well-known CP starβ CrB, the only star for which all kindsof magnetic data are available. We find thatβ CrB, like mostCP stars, exhibits a complex magnetic structure, which may notbe fully accounted for even by a second-order multipolar ex-pansion. However, we are able to suggest two models which arecompatible with all the magnetic data.

Key words: stars: magnetic fields – polarization – stars: chem-ically peculiar – stars: individual:β CrB

1. Introduction

Spectra of chemically peculiar (CP) stars of the upper main se-quence exhibit a wealth of features uncommon to those of themajority of the objects located in the same region of the H-R diagram. Spectral lines of certain elements appear stronger,or weaker, than those of most of the stars of similar spectraltype. In many cases, the spectral lines are polarised, and bothshape and strength of the observed Stokes profiles change peri-

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odically with time, with the same period as the light variations.These observations suggest that the photospheres of CP stars arecharacterised by non-cosmic element abundances, and that theregions of line formation are often permeated by a large-scaleorganised magnetic field (with a typical strength of few hundredto few ten thousand G). A magnetic morphology not symmetricaround the rotation axis accounts for the periodic variation ofthe observed spectra, although, in some cases, abundance inho-mogeneities must also be invoked to explain dramatic changesof the line strengths.

Abundance anomalies and presence of a magnetic field arecorrelated phenomena. Diffusing elements are guided by themagnetic field, some of them (like rare earths) may be concen-trated where the field lines are horizontal, some others (e.g.,Sr) where the field lines are vertical (Michaud et al. 1981). Thesituation is actually extremely complex, and predictions for dif-fusion of the various elements vary from star to star. It is clearthat obtaining realistic models for the magnetic morphologiescan provide very useful constraints to the diffusion theory, andcan also help to understand the origin of the magnetic field inCP stars.

Although the signature of the Zeeman effect is quite clear inthe observed spectra, it proves unpractical to try to recover themagnetic configuration via computation of straight fits to theobserved spectra. The limited spectral resolution and signal-to-noise ratios that can be achieved in spectropolarimetric obser-vations generally hamper the plain application of an inversionprocedure, which they tend to render unstable. During the lastyears, however, a number of techniques have been introduced formeasuring certain quantities related to the magnetic field, basedon the combined use of the information contained in severalspectral lines. Such quantities are usually determined with bet-ter accuracy, so that, at the moment, they look more adequate asdiagnostic tools for recovering the magnetic configuration. Thisapproach also has the advantage of being relatively independentof the various stellar parameters such as effective temperature,gravity and element abundances, and the inversion proceduredoes not require to perform time consuming spectral synthesis.

930 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

So far, four such quantities have been determined via theanalysis of StokesI andV line profiles, namely,i) the meanlongitudinal field (e.g., Babcock 1947; Mathys 1991),ii ) thecrossover (Mathys 1995a),iii ) the mean square field (Mathys1995b), andiv) the mean field modulus (Babcock 1960; Mathyset al. 1997). Magnetic signatures in StokesQ andU are consid-erably smaller than in StokesI andV , so that noise limitationsare even more penalising for the former two parameters than forthe latter. As a result, the first unquestioned measurements oflinear polarisation in spectral lines have been obtained in the lastcouple of years only, thanks to the combination of progress in theachievable quality of the observations and of new data analysismethods such as least-squares deconvolution (Wade et al. 1998)or the moment technique (Mathys 1999). Yet, such determina-tions are still too scarce to be useful for modelling purposes.By contrast, constraints to the magnetic field can be obtainedvia observations of broadband linear polarisation (BBLP) (e.g.,Landi Degl’Innocenti et al. 1981; Landolfi et al. 1993a; Leroy1995a).

The most favourable situation where all the magnetic quan-tities are well observed for one star is rarely encountered in realcases. This is partly due to intrinsic limits of the different ob-servational techniques. Observations of magnetically split lines(from which one can derive the mean field modulus) are onlypossible for stars with small values of the projected equato-rial velocity. On the other hand, for such stars observations ofcrossover (which is proportional tove sin i) are unlikely to besignificantly different from zero. Finally, observations of BBLPare only possible if the star’s intrinsic signal is not washed out byinterstellar polarisation: this limits the observations of BBLP tostars typically nearer than 100–200 pc. However, observationsof all the five quantities mentioned above are available for onemagnetic CP star (β CrB), and observations of three or fourquantities are available for several others. Obviously, a reliablemagnetic model should be able to account forall the differentobservables.

So far, most of the modelling was confined to one or twoobservable quantities, e.g., BBLP alone (Landolfi et al. 1993b),mean longitudinal field and mean field modulus (Hensberge etal. 1979; Wade et al. 1997), mean longitudinal field and BBLP(Bagnulo et al. 1995). More recently, Landolfi et al. (1998, here-after referred to as Paper I) and Bagnulo et al. (1999a) tried togive a combined interpretation of three observational quantities(mean longitudinal field, crossover, and mean square field). Inthis paper we further generalise the technique proposed in Pa-per I in order to include the two remaining observables, BBLPand mean field modulus. Though straightforward in principle,such generalisation involves some technical difficulties that willbe discussed in the following.

It has been known for a long time (e.g., Landstreet 1992)that the simple oblique rotator model with centred dipole fieldcannot explain the observations of magnetic CP stars in detail.The most direct extension of that model is the inclusion of a(non-linear) quadrupole contribution, which can be regarded asthe first correction in the expansion of an arbitrary magneticfield into multipoles. The dipole plus quadrupole model has

already proved successful in several cases (Bagnulo et al. 1999a;Bagnulo & Landolfi 1999), so it has again been adopted in thispaper.

As an application of the technique proposed in this paper,we present the results of the modelling of the magnetic CPstarβ CrB. For this star, two distinct dipole plus quadrupolemodels are found to be compatible with all the observationaldata. We present the corresponding magnetic maps, and discusstheir reliability.

The paper is organised as follows. In Sect. 2 we give theexplicit expressions for the BBLP and the mean field modulus.In Sect. 3 we outline the inversion procedure and we discusswhich combinations of observable quantities are in principlesufficient to recover the magnetic configuration. In Sect. 4 wepresent the analysis ofβ CrB.

2. Expressions of the broadband linear polarisationand of the mean field modulus for the dipoleplus quadrupole magnetic configuration

We assume that the magnetic configuration on the stellar surfaceis produced by the superposition of a dipole and a (non-linear)quadrupole located at the star’s centre. The model is illustrated inPaper I; here we just recall the meaning of the different symbols,referring the reader to that paper for a detailed description ofthe magnetic geometry.

In the observer’s reference frame(x, y, z) – see Fig. 1 ofPaper I – the direction of the star’s rotation axis is specified bytwo angles (inclination and azimuth) which are denoted byi andΘ respectively. The dipole is characterised by the amplitudeBdand by the unit vectoru ≡ (β, f − f0), whereβ is the anglebetweenu and the rotation axis andf is the rotational phase.Similarly, the quadrupole is characterised by the amplitudeBqand by the unit vectorsu1 ≡ (β1, γ1) andu2 ≡ (β2, γ2). Foran assigned magnetic configuration and an assigned value of therotational phase, the magnetic fieldB at the pointr ≡ (R∗, θ, χ)of the stellar surface is given by the vector sumB = Bd + Bq,whereBd andBq are the dipole and quadrupole fields given byEqs. (1) and (2), respectively, of Paper I. In the frame(x, y, z)the direction ofB(r) is specified by the anglesψ andφ (seeFig. 1), which can easily be expressed in terms of the anglesi,β, f − f0, β1, β2, γ1, γ2, Θ, θ, χ (see below).

In Paper I we have introduced the quantitiesF (1),F (2),F (3)

representing the mean longitudinal field, the crossover, and themean square field respectively

F (1) = 〈Bz〉F (2) = ve sin i 〈dBz〉F (3) = 〈B2 + B2z〉1/2 ,

(1)

whereve is the star’s equatorial velocity. They are functions ofthe rotational phasef and depend on the parameters specifyingthe magnetic configuration: their expressions for a dipole plusquadrupole field are given by Eqs. (4)-(7) of Paper I.

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 931

Fig. 1. In the observer’s reference frame(x, y, z) the magnetic field vectorB at thepointr ≡ (R∗, θ, χ) of the stellar surface isspecified by the inclination angleψ and theazimuth angleφ

Similarly, we now define the functions

F (4) = PQ + P intQ

F (5) = PU + P intU

F (6) = 〈B〉 ,(2)

wherePQ,PU are the disk averages of the Stokes parametersQ,U integrated in frequency across a given interval∆ν and nor-malised to the intensity flux in the same interval,P intQ , P

intU are

the contributions to these quantities due to interstellar polarisa-tion, and〈B〉 is the mean field modulus. We want to establish theexpressions ofF (4), F (5), F (6) for the dipole plus quadrupolemagnetic configuration.

Consider first the quantitiesPQ, PU (i.e., the BBLP). Wehave by definition

PQ =

2π∫

0dχ

π/2∫

0dθ sin θ cos θ

∫

∆νdν Qν(θ,χ)

2π∫

0dχ

π/2∫

0dθ sin θ cos θ

∫

∆νdν Iν(θ,χ)

PU =

2π∫

0dχ

π/2∫

0dθ sin θ cos θ

∫

∆νdν Uν(θ,χ)

2π∫

0dχ

π/2∫

0dθ sin θ cos θ

∫

∆νdν Iν(θ,χ)

,

(3)

whereIν , Qν , Uν are the Stokes parameters (hereafter definedas in Shurcliff 1962) of the radiation of frequencyν comingfrom the pointr.

Our basic assumption is that the BBLP actually observedin several CP stars (see Leroy 1995a for a comprehensive pre-sentation of measurements) is due to the differential saturationmechanism. According to this mechanism – first introduced byLeroy (1962) to interpret the BBLP observed in sunspots – thepolarisation results from the cumulative effect of all the magnet-ically sensitive lines contained in the observed spectral interval.The contribution of each single line arises from the combinationof two effects: the splitting (due to the presence of a magnetic

field) and the saturation (due to the transfer of radiation) which isgenerally different forσ andπ Zeeman components. Since botheffects are obviously present in the atmosphere of a magneticCP star, it looks quite natural to ascribe the observed BBLP tosuch a mechanism. This approach has been followed in severalworks (e.g., Landi Degl’Innocenti et al. 1981; Landolfi et al.1993a,b; Bagnulo et al. 1995).

Then we assume that the element abundances on the stellarsurface are constant, and thatN identical, well-separated (un-blended) lines are contained in the interval∆ν: in other words,we assume that the observed BBLP is produced by an ‘aver-age’ line, whose properties are the same throughout the wholesurface.

The above assumptions allow us to replace in Eqs. (3)∫∆ν

dν Qν(θ, χ) → N∫

linedν Qν(θ, χ)∫

∆νdν Uν(θ, χ) → N

∫line

dν Uν(θ, χ)∫∆ν

dν Iν(θ, χ) →∫

∆νdν Ic(θ, χ)

−N ∫line

dν[Ic(θ, χ) − Iν(θ, χ)

],

(4)

whereIc is the continuum intensity andline means that theintegrals extend over the profile of the ‘average’ line.

Finally, we assume that the atmosphere is described by theMilne-Eddington model, which implies that the Stokes parame-ters of the emitted radiation are given by the well-known Unno-Rachkovsky formulas. However, in accordance with Paper I, wereplace the limb-darkening law incos θ with the more realisticlaw (1 − u+ u cos θ). Thus we use the expressionsIν(θ, χ) = B0 + (1 − u+ u cos θ)B1 i(θ, χ)Qν(θ, χ) = − (1 − u+ u cos θ)B1 q(θ, χ)Uν(θ, χ) = − (1 − u+ u cos θ)B1 u(θ, χ) ,

(5)

932 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

whereB0 andB1 characterize the dependence of the PlanckfunctionBP on optical depth (BP(τ) = B0 +B1τ ), and where

i(θ, χ) = ∆−1(1 + ηI)[(1 + ηI)2 + ρ2Q + ρ

2U + ρ

2V

]q(θ, χ) = ∆−1

[(1 + ηI)2ηQ + (1 + ηI)(ηV ρU − ηUρV )+ ρQ(ηQρQ + ηUρU + ηV ρV )

]u(θ, χ) = ∆−1

[(1 + ηI)2ηU + (1 + ηI)(ηQρV − ηV ρQ)+ ρU (ηQρQ + ηUρU + ηV ρV )

],

(6)

with

∆ = (1 + ηI)2

×[(1 + ηI)2 − η2Q − η2U − η2V + ρ2Q + ρ2U + ρ2V ]−(ηQρQ + ηUρU + ηV ρV )2 .

The quantitiesηI,Q,U,V andρQ,U,V have their usual meaning(see e.g. Landolfi & Landi Degl’Innocenti 1982). Since theydepend onν via the reduced frequency

v = (ν0 − ν)/∆νD , (7)whereν0 is the line centre frequency and∆νD the Dopplerwidth, it is convenient to change the integration variable inEqs. (4) fromν to v. From Eqs. (2)-(5) we finally obtain

F (4) = −α

2π∫

0dχ

π/2∫

0dθ c(θ;u)

∫

linedv q(θ,χ)

1−α2π∫

0dχ

π/2∫

0dθ c(θ;u)

∫

linedv

[1−i(θ,χ)

] + P intQ

F (5) = −α

2π∫

0dχ

π/2∫

0dθ c(θ;u)

∫

linedv u(θ,χ)

1−α2π∫

0dχ

π/2∫

0dθ c(θ;u)

∫

linedv

[1−i(θ,χ)

] + P intU ,(8)

where

c(θ;u) = sin θ cos θ (1 − u+ u cos θ) (9)and

α =N ∆νD βP

π∆ν[1 + 3−u3 βP

] , (10)with βP = B1/B0.

In several previous papers dealing with the interpretationof BBLP observations in CP stars, the so-called weak field ap-proximation has been used. This approximation entails a majorsimplification, because it allows one to evaluate analytically thedisk integrals appearing in Eqs. (8). Such calculations have beencarried out in Landolfi et al. (1993a) for the dipole field (and aspecial type of dipole plus quadrupole, the quadrupole being lin-ear and aligned with the dipole), and in Bagnulo et al. (1996) foran arbitrary multipole field. In the former paper, and in Bagnuloet al. (1995), it was shown that the weak field approximationdoes not introduce large errors for the magnetic configurationsconsidered, up to field strengths of about 20 kG. At the sametime, it was suggested that this property (which is not obvious,since the weak field approximation is verified, in metallic lines,for a strength up to' 1 kG) is most likely related to the factthat, for those configurations, the variation of the field modulusacross the stellar surface is small. This isnot the case, however,

for the superposition of a dipole and a non-linear quadrupole.In fact we found, by comparing the polarisation predicted byEqs. (8) and by the analytical expressions given in Bagnulo etal. (1996), that the weak field approximation isnot adequate forthe dipole plus non-linear quadrupole configuration consideredin the present paper. Hence we are forced to give up this approx-imation, and to evaluate numerically the integrals in Eqs. (8).

The quantitiesηI,Q,U,V andρQ,U,V depend on the strengthη0 and on the damping constanta of the ‘average’ line. Fur-thermore they depend on the magnetic field vectorB(r) via theZeeman splitting normalised to the Doppler width1

vH = B/b (11)with

b =4πmc∆νD

eg(12)

(whereg is the Land́e factor of the line ande,m, c the electroncharge and mass and the velocity of light, respectively), and viathe angular factorssin2ψ cos 2φ, sin2ψ sin 2φ, cosψ, whereψandφ are the angles defined in Fig. 1. Such quantities can easilybe expressed in terms of the components ofB in the referenceframe(x, y, z)

B =√

B2x + B2y + B2zsin2ψ cos 2φ = (B2x − B2y)/B2sin2ψ sin 2φ = 2BxBy/B2cosψ = Bz/B .

(13)

The componentsBx, By, Bz can in turn be calculated fromEqs. (1) and (2) of Paper I once the components of the unit vec-torsu, u1, u2 in the frame(x, y, z) are known. Denoting suchcomponents by(X,Y, Z), (X1, Y1, Z1), (X2, Y2, Z2) respec-tively, we have (cf. Eqs. (30) of Bagnulo et al. 1996 and Eqs. (7)of Paper I)2

X = cos Θ[sin i cosβ− cos i sinβ cos(f − f0)

]+ sin Θ sinβ sin(f − f0)

X` = cos Θ[sin i cosβ`− cos i sinβ` cos(f − f0 + γ`)

]+ sin Θ sinβ` sin(f − f0 + γ`)

Y = sin Θ[sin i cosβ− cos i sinβ cos(f − f0)

]− cos Θ sinβ sin(f − f0)

Y` = sin Θ[sin i cosβ`− cos i sinβ` cos(f − f0 + γ`)

]− cos Θ sinβ` sin(f − f0 + γ`)

Z = cos i cosβ + sin i sinβ cos(f − f0)Z` = cos i cosβ` + sin i sinβ` cos(f − f0 + γ`) ,

(14)

with ` = 1, 2.1 For simplicity, the ‘average’ line is assumed to be a Zeeman triplet.2 We recall that, contrary to the functionsF (1),F (2),F (3) andF (6),

the BBLP also depends on the angleΘ.

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 933

It should be noticed that the denominator of the first termsin the r.h.s. of Eqs. (8) can be written in the form(1− ξ), whereξ is the line-blocking factor representing the fraction of contin-uum radiation subtracted by the spectral lines contained in theinterval∆ν,

ξ = (Fc − FI)/Fc , (15)whereFI andFc are the fluxes

FI =2π∫0

dχπ/2∫0

dθ sin θ cos θ∫

∆νdν Iν(θ, χ)

Fc =2π∫0

dχπ/2∫0

dθ sin θ cos θ∫

∆νdν Ic(θ, χ) .

(16)

Owing to magnetic intensification,ξ varies with the rotationalphasef because the magnetic configuration on the visible hemi-sphere changes. Such variation has usually been neglected (see,e.g., Landi Degl’Innocenti et al. 1981; Landolfi et al. 1993a).Although it is not expected to affect deeply the BBLP signal(Leroy et al. 1995), in this paper we take it explicitly into ac-count, in order to minimize the number of approximations.

The expression of the functionF (6), the mean field modulus,is much more direct. From Eqs. (3), (4), (7) of Bagnulo et al.(1996) we have

F (6) = 3π(3−u)

×2π∫0

dχπ/2∫0

dθ c(θ;u)√

B2x + B2y + B2z ,(17)

wherec(θ;u) is defined in Eq. (9). As noticed in that paper, theintegral has to be evaluated numerically. For a given magneticconfiguration and rotational phase, the componentsBx, By, Bzcan be calculated as illustrated above.

It is thus seen that, when the dipole plus non-linear quadru-pole configuration is considered, a common feature of the BBLPand the mean field modulus is that both require the numericalevaluation of certain integrals – contrary to the functionsF (1),F (2), F (3) for which simple analytical expressions have beenworked out. This is an obvious drawback from a practical pointof view, when real observational data are to be interpreted.

3. The inversion procedure

Once the expressions of the quantitiesF (k) (with k = 1, . . . , 6)are established, we can try to interpret simultaneously all the dif-ferent kinds of measurement available for a given star in terms ofthe dipole plus quadrupole magnetic model. It is however con-venient to introduce, as further constraints to the fit, the valuesof the stellar radiusR∗ (which can always be estimated withinsome uncertainty) and – when known from Doppler broadeningmeasurements – the projected equatorial velocityve sin i. Werecall thatR∗ is related to the rotation periodP and the equato-rial velocityve byR∗ = Pve/(2π). It has been shown (Bagnulo& Landolfi 1999) that inclusion of these constraints may helpin discriminating among several possible models.

The search for the best model can be carried out by lookingfor the minimum, in the parameters space, of aχ2 of the form

χ2 =∑6

k=1∑nk

i=1

[F

(k)i −F (k)

(t(k)i

)]2[

σ(k)i

]2

+

[R(0)∗ −R∗

]2[

σR∗

]2 +[

ve sin i(0)−ve sin i]2

[σve sin i

]2 ,(18)

whereF (k)i are the measurements of thekth observable (per-

formed at timest(k)i , i.e. at rotational phasesf(k)i = f0 +

2π t(k)i /P ), R(0)∗ andve sin i(0) the estimated/measured values

of the stellar radius and the projected equatorial velocity, andσ the corresponding errors. The functionsF (1), F (2), F (3) aregiven by Eqs. (4) of Paper I, the functionsF (4), F (5), F (6) byEqs. (8) and (17) of this paper.

When the observations of a given star are to be interpreted,it is appropriate to look both for the best dipole plus quadrupolemodel and for the best pure dipole model. In the latter case, theexpressions of the functionsF (k)(f) are readily obtained byneglecting, in the general expressions established above, all theterms involvingBq. Theχ2 defined in Eq. (18) depends on theparameters

i, β, f0, Bd, ve, α, Θ, P intQ , PintU (19)

for the pure dipole configuration, and

i, β, β1, β2, f0, γ1, γ2, Bd, Bq, ve, α, Θ, P intQ , PintU (20)

for the dipole plus quadrupole configuration. The last four pa-rameters of each set are only present when BBLP measurementsare included.

We developed two distinct codes for the search of theχ2

minimum in the pure dipole and the dipole plus quadrupolecase. These codes, which are based on the Marquardt algorithm(described, e.g., in Bevington 1969), are a direct generalisationof the codes illustrated in Paper I to include BBLP and meanfield modulus observations (as well as the values ofR∗ andve sin i). The same general remarks about the fitting method(including the existence of ‘unphysical’ sets of parameters andof relative minima on theχ2 hypersurface) apply here as well.Obviously, the new codes require a much longer execution timebecause of the numerical evaluation of the integrals in Eqs. (8)and (17). We recall that the rotation periodP and the limb-darkening coefficientu are treated as fixed parameters. We alsorecall, from the preceding section, that the BBLP depends onthe additional parametersη0 anda (the strength and dampingconstant of the ‘average’ line) and on the quantityb (having di-mensions of magnetic field) defined in Eq. (12). Such quantitiesare also treated as fixed data, to avoid an exceedingly large num-ber of free parameters. When BBLP observations are present,the fitting procedure is repeated for different values ofη0, a, b(chosen within a physically reasonable range, see Sect. 4) untilthe absolute minimum ofχ2 is reached.

In principle, the method presented in this paper allows one torecover the magnetic configuration of a given star by usingany

934 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

combination of observables (e.g. longitudinal field and BBLP,field modulus andve sin i, BBLP alone, etc.). On the other hand,it is well-known that the knowledge of certain observables is notsufficient (whatever the precision and the phase coverage of themeasurements) to recover the configuration: for example, eventhe pure dipole configuration cannot be retrieved from longitu-dinal field measurements alone. Therefore, the question arises asto which combinations of observables are in principle sufficientto recover the magnetic configuration.

A related question concerns the uniqueness of the recov-ered configuration. All the functionsF (k) are, to some extent,invariant under certain transformations of the angles listed inEqs. (19) and (20). Hence, according to the specific combina-tion of observables considered, the recovered configuration willbe more or less degenerate.3

The answer to such questions can be obtained by inspec-tion of the expressions for the functionsF (k), and can easilybe checked via numerical simulations with the two codes men-tioned above. The results are summarised in Table 1.

Consider first the pure dipole configuration. We label thedegenerate magnetic configurations by the letters A through Eaccording to the following scheme:

A1 — unique configuration (no degeneracy);B1 — unique configuration (see below);C2 — two different configurations characterised by exchangedvalues fori andβ (cf. Paper I, line d of Table 1):

( i, β, f0 ),( β, i, f0 );

D2 — two different configurations characterised by the reverseddirection of the magnetic field vector at any point of the stellarsurface (cf. Paper I, line b of Table 1):

( i, β, f0 ),( i, π − β, π + f0 );

E4 — four different configurations resulting from the combina-tion of C2 andD2 :

( i, β, f0 ), ( β, i, f0 );( i, π − β, π + f0 ), ( β, π − i, π + f0 ).

Furthermore, in caseA1 both the star’s rotation direction (clock-wise or counterclockwise) and the azimuthΘ of the projectedrotation axis are univocally determined (except for the ambigu-ity betweenΘ andΘ + π), whereas in casesB1, C2, D2, E4both are undetermined.4

3 Contrary to Paper I, in the following analysis we disregard the‘pseudo-degeneracies’ arising from the exchange of the unit vectorsu1,u2 and from the inversion of their directions, since both transformationsleave the magnetic configuration unchanged.

4 The functionsF (1), F (2), F (3), F (6) are both independent ofΘand invariant under the transformation(i → π − i, β → π − β),which describes two stars having the same magnetic configuration androtating in opposite directions (cf. Paper I, line a of Table 1).

Table 1.The different combinations of observable quantities sufficientto recover the magnetic configuration, and the corresponding degener-acy

F (1) F (2) F (3) F (4,5) F (6) ve sin i d d + q

∗ A1 A′1∗

∗∗ E4

∗ E4∗ ∗ B1

∗ ∗∗ ∗ D2

∗ ∗ D2 C′4∗ ∗∗ ∗ C2 B′2∗ ∗ C2 B′2

∗ ∗ C2 B′2∗ ∗ C2 B′2

∗ ∗ E4 C′4∗ ∗ ∗ B1∗ ∗ ∗ B1 B′2∗ ∗ ∗ B1 B′2

∗ ∗ ∗ B1 B′2∗ ∗ ∗ B1 B′2

∗ ∗ ∗ D2 C′4∗ ∗ ∗ C2 B′2∗ ∗ ∗ C2 B′2∗ ∗ ∗ C2 B′2

∗ ∗ ∗ C2 B′2∗ ∗ ∗ ∗ B1 B′2∗ ∗ ∗ ∗ B1 B′2∗ ∗ ∗ ∗ B1 B′2

∗ ∗ ∗ ∗ B1 B′2∗ ∗ ∗ ∗ C2 B′2∗ ∗ ∗ ∗ ∗ B1 B′2

For the dipole plus quadrupole configuration we label thedegenerate magnetic configurations as follows:

A′1 — unique configuration (including the star’s rotation direc-tion and the angleΘ as in caseA1);B′2 — two different configurations, symmetrical about the planecontaining the dipole axis and the rotation axis (cf. Paper I, linea of Table 3):

( i, β, β1, β2, f0, γ1, γ2 ),( π − i, π − β, π − β1, π − β2, f0, γ1, γ2 );

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 935

C′4 — four different configurations resulting from combiningB′2 with the reversed direction of the magnetic field vector atany point of the stellar surface (cf. Paper I, lineb1 of Table 3):

( i, β, β1, β2, f0, γ1, γ2 ),( π − i, π − β, π − β1, π − β2, f0, γ1, γ2 ),( i, π − β, β1, π − β2, π + f0, π + γ1, γ2 ),( π − i, β, π − β1, β2, π + f0, π + γ1, γ2 ).

In Table 1, an asterisk means that the corresponding observ-able is known, an empty space that it is not known. The twolast columns show the kind of degeneracy according to the pre-ceding schemes, for the pure dipole (d) and the dipole plusquadrupole (d+q) configuration, respectively; an empty spacein these columns means that the knowledge of the associatedobservable(s) isnot sufficient to recover the configuration.

In the dipole case, it appears that the BBLP, the mean squarefield and the mean field modulus are sufficient by themselves toretrieve the magnetic configuration; the mean longitudinal fieldis sufficient only ifve sin i is known. By contrast, the crossoveris not sufficient, even if combined withve sin i or the longitudi-nal field. The only observable which enables one to determineunivocally the configuration is the BBLP; obviously, any com-bination of observables which includes the BBLP has the sameproperty (such combinations are not shown in Table 1). Omittingthe BBLP, the number of degenerate configurations obviouslydecreases (changing fromE4 toC2/D2 and toB1) as the numberof known observables is increased; in particular, kindB1 canonly be obtained ifve sin i is known.

Similar remarks apply to the dipole plus quadrupole case.The minimum combinations of observables sufficient to recoverthe magnetic configuration are: the BBLP by itself; the meanfield modulus providedve sin i is known; any two observablesamong the longitudinal field, crossover, square field and fieldmodulus (even ifve sin i is not known), except longitudinal fieldand crossover: this combination is not sufficient even ifve sin iis known.

Altogether, the BBLP turns out to be the richest in diagnosticcontent, the crossover the poorest. This does not mean that theactual localisation of theχ2 minimum is easier when BBLPmeasurements are available: in fact the opposite is usually true,as expected because of the larger number of free parametersinvolved, which implies a more complex structure of theχ2

hypersurface.

4. Application: Modelling of the magnetic field ofβ Coronae Borealis

4.1. Observations

β CrB (= HD 137909) is one of the best observed CP stars, andmeasurements of magnetic field have been performed by severalauthors. As for the mean longitudinal field, a list of references isgiven in Mathys (1991); for the mean field modulus, see Mathyset al. (1997). In this work we considered 21 determinations ofmean longitudinal field, crossover, and mean square field, 46determinations of mean field modulus, and 27 observations of

BBLP. Part of the determinations of longitudinal field, crossoverand square field were published by Mathys (1994, 1995a) andMathys & Hubrig (1997), while 32 determinations of field mod-ulus were presented by Mathys et al. (1997). Work in progressby Mathys led to the revision of his earlier determinations of thesquare field (Mathys 1995b); these revised values are used here.The remaining measurements of all 4 quantities will be reportedin Mathys et al. (in preparation). Finally, all the observations ofBBLP, obtained in a 100̊A band centred at 4200̊A, were takenby Leroy (1995a).

It should be noted that measurements of longitudinal fieldcarried out by other authors show discrepancies with the datasetthat we have adopted for this analysis. As pointed out in Mathys(1991), other authors (e.g., Babcock 1958; Preston & Sturch1967; Borra & Landstreet 1980) measured a weaker magneticminimum than exhibited by our dataset, whereas Vogt et al.(1980) observed a magnetic minimum about 200 G more nega-tive.

Longitudinal field determinations by Mathys (1994) rest onthe analysis of spectral lines between 5800 and 6400Å, a largefraction of which are located bluewards of 6000Å, Vogt et al.’s(1980) work is based on observations at 6000–6100Å, while al-most all other determinations of longitudinal field are obtainedfrom observations in the blue spectral region. Exceptions arethe works of Wolff (1978) and Romanyuk (1986), who in factperformed a comparison of the curves of longitudinal field asobtained from observations on either side of the Balmer jump.Their results show that a systematic difference between the de-terminations of the longitudinal field obtained in the two dif-ferent spectral regions may indeed be suspected, although theobserved differences are probably smaller than the uncertaintiesinherent in the measurements.

Observations performed at different wavelengths imply adifferent sampling in depth of the stellar photosphere, and onemay suspect that the field strength varies with depth, as alreadysuggested by Wolff (1978) and Leroy (1995b).

The quality of the currently available observations is prob-ably not sufficient to confirm whether this phenomenon is real,and dealing with a depth-dependent magnetic structure is outof the scope of the present modelling. In this paper we im-plicitly assume that discrepancies among different datasets aredue to systematic effects introduced by the different observ-ing techniques. The choice of Mathys (1994) dataset is almostmandatory for the sake of consistency with the available deter-minations of crossover and square field, which are performedon the same spectra used for the determinations of longitudi-nal field. Furthermore, this dataset is based on spectra obtainedwith an instrument (CASPEC) mounted at the Cassegrain fo-cus, hence devoid of significant instrumental polarisation. Byconstrast, most other works are based on Coudé spectra, and therequired compensation for instrumental polarisation is a sourceof uncertainty. This may be particularly critical and delicate forβCrB, as illustrated by the shift between the mean field moduluscurves obtained from KPNO and AURELIE spectra reported inMathys et al. (1997) (see also Fig. 2).

936 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

Fig. 2. The best-fits to the curves of longitudinal field, crossover, square field, field modulus, and BBLP (as customary, the so-called quadraticfield

√F (3) is shown in the place ofF (3)). Dotted lines show the best-fits obtained by neglecting the observations of BBLP. The dashed lines

refer to the model of Eq. (22), the solid lines to the model of Eq. (23). Different symbols are used to distinguish various sets of measurements,between which systematic differences exist (see the references with the original measurements for details). In the left three panels, open circlescorrespond to the data from Mathys (1994, 1995a, 1995b), filled circles to those from Mathys & Hubrig (1997), and filled squares to those fromMathys et al. (in preparation). (In fact, early determinations of the mean square field were revised as explained in the text.) In the upper rightpanel, data obtained at Haute-Provence Observatory with the AURELIE spectrograph are represented by filled circles, and data obtained at KittPeak National Observatory with the cross-dispersed echelle spectrograph by open circles (see Mathys et al. 1997 for details)

β CrB is known as a binary, both by spectroscopy andspeckle interferometry. According to Tokovinin (1985) its com-panion is 1.7 mag fainter in theV filter. For the stellar temper-atures North et al. (1998) estimateTeff = 7750 K and 7200 K,where the lower temperature refers to the fainter component,which in the following we will assume to be non-magnetic. Ac-cording to these estimates, the contribution of the fainter com-

ponent to the observed flux in the visible is five times weakerthan that due to the magnetic star. Close inspection of the spectraobtained at various orbital phases does not reveal any evidenceof significant contribution of thelines of the secondary to theprofiles of the diagnostic lines of the primary. Accordingly, themoments of those lines, and the magnetic quantities derivedfrom their measurements, are unaffected by the fact thatβ CrB

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 937

is a binary system. The impact of the presence of a compan-ion on the interpretation of the observations of BBLP will bediscussed in Sect. 4.3.

For the rotation period we adopted the value of18.4866 ±0.001 d, which was derived via a Fourier analysis of the mea-surements of longitudinal field, field modulus, and BBLP. Asconstraint to the stellar radius we adopted the valueR(0)∗ =3.25±0.31R�, obtained from Hipparcos astrometry by Hubriget al. (2000), whereas for the projected equatorial velocity weadopted the valueve sin i(0) = 3.5± 1.5 km s−1given by Wade(1996). For the limb-darkening coefficient we adoptedu = 0.5.This choice is not particularly crucial, since the observable quan-tities do not strongly depend on it: this was verifieda posterioriby repeating the calculations withu = 0.3 andu = 0.7: theresults were found to be consistent within the parameter errors.

4.2. Analysis

As explained in Sect. 3, a magnetic model for the star could bederived by the analysis of several different subsets of observablequantities (see Table 1). However, this possibility is effectiveonly for data which areexactlymeasured, and the problem ofthe actual invertibility of the observations should be revisited bytaking into account the observational errors. Such a problem wasapproached by means of numerical simulations in Paper I. Wefound that under certain circumstances, theχ2 hypersurface ischaracterised by two or more relative minima. In these cases, thebest-model is usually representative of the actual magnetic con-figuration, but not always. Sometimes the true magnetic topol-ogy is best described by the model corresponding to a secondaryminimum, or, in the most extreme cases, it is not recovered at allby the inversion algorithm. The occurrence of multiple solutions(that is, several relative minima characterised by similar valuesfor theχ2 – which ought not be confused with the degeneracy ofthe magnetic configuration discussed in Sect. 3) is related to themagnitude of the observational errors, but depends also on theregion of the parameter space where the magnetic configurationis located. We will shortly see how this reflects on the analysisof the magnetic configuration ofβ CrB.

A modelling which includes observations of BBLP requiresone to introduce a larger number of approximations and freeparameters than the analysis of the magnetic quantities derivedfrom StokesI andV profiles. In particular, the observations ofBBLP are more sensitive to the possible presence of abundancepatches at the surface of the star. On the contrary, when measur-ing I andV profiles, it is possible to select spectral lines whoseshape does not appear affected by an abundance modulation ofthe element from which they originate. Therefore, we decided tofollow a two-step procedure. First we modelled the observationsof longitudinal field, crossover, square field and field modulus(together with the explicit constraints tove sin i andR∗). Thenwe repeated the calculations by including the observations ofBBLP, and we compared the results.

Even by neglecting the observations of BBLP, it is clear thatthe simple dipole field is not sufficient to describe the magneticconfiguration ofβ CrB. In Bagnulo et al. (1999a) we found that

the observations of longitudinal field, crossover and square fieldcould be well explained by a dipole field, but the best-modelcould not account for the curve of field modulus. By applyingthe new technique presented in this paper, that is, by explicitlyincluding the field modulus in the inversion algorithm, we foundthat the best-fit to the magnetic data has a reducedχ2 = 7.7.The fact that the magnetic morphology ofβ CrB cannot beapproximated by a dipole field has been well-known for a longtime (see, e.g., Stift 1975). The two extrema of the curve offield modulus are not in phase with the extrema of the curve oflongitudinal field, but almost in phase quadrature (see Fig. 2).This feature cannot be reproduced with a simple dipolar field(Landolfi et al. 1997; Mathys et al. 1997).

The situation radically improves (even too much!) when in-cluding a quadrupolar component. We foundfour models, withreducedχ2 ' 1.4 − 1.8, that account for the curves of longitu-dinal field, crossover, square field and field modulus (the predic-tions of the best one are shown in Fig. 2 with dotted lines). Veryfew characteristics are common to these models: a low valuefor the inclination anglei and a value forβ near90◦, whichalso explains why the remaining parameters are so scattered:the observer sees only a limited fraction of the stellar surface,as the rotation axis is almost pole-on.5 This greatly weakens theconstraints to the magnetic morphology, especially because themeasurements derived via StokesI andV are basically sensitiveto the modulus and the longitudinal component of the magneticfield. It is clear that the observations of BBLP, sensitive to thetransverse component of the field, may be of great help to con-strain the magnetic configuration of the star, although, as weshall discuss in Sect. 4.3, their interpretation requires some cau-tion.

As explained in Sect. 3, when considering the observationsof BBLP, we have to introduce a number of additional parame-ters. Four of them (α, Θ,P intQ ,P

intU ) are free, that is, their values

are recovered by the inversion algorithm. Three parameters (η0,a, b) are treated as fixed.

The choice ofη0 andadepends on which features we believeshould pertain to the ‘average’ line that we deem as representa-tive of the stellar spectrum. One could assume that the ‘average’line should simply account for the magnetic observations, andselect its parameters in order to produce the best-fit to the BBLP.We soon realised that this approach would lead to some clashinginconsistency, since the best-fits were obtained forη0 < 1: a linecharacterised by such a small strength would in fact be invisiblein the spectrum, which by contrast appears to be crowded ofstrong lines (see, e.g., Hiltner 1945). We thus decided to fol-low a more realistic approach, that is, to fixη0 anda such thatthe resulting line be representative of the observed stellar spec-trum, and account for the overall line-blocking factor,ξ, in thewavelength region of the observations of BBLP (ξ was mea-

5 These features of the magnetic morphology ofβ CrB, that is, arotation axis almost pole-on and the dipolar axis almost perpendicularto the rotation axis, are also common to the best pure dipole model, andwere found in previous works by several authors, e.g., Wolff & Wolff(1970), Stift (1975).

938 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

sured at 4200̊A by Wolff 1967 as 0.26). At the same time, theline parameters to be adopted should also be consistent withour assumption that the spectral lines are not blended. Underthe hypothesis of Milne-Eddington atmosphere we calculated,by numerical integration over the stellar disk, the line profile forseveral combinations ofη0, a, andβP. We concluded that theappropriate value ofη0 for the ‘average’ line should be com-prised in the interval2

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 939

any case, the points near the minimum of the curve showingthe largest deviations are few, so that their weight is low com-pared to the whole set of observations. Indeed, we found similarmodelling results even by neglecting them.

It is also evident that many determinations of square fieldare below the model predictions. This seems to be intrinsic tothe way the various magnetic quantities are derived, rather thanto indicate a shortcoming of the model. Plain comparison ofthe values of the longitudinal field, of the field modulus, and ofthe square field throughout the rotation cycle of stars for whichthese three magnetic quantities have been repeatedly determinedshows that for many of them, the derived values are at the limitof mutual inconsistency, in the sense that the inequality

(〈B〉2 + 〈Bz〉2)1/2 ≤ 〈B2 + B2z〉1/2

is often only marginally satisfied (Mathys et al. in preparation).Although no definite incompatibility has been found for any star,the fact that in many instances the square field hardly exceeds thesum of the square of the longitudinal field and of the square ofthe field modulus suggests that the technique used to derive themagnetic quantities tends to somewhat underestimate the formerwith respect to the latter two (or conversely, to overestimate oneor both of the latter two with respect to the former). The error, ifreal, is rather small compared to the values of the correspondingmagnetic quantities, but it may be sufficient to account for thekind of difficulty which appears here when trying to use themall simultaneously to establish a best magnetic model.

The determinations of mean field modulus are definitelybetter reproduced by the model obtained without taking intoaccount the observations of BBLP (dotted lines) than by themodels of Eq. (22) or (23). Similarly to what is discussed for thelongitudinal field, this suggests either that the magnetic structureof β CrB is more complex than a dipole plus quadrupole, or thatBBLP introduces some artifact in the modelling.

In fact, the observations of BBLP are well reproduced byboth models of Eqs. (22) and (23). Nevertheless, a few com-ments are still needed.

One of the problems to be addressed is related to the presenceof a companion. Assuming that the radiation arising from thecompanion is not polarised, the binarity simply causes a dilutionof the signal of BBLP by a factor

940 S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III

Fig. 3. The magnetic configuration ofβ CrB according to the model of Eq. (22). The figure is explained the text

Fig. 4. The magnetic configuration ofβ CrB according to the model of Eq. (23). The figure is explained the text

S. Bagnulo et al.: Modelling of magnetic fields of CP stars. III 941

The remaining panels display the magnetic configurationover the entire stellar surface. The rotation axis is taken as pre-ferred direction. The two middle panels (beside the legend) showthe magnetic structure as seen from the North, and from theSouth rotation pole (i = 0◦ andi = 180◦, respectively). Thefour bottom panels show the star with the rotation axis perpen-dicular to the line of sight (i = 90◦), at four different rotationphases specified in the figure. The visible rotation pole is arbi-trarily set toward the top of the page (Θ = 0◦).

The magnetic maps clearly show that, according to bothmodels, the morphology ofβ CrB exhibits magnetic patches ofstrong field right on the side of the stellar surface that isnotvisible to the observer. A similar result was found in a previouswork for HD 119419 (Bagnulo & Landolfi 1999). It would prob-ably be näıve to believe that this represents a physical reality.It rather suggests that the magnetic morphology ofβ CrB (aswell as that of other CP stars) is more complex than what can berepresented by means of a second-order multipolar expansion.The strong field gradients in the non visible stellar hemisphere,especially those of the model of Eq. (22), look like a mathemat-ical artifact of our method, which is ‘forced’ to represent theactual magnetic structure with an insufficient number of freeparameters. The same remarks are also suggested by the rela-tively high value of the reducedχ2 of the best-models, and bythe suspiciously low values for the errors associated to the modelparameters (see Sect. 4.2). However, it makes sense to supposethat the magnetic morphology as described over the visible stel-lar hemisphere is similar to the actual one. New determinationsof the observable quantities obtained via higher-resolution spec-tra and exploitation of additional constraints derived from theconsideration of line profile moments beyond the second orderwill certainly help to obtain less ambiguous results.

5. Conclusions

We have presented a method to recover the magnetic morphol-ogy of CP stars from the modelling of mean longitudinal field,crossover, mean square field, mean field modulus, and broad-band linear polarisation observations, under the assumption ofa centred dipole plus non-linear quadrupole field. Constraints tothe stellar radius and the projected equatorial velocity are alsoexplicitly taken into account.

We have applied the method to the well-known magnetic starβ CrB. This choice is appropriate in the sense thatβ CrB is theonly object for which observations of all the magnetic quanti-ties are available. On the other hand, the particular orientation ofthe star is not fully suitable to obtain unambiguous conclusions.Since the rotation axis is almost pole-on, the observer sees onlyabout half of the entire stellar surface throughout the rotationcycle, which weakens the constraints to the magnetic morphol-ogy. In fact, we found two different magnetic configurationsthat can explain the observations of all the magnetic quantities,and account for the values of the stellar radius, the projectedequatorial velocity, and the line-blocking factor. In both cases,the magnetic morphology exhibits strong field gradients on theside of the stellar hemisphere which is not visible to the ob-

server. This suggests that the magnetic configuration ofβ CrBcould be even more complicated than what can be describedby a second-order multipolar expansion. The modelling may berevisited either when new high-quality data become available,or by taking into account additional magnetic parameters thatcould be determined from further analysis of the line profiles.Nevertheless, ultimately, any model that is derived will haveto be used to build synthetic spectra for comparison with thespectra observed in the various Stokes parameters, as a consis-tency check in order to validate (and possibly refine) the modelsand to establish that the derived magnetic morphology is fullycompatible with the existing observational constraints.

Acknowledgements.Stefano Bagnulo has been supported by the Aus-trian Fonds zur F̈orderung der Wissenschaftlichen Forschung, projectP12101-AST.

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