The pecular magnetic field morphology of the white dwarf WD 1953-011: evidence for a large-scale magnetic flux tube?
G. Valyavin, G.A.Wade, S. Bagnulo, T.Szeifert, J.D.Landstreet, Inwoo Han, A.Burenkov
aa r X i v : . [ a s t r o - ph ] A p r The pecular magnetic field morphology of the white dwarfWD 1953-011: evidence for a large-scale magnetic flux tube?
G. Valyavin , G.A. Wade , S. Bagnulo , T. Szeifert , J.D. Landstreet , Inwoo Han ,A. Burenkov ABSTRACT
We present and interpret new spectropolarimetric observations of the mag-netic white dwarf WD 1953-011. Circular polarization and intensity spectra ofthe H α spectral line demonstrate the presence of two-component magnetic field inthe photosphere of this star. The geometry consists of a weak, large scale compo-nent, and a strong, localized component. Analyzing the rotationally modulatedlow-field component, we establish a rotation period P rot = 1 . ± . i ≈ ◦ , and the angle between the rotation axis and thedipolar axis is β ≈ ◦ . The dipole strength at the pole is about 180 kG, and thequadrupolar strength is about 230 kG. These data suggest a fossil origin of thelow-field component. In contrast, the strong-field component exhibits a peculiar,localized structure (“magnetic spot”) that confirms the conclusions of Maxtedand co-workers. The mean field modulus of the spot ( | B spot | = 520 ± Korea Astronomy and Space Science Institute, 61-1, Whaam-Dong, Youseong-Gu, Taejeon, Republic ofKorea 305-348 Physics Department, Royal Military College of Canada, Kingston, Ontario, Canada Armagh Observatory, Northern Irland European Southern Observatory, Alonso de C´ordova 3107, Santiago, Chile Physics & Astronomy Department, University of Western Ontario, London, Canada Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnii Arkhyz, Karachai CherkessRepublic, 357147, Russia
Subject headings: stars: individual (WD1953-011 — stars: magnetic fields —stars: white dwarfs
1. Introduction
At present, there are more than one hundred known isolated magnetic white dwarfs(MWDs) with magnetic field strengths from a few tens of kilogauss to several hundreds ofmegagauss (Angel et al. 1981; Schmidt & Smith 1995; Liebert et al. 2003; Valyavin et al.2003; Aznar Cuadrado et al. 2004; Valyavin et al. 2006; Kawka et al. 2007; Jordan et al.2007). It is generally assumed that the magnetic fields of the strong-magnetic MWDs (thosewith MG-strength fields) are organized as low-order multipolar fields with dominating dipolarcomponents (Putney 1999). The rotation periods and surface magnetic fields of the strong-magnetic MWDs are believed to be stable on long time scales (Schmidt & Northworthy1991), suggesting that their fields are fossil remnants of the fields of their progenitor stars.A comparison of the field strengths and incidence statistics of the strong-magnetic MWDswith magnetic fields of Ap/Bp main sequence stars support this assumption (Angel et al.1981).Despite the progress with the strong-magnetic MWDs, the magnetic properties of theweak-field degenerates are only poorly known. Presently, only a few white dwarfs withkilogauss magnetic fields has been identified (Schmidt & Smith 1995; Fabrika et al. 2003;Valyavin et al. 2003; Aznar Cuadrado et al. 2004; Valyavin et al. 2006; Kawka et al. 2007;Jordan et al. 2007). Their rotation and field geometries are poorly studied, although someprogress has been achieved by Maxted et al. (2000) and Wade et al. (2003) with a studyof WD 1953-011 and by Valyavin et al. (2005) with WD 0009+501. Maxted et al. (2000)established that the magnetic morphology of WD 1953-011 can be described by both low-field( B ∼
90 kG) and strong-field ( B ∼
500 kG) components. Some evidence for the presence ofa non-dipolar (quadrupolar) component was also found in WD 0009+501 by Valyavin et al.(2005).Motivated by the results of Maxted et al. (2000), we have undertaken collaborative spec-tropolarimetric monitoring of WD1953-011. In this paper we report results of these observa-tions and analyse them in the manner presented by Wade et al. (2003) and Valyavin et al.(2005). Our goal is to determine precisely the magnetic morphology of this degenerate. 3 –
2. A few preliminary remarks
Our preliminary analysis of the spectropolarimetric data obtained with FORS1 at theVLT (Wade et al. 2003) revealed significant variability of the Stokes I and V spectra ofWD 1953-011 due to rotation, with a period of about 1.45 days. In Fig. 1 we show thoseresults which illustrate the variation of the Stokes I, V, Q and U profiles of the H α line withrotational phase (phase increases from top to bottom in the figure). As can be seen, theStokes I profile is strongly variable. The central S-wave of the Stokes V profile is almostconstant. However, near rotational phase 0.6 additional broadened Stokes V signatures ap-pear in the H α wings. These signatures correspond to the weak “satellite features” observedby Maxted et al. (2000) in the wings of the H α profile at these phases, and were attributedto the presence of a high-field magnetic structure. Linear polarization Stokes Q and U signatures are only marginally detected at several rotational phases.In this paper we extend this analysis using addition observational material obtainedwith the AAT (Maxted et al. 2000) and with the 6-m Russian telescope BTA, and usingmore sophisticated modeling techniques.
3. Observations
Spectropolarimetric observations of WD 1953 −
011 were obtained in service mode be-tween May and June 2001 with FORS1 on the ESO VLT. FORS1 is a multi-mode instru-ment for imaging and multi-object spectroscopy equipped with polarimetric optics, andis described by Appenzeller et al. (1998). For this work, FORS1 was used to measureStokes
IQUV profiles of WD 1953 −
011 at 12 different rotation phases, using grism 600 R(plus order separation filter GG 435), which covers the interval 5250 ˚A– 7450 ˚A. With a slitwidth of 0.7”, the spectral resolving power was about 1650. To perform circular polarizationmeasurements, a λ/ λ/ ◦ steps.To perform linear polarization measurements, a λ/ . ◦ steps. At each epoch, Stokes V was measured taking two 420 sexposures: one with the λ/ − ◦ , and one with the λ/ ◦ . Stokes Q and U were measured taking four 600 s exposures with the λ/ ◦ , 22 . ◦ , 45 ◦ , and 67 . ◦ . The Stokes V /I circular polarization 4 –spectrum was then obtained by calculating P V = VI = 12 ( r − − r +45 ) (1)where r α = f o − f e f o + f e . (2)In Eq.(2) f o is the flux measured in the ordinary beam and f e is the flux measured in theextra-ordinary beam, obtained with the λ/ α . Similarly, thelinear polarization was obtained by calculating P Q = QI = ( r − r ) P U = UI = ( r . − r . ) (3)where r β is defined by Eq. 2, except that β refers to the position angle of the λ/ f o and f e were obtained from the raw data after bias correction andwavelength calibrations performed using standard IRAF routines.These observations were supported by a short observing run at the 6-m Russian telescopeBTA where we obtained additional I, V series of spectra of WD 1953-011 using the UAGSspectropolarimeter, with nearly the same resolving power as in the observations with FORS1.The instrument is described in detail by Afanasief et al. (1995) and by Naydenov et al.(2002). The observational technique and data reduction are similar to those described byBagnulo et al. (2000, 2002) and by Valyavin et al. (2005). A comparative analysis of thespectropolarimetric data obtained from the different telescopes showed identical results. Acomparison of the Stokes V spectra obtained with the VLT and BTA is illustrated in Fig. 2.The spectra are obtained at different times but similar rotational phases.In addition to the spectropolarimetric data from the VLT and BTA, in this paper wealso use high-resolution spectroscopic data (Stokes I ) obtained at the AAT and described byMaxted et al. (2000). Together with the spectropolarimetry, these data extend the analysispresented by Wade et al. (2003) to a much longer time base. Table 1 gives an overview of allthe observations. In the table: JD is the Julian Date; Exp is an equivalent exposure timeof an observation;
Stokes is the observed Stokes parameter (
I, V, Q or U ); and Telescope is telescope used (VLT, AAT or BTA).
4. Mean field modulus of the magnetic field of WD 1953-011
We begin with an analysis of the low and high-resolution Stokes I spectra, extendingover 5 years. These spectra are used for establishing the rotation period of the star, as wellas the mean field modulus of the low- and strong-field components. 5 – The low-field component of WD 1953-011 was first discovered spectroscopically by Koester et al.(1998) and described in detail by Maxted et al. (2000). The mean field modulus, | B G | ex-hibits a low-amplitude variation due to the star’s rotation, with a period estimated betweenhours and days (Maxted et al. 2000). These conclusions were made on the basis of thehigh-resolution spectroscopy of the Zeeman pattern in the H α core.In our low-resolution FORS1 and BTA observations, Zeeman splitting attributed to thelow-field component cannot be resolved spectroscopically. In the spectra the splitting is re-vealed as an additional variable broadening and desaturation of the H α core. In this case,measurements of the low-field component can be carried out by an analysis of the equivalentwidths ( EW core ) of the H α core. Using field modulus | B G | values determined by Maxted et al.(2000) from an analysis of individual high-resolution H α line profiles and measuring equiva-lent widths of the H α cores, we may try to calibrate the relationship EW core – | B G | to allowus to determine | B G | in the low-resolution spectra. To obtain the required calibration, weestimated equivalent widths of the H α core from the high resolution spectra obtained byMaxted et al. (2000). In order to work with measurements having a uniform resolution, allhigh-resolution spectra were convolved with a gaussian instrumental profile to reproduce thespectral resolution of FORS1 and UAGS. The resultant spectra are presented in Fig. 3.As one can see in Fig. 3, the profiles are strongly variable. The central intensity of thecore also correlates with the intensity of the strong-field Zeeman features which are found inthe wings of the H α profile (Fig. 3: the two satellite features at ±
10 ˚A around the H α core).This correlation (the higher the intensity of the features, the weaker the central intensity) isdue to the fact that the spot, which appears periodically on the visible disc due to rotation,redistributes the flux according its projected area. It is also seen that the width of the H α core is variable itself due to the variable Zeeman pattern of the line core. Therefore, tominimize the influence of the variable high-field spectral features in measurements of thecentral Zeeman pattern attributed to the low-field component, we artificially re-normalizedall the profiles to equal residual intensities ( r c = 0 .
47 at the line center) and measuredequivalent widths of the central narrow portion ( r c ≤ .
6) of the resultant H α profiles. Inthese conditions, the variation of the Hα core is attributed only to the rotationally modulatedlow-field component. As hoped, we find a close correlation between the EW core measured inthis way and the value of | B G | measured by Maxted. This relation is shown in Fig. 4. For distinctness we label all magnetic observables related to the low-field component with the subscript“ G ”, assuming its large scale (Global ) geometry. Observables related to the strong-field component will belabeled with the subscript “ Spot ” or “ S ”. | B G | − EW relationship derived from the high-resolution spectra asillustrated in Fig. 4, we inferred the field modulus | B G | associated with each of the low-resolution spectra (see Table 2). In order to measure the magnetic field modulus | B Spot | of the strong-field component wedeblended the H α profile by means of a simultaneous fit of five Gaussian profiles (three centralprofiles used to fit the H α core, and two satellite gaussians to reproduce the strong-fieldZeeman pattern). This method enabled us to reproduce the Zeeman splitting of the strong-field component and the corresponding magnetic field strength in those spectra where thestrong-field spectral features are seen. The method also allows us to estimate the projectedfractional area S of the strong-field area on the disc. Reconstructing by gaussians andextracting the Zeeman pattern of the strong field component from the observed H α lineprofiles, we determined S , the fraction of the flux absorbed by the strong-field patternrelative to the total H α absorption. The method is rather rough and can be considered as afirst-guess approximation that is necessary for the analysis described below. A more realisticcalculation of the size of the strong-field area is performed in Sect. 8 where we model thespectra. The results ( | B spot | and S ) are presented in Table 3. S is given in per cent of thedisc area.
5. Mean longitudinal magnetic field of WD1953-011
From the Stokes I and V spectra obtained with the VLT and BTA we determinedlongitudinal fields through the weak-field approximation (Angel et al. 1973) modified to theanalysis of the two-component Stokes V spectra: V ( λ ) ∼ (1 − S ) B lG (cid:16) λλ (cid:17) I ( λ ) d I ( λ )d λ + SV ( λ ) Spot (4)where B lG is the longitudinal field of the low-field component, dI ( λ ) dλ describes the gradient ofthe flux profile, S is the relative area of the spot projected on the disc, λ is the H α restwavelength and V ( λ ) Spot is the Stokes V profile from the strong-field component observedin the H α wings.In Eq.(4) we effectively separate the disc into two equivalent areas with different averagedmagnetic field strengths. The first term in the equation describes the weak-field area and the 7 –second term is attributed to the strong-field component. The first term is used in the usualmanner according to which the flux and its gradient are taken directly from the observedspectra. (Inaccuracies due to the presence of the strong-field features in the wings arecomparatively weak: these features are located quite far from the line core and do not affectthe narrow central low-field polarization profile). However, circular polarization V ( λ ) Spot from the strong-field component (the second term in the equation) cannot be fitted in thesame way.In order to fit V ( λ ) Spot and estimate the longitudinal field of the strong-field area, wecompute: V ( λ ) Spot = I ( λ ) LSpot − I ( λ ) RSpot I ( λ ) (5)where the flux I ( λ ) is the observed H α flux profile, and I ( λ ) LSpot and I ( λ ) RSpot are the left-and right-hand polarized parts of the H α profile from the strong-field equivalent area of thedisc. In the observed polarization spectra I ( λ ) LSpot and I ( λ ) RSpot are mixed with fluxes fromthe weak-field equivalent area and therefore can not be extracted directly. However, we mayestimate them with some simplifications.Individually, I ( λ ) LSpot and I ( λ ) RSpot are Zeeman-split profiles of the circularly polarizedsatellite σ components. Due to the fact that the σ − component is absent in I ( λ ) RSpot , andthe σ + component is absent in I ( λ ) LSpot , their centers of gravity are displaced, indicating thepresence of the longitudinal field from the strong-field area. Their difference provides thenon-zero circular polarization (Eqn. 5).Because I ( λ ) LSpot and I ( λ ) RSpot are not resolved in the total left- and right-circularly po-larized observed fluxes, the determination of their true shapes requires detailed modeling thefield geometry. However, as a first-guess approximation we may describe them by simulat-ing an equivalent “mean” Zeeman-broadened H α profile magnetically displaced to the left-and right-sides from the rest wavelength . With this simplification only two parameters –Zeeman broadening and their magnetic displacement due to the averaged longitudinal fieldfrom the spot should be varied to reproduce the observed circular polarization. The low resolving power of the FORS1 and UAGS, as well as unresolved circular polarization featuresattributed to the strong-field area, enable us to consider the problem in terms of Zeeman broadening insteadof detailed analysis of the strong-field Zeeman pattern. α profile, artificially broadened tounresolved Zeeman patterns typical for I ( λ ) RSpot and I ( λ ) LSpot and magnetically displaced.This template can be taken from a zero magnetic field solution for the atmosphere ofWD 1953-011 or from the observed spectra. For example, assuming the pressure-temperatureconditions in the spot area to be similar to conditions in the other parts of the white dwarf’ssurface we may choose as the template one of the observed weak-field H α profiles (obtainedat those moments when the spot is not seen). In our analysis we proceed this way.Thus, to simulate I ( λ ) LSpot and I ( λ ) RSpot in order to fit the circular polarization (5) fromthe strong-field area and estimate its longitudinal magnetic field we used the following iter-ative method: • Step-1:
We construct the reference weak-field “template” H α profile from the observedI-profiles obtained at those rotational phases where the strong-field Zeeman pattern isnot seen. • Step-2:
We artificially broaden the template profile by a gaussian filter with an arbi-trary half-width to an unresolved strong-field Zeeman pattern and displace the resultby the magnetic displacement factor ∆ λ to the shorter / longer wavelengths to estimatethe I ( λ ) LSpot and I ( λ ) RSpot profiles. • Step-3:
Varying the magnetic broadening of the estimated profiles I ( λ ) LSpot and I ( λ ) RSpot and their Zeeman displacement we finally fit the strong-field circular polarization (5) inthe working equation (4). The displacement found ∆ λ = 4 . · − B lS λ (Landstreet1980), gives an estimate of the longitudinal field B lS . (In other words, taking S mea-sured from the Stokes I spectra and presented in Table 3 we simultaneously fit theobserved combined Stokes V (4) varying B lG and circular polarization (5) of the strong-field component, where B lS is one of the parameters.)In the fit procedure, the associated error bars are obtained using the Monte Carlomodeling method presented by Schmidt & Smith (1994). An example of the fit is presentedin Fig. 5. The results are collected in Table 4.This method gives quite robust estimates of the longitudinal magnetic field of the low-field component. In the case of the strong-field component, the real intensities of the fieldscould be slightly over or underestimated due to the simplifications described above. Forthese reasons, estimates of the strong-field component given here could be considered to be 9 –approximate. As an alternative, the two-component circular polarization spectra could beanalyzed by using Zeeman tomography (Euchner et al. 2002, for instance). To provide moreprecise modeling, below (Sec. 8) we analyze our data again in the framework of simplifiedZeeman tomography.
6. Period determination
To search for the star’s rotation period we used the equivalent widths EW core of theH α core determined in Sect. 4 . This observable is the most sensitive indicator for thedetermination of the rotation period. To determine the rotation period we applied theLafler-Kinman method (Lafler & Kinman 1965), as modified by Goransky (2004). Analysisof the power spectrum of the data revealed a signal indicating a probable period between1.4 and 1.5 days (Fig. 6). This is consistent with the period estimate ( P ≈ .
45 days) givenby Wade et al. (2003) and Brinkworth et al. (2005).Detailed study of the periodogram showed that the most significant sinusoidal signalcorresponds to a period P = 1.4480 ± α core equivalent widths EW core derived with this periodis presented in Fig. 7. The derived period shows a very good agreement among all theobservations taken from different telescopes (the VLT, BTA and AAT). For the minimum of EW core we obtain the following ephemeris:JD = 2452048 . ± .
03 + 1 d · ± . | B G | and longitudinal field | B lG | of the weak-field components are presented in Fig. 8. The phase curves are almostsinusoidal, and symmetric about the values of about | B G | = +87 kG and | B lG | = −
43 kG.The modulus of the weak-field component varies from +77 ± . ± . − ± − ±
7. Modeling the weak-field component of the magnetic field of WD 1953-011
To verify that the behavior of the weak-field component of WD 1953-011 is consis-tent with a nearly dipolar geometry, we have followed the schematic method proposed byLandolfi et al. (1997). This method has already been described and applied to establish themagnetic field morphology of the weak field white dwarf WD0009+501 (see Valyavin et al.(2005) for details). For this reason here we do not explain all the modeling details, butrestrict ourselves to the presentation of the results.In this paper we model the phase-resolved measurements of the mean longitudinal fieldand mean field modulus of the weak-field component within the framework of a pure dipoleand dipole+quadrupole field. The phase-resolved observables for the weak-field componentwhich we use as input data are obtained by binning measurements in phase and averaging.The binned data are presented in Table 5.The dipole or dipole plus quadrupole models depend on the following 10 parameters:– B d and B q , the dipole and quadrupole strength, respectively;– v e , the stellar equatorial velocity;– i , the inclination of the stellar rotation axis to the line of sight;– β , the angle between the dipolar axis and the rotation axis;– β and β , the analogues of β for the directions identified by the quadrupole;– γ and γ , the azimuthal angles of the unit vectors of the quadrupole;– f , the “reference” rotational phase of the model;– v e sini , the projected stellar rotation velocity.The angles i , β , β , β range from 0 ◦ to 180 ◦ , while γ , γ , f range from 0 ◦ to 360 ◦ .The rotational period P = 1 .
448 and the limb-darkening constant u , which also affects theexpressions of the magnetic observables, are taken as fixed. (Note that the pure dipole modelwould retain as free parameters only B d , v e , i , β , and f .) 11 –For the stellar mass, Bragaglia et al. (1995) gave the value of 0 . M ⊙ , which togetherwith known surface gravity of WD 1953-011 ( log g = 8 . . R ⊙ . This parameter and the period were then usedto estimate the equatorial and projected velocities of the star.For the limb-darkening coefficient, we adopted the value of u = 0 .
5. Note that, as dis-cussed by Bagnulo et al. (2000), the results of the modeling are only slightly influenced bythe u value. The best-fit parameters are: A) Dipole i = 14 ◦ ± ◦ β = 14 ◦ ± ◦ f ≈ ◦ B d = 108 ± kGv e = 0 . ± . km s − v e sini = 0 . ± . km s − B) Dipole + quadrupole i = 18 ◦ ± ◦ β = 8 ◦ ± ◦ f ≈ β = 22 ◦ ± ◦ β = 24 ◦ ± ◦ γ ≈ ◦ γ ≈ ◦ B d = 178 ± kGB q = 233 ± kGv e = 0 . ± . km s − v e sini = 0 . ± . km s − Note that, as explained by Bagnulo et al. (2000), the available observations do notallow one to distinguish between two magnetic configurations symmetrical about the planecontaining the rotation axis and the dipole axis. Such configurations are characterized by 12 –the same values of B d , B q , v e , γ , γ , f , while the remaining angles are related by( i, β, β , β )( 180 ◦ − i, ◦ − β, ◦ − β , ◦ − β ) . Due to the fact that the spin axis angle is close to a pole-on orientation and due tothe small number of available observables, the error bars on the derived quantities are fairlylarge. For the same reasons, there are some uncertainties in the determinations of all theparameters considered together. However, despite these weakness, we are able to obtainsome conclusions about the most probable geometry of the white dwarf’s global field.The best fit of the dipole+quadrupole model applied to the observations is shown by solidlines in Fig. 9. For comparison, the dashed line shows the fit obtained using the pure dipolarmorphology. As is evident, the dipolar model does not reproduce the observations well.Examination of the reduced χ r statistics shows that the quality of the dipole+quadrupolefit ( χ r = 0 . , .
43 for longitudinal field and field modulus, respectively) is significantlybetter than the pure dipole fit ( χ r = 1 . , .
8. High-field component8.1. Migrating magnetic flux tube?
In contrast to the well-organized, nearly sinusoidal variation of the weak-field compo-nent, the phase behavior of the high-field structure exhibits a number of peculiar featuresthat made it impossible to model these data as a simple low-order multipole: • According to the measurements of the Zeeman-split satellite spectral features in theH α wings, the mean field modulus of the strong-field component does not show anynoticeable variation during the star’s rotation. (Due to rotation, we see variation of theflux intensities from the strong-field area, but the corresponding Zeeman displacementis nearly constant.) The most likely explanation (Maxted et al. 2000) is that there isan area with a nearly uniformly distributed strong magnetic field. This explanation, iftrue, suggests that the strong-field component has a localized geometry and cannot beunderstood as a high-field term in the multipolar expansion of the star’s general field.Averaging all the data we determine < | B spot | > = 515 ± • The Zeeman pattern attributed to the strong-field component becomes visible at ro-tational phases φ = 0 . − . S of the magnetic spot.The projected area of the strong-field structure varies from zero to about 12% of thedisk, consistent with the study of Maxted et al. (2000). This observable can be used asan additional parameter to test the rotational period of the star. However, using thisquantity to search for the period we did not find a regular signal at any period withinthe tested 5-year time base. Moreover, phasing the data with the magnetic ephemerischaracterized by the rotational period of 1.448 days, the resultant phase curve of thespot size variation (Fig. 10) appears distorted in comparison to the well-organized be-havior of the weak-field component phased with the same ephemeris. We observe asmall relative phase shift between the data obtained with different telescopes that mayindicate a possible secular longitudinal drift of the strong-field component. • The averaged longitudinal field of the strong field area varies from zero (when thearea is invisible) to about 450 kG (Table 4) which is comparable to the averagedmean field modulus ( ≈
515 kG) of the field. This suggests a deviation of the strongfield component from any of low-order multipolar geometries for which the differencebetween the full vector and its longitudinal projection should much larger (for example,for a centered dipole field the difference should be at least 2.5 times, Stibbs (1950)).The last point suggests the presence of an essentially vertical orientation of the magneticfield lines relative to the star’s surface, typical for local magnetic flux tubes in cool, convectivestars (the Sun for example). If the geometry is a tube seen as a local magnetic spot in thephotosphere, we may also expect the above-mentioned secular drift. To our knowledgeand by an analogy to the Sun, such fields are expected to show dynamical activity likemigration over the star’s surface and be associated with dark spots that might producephotometric variability of the star. Significant photometric variability of WD 1953-011 hasbeen established (Wade et al. 2003; Brinkworth et al. 2005). However, in this paper we areunable to establish the association of the darkness and magnetic spots for reasons which wediscuss below.
Measuring the longitudinal magnetic field of the strong-field component as described inSect. 5 we noted that the estimates of the longitudinal field of the strong-field area mightbe affected by our simplifying assumptions. In order to control these measurements we have 14 –also directly modeled the observed polarization and intensity H α spectra of the star withoutparametrizing the surface field components. In addition to the analysis of the Stokes I and V spectra, the observed Q and U spectra were also taken into consideration. As can beseen in Fig. 1, linear polarization Stokes Q, U signatures are detected only marginally at afew rotational phases. However, this information can also be used to constrain the magneticgeometry of the degenerate. Note that different magnetic geometries may produce similarStokes V spectral features due to axial symmetry of circular polarization provided by thelongitudinal projection of the field. Linear polarization restricts the strength and orientationof the transverse field that, together with circular polarization, makes it possible to resolvethe geometry spatially. The observed Q and U spectra are mainly noise, but we may try touse them in terms of the upper limits.In our model we examined several low order multipolar magnetic field geometries, inte-grating over the surface elementary (taken at a single surface element) Stokes I, V, Q , and U spectra calculated for various field strengths and orientations of the magnetic field lines.The technique we used to calculate the elementary spectra deserves some special explanation. a) Simulation of the Stokes I, V, Q and U synthetic H α spectra. Generally, accurate simulation of the split Balmer profiles in spectra of strong magneticwhite dwarfs requires detailed computations of the main opacity sources under the influenceof strong magnetic fields. To our knowledge, these computations have not yet been tabulatedfor practical use. For this reason, a self-consistent solution of the transfer equation for theline profiles in spectra of strong-magnetic white dwarfs cannot be performed without specialconsideration of additional parameters (related, for example, to the Stark broadening inthe presence of a strong magnetic field, Jordan (1992)). However, in case of the weak-fielddegenerates we may restrict ourself to a zero-field solution similar to that presented byWickramasinghe & Martin (1979) or by Schmidt et al. (1992). The method assumes that ifthe Stark broadening dominates the line opacity, the total opacity can be calculated as thesum of individual Stark-broadened Zeeman components. The Stark broadening is suggestedto be taken as “non-magnetic” in this case. Under these simplifying assumptions we maysimulate the local, elementary Zeeman spectra using one of the following two ways:i to compute the transfer equation for all Stokes parameters at given strength and ori-entation of a local magnetic field line calculating the H α opacities as described, or(alternatively)ii to select a “template” H α profile typical for a zero-field white dwarf with the same 15 –pressure-temperature conditions as in WD 1953-011 for construction of the elementaryZeeman spectra. (In other words, we may try to construct from this template profileindividual Zeeman π − and σ − components, parametrizing their magnetic displace-ment and relative intensities, and additively combine them to obtain the elementary I, V, Q, U H α profiles.)The first, direct method of atmospheric calculations for WD 1953-011 requires specialtheoretical tools which are outside the scope of this observational paper. The second, simpli-fied method, which we will use, seems to be rather rough due to the fact that the fluxes fromthe individual π − and σ − components obtained by using the zero-field template profile aregenerally not additive (whereas their corresponding opacities can be added in the transferequation). Nevertheless, in the linear guess approximation they can be taken as additiveand the method can also be applied. Besides, testing this method on some standard, well-studied magnetic Ap/Bp stars we have obtained satisfactory results modeling their observedpolarizations. This allowed us to conclude that the method is reasonably accurate.Thus, to calculate the elementary I, V, Q, U H α profiles we adopted the use of the zero-field H α template profile which was constructed from the observed I-spectra obtained atthose moments, when the strong-field Zeeman pattern is not seen. The Stark parts of theprofile were obtained by averaging the I profiles at the rotation phases 0 and 0.91 (seeFig. 1) in which the strong-field features are not seen. The central “zero-field” Dopplerprofile was adopted to reproduce in the model procedure the observed low-field magneticbroadening of the H α cores at phases 0 and 0.91. The necessary individual profiles ofthe non-displaced π − and displaced σ − components were obtained by entering the normalZeeman displacements according to the orientation of the local magnetic field: the circularlypolarized σ − components are displaced according to the longitudinal projection of the localmagnetic field, and the linearly polarized σ − components are displaced by the transversefield.In order to model the polarization H α profiles, the relative intensities of the central π − and displaced σ ± components were computed from the prjection of the local field vectoronto the plane on the sky and on the line of sight as described by Unno (1956) (for aqualitative explanation see also Landstreet (1980)). The final intensities of the π − and σ − components were obtained by renormalisation such that the total sum of the fluxes from allthe components be equal to the flux from the zero-field template H α profile.Finally, the elementary π − and σ − components were combined to construct from themthe elementary I, V, Q, U spectra by simulation of the ordinary and extraordinary beamsgiven by a polarimetric analyzer. For example, simulating the H α Stokes V profile, all thecomponents except the circularly polarized σ − components are equally distributed between 16 –the beams. The circularly polarized σ + components are absent in one of the beams, and theoppositely polarized σ − components are absent in the other beam. The final Stokes V H α profile was obtained by subtraction of the ordinary from extraordinary beams and devisionof the result by the total flux. The Q and U spectra were obtained in a similar way: thelinearly polarized σ components and central π component are distributed between the beamsaccording to the projection of the local magnetic field onto the plane of the sky.After the determination of the elementary I, V, Q, U spectra given by a magnetic geom-etry in all surface elements, we finally integrated and averaged them over the disc. For thelimb-darkening coefficient we adopted the value of u = 0 . b) Modeling the field geometry in WD 1953-011 by simulation of the observed polarizationspectra. In Sect. 5) we have concluded, that estimates of the longitudinal magnetic field based onthe weak-field approximation are accurate enough to make it possible to model the geometryof the low-field component separately from the strong-field component in the manner asdemostrated in Sect. 7. For this reason, and in order to reduce a number of variables, we usethose results (obtained in Sect. 7) as input and non-changeable parameters in the tomography.We just note, that modeling the observed spectra obtained at those time moments, wherethe spot is not seen, we have confirmed, that the observations (Stokes-V spectra) can bebetter fit by the dipole+quadrupole geometry of the low-field component with parametersperformed in Sect. 7 ( case B ). Examination of the pure dipole model (case A ) gives nosatisfactory results and we do not use this case here.Modeling the strong-field area as an additional harmonic in the low-order (lower thanoctupole) multipolar expansion we were unable to reproduce the observations. The fit doesnot provide the necessary contrast in the observed Zeeman patterns at those phases where thepolarization and intensity spectra demonstrate the weak- and strong-field Zeeman featurestogether. However, assuming the strong-field component to be concentrated into a localizedarea having maximum projected size of about 12 per cent of the disk, the strong-field Zeemanspectral features can be well-reproduced with an average magnetic field of 550 ±
50 kG.Practically the same result has been obtained by Maxted et al. (2000).To model the strong-field area we tested two simplified localized geometries: a “contrastspot” with a homogeneously distributed, essentially vertical magnetic field, and a “sagittal”geometry with a strong vertically-oriented central magnetic field, that smoothly decreasedto zero at the spot edges. Generally, both geometries are able to describe the Stokes I and V spectra with more or less acceptable accuracy. The first model, however, does not provide us 17 –with a good fit of the Q and U spectra due to the presence of the sharp (and non-physical)jump of the field intensity at the edges of the strong-field area. For this reason we do notdiscuss this case in detail.The “sagittal” geometry of the strong-field area was constructed by using a modifiedmodel of a centered dipole: about 45% of the spot’s area (central parts) have the dipolardistribution with polar field Bp = +810 kG at the center. The remaining 55% of the externaldipolar field is artificially modulated to have a gradual decrease to zero at the edges of thearea. This model provides a good fit of the Stokes I, V spectra and reasonable reproductionof the linear polarization Q-,U-spectra, as shown in Fig. 11 where we also illustrate thetomographic portrait of the white dwarf’s magnetosphere.Despite the fact that we obtain such a good agreement of the “sagittal” geometry withthe observables from the strong-field area, we do not claim that this geometry is fully correctin all details (for example, the model does not control conservation of magnetic flux). Similarto the case of the weak-field component, the most natural way to study the strong-field areais to describe it as a strong-field feature resulting from the superposition of several high-orderharmonics in the multipolar expansion. At this time we are unable to study this term usingany combination of the first hamonics higher than octupole, but we do not exclude that theuse of the highest terms of different polarities and intensities will resolve the problem.However, this result clearly demonstrates a qualitative difference in the morphologies ofthe strong-field area and the global field of the white dwarf. From the model we establish witha very high probability that the strong-field area has a localized structure with essentiallyvertical orientation of the magnetic field lines. The physical size of the area is about 20%of the star’s surface giving maximum 12% projection on the disc. The spot is located atan angle of about 67 ◦ with respect to the spin axis, providing a maximum longitudinal fieldstrength of about 400 kG. These results are in good agreement with the measurements of thelongitudinal field of the spot.
9. Discussion
We have presented new low-resolution spectropolarimetric observations of the magneticwhite dwarf WD 1953-011. From these observations and observations of previous authorswe have determined the star’s rotation period, mean longitudinal field, mean field modulus,and surface field morphology. Let us finally summarize these results.1) Our present picture of WD 1953-011 consists of a MWD with relatively smooth, low-field global magnetic field component, and a high-field magnetic area. 18 –2) The low-field component demonstrates regular periodicity with period P = 1 . ± . i = 18 ◦ ± ◦ ;2) the angle between the dipolar axis and the rotation axis β = 8 ◦ ± ◦ ;3) the dipole strength B d = 178 ±
30 kG;4) the quadrupole strength B q = 233 ±
30 kG.3) The strong-field component exhibits a peculiar localized structure. The mean fieldmodulus of the spot | B spot | is estimated to be 515 ± <
300 kG to about 400 kG. Comparing the meanfield modulus with the maximum longitudinal field we suggest that the geometry of thehigh-field spot may be similar to a magnetic flux tube with vertically-oriented magneticfield lines. The spot is located at an angle of ≈ ◦ with respect to the spin axis.Our results suggest that the magnetic field of WD 1953-011 consists of two physicallydifferent morphologies - the fossil poloidal field and an apparently induced magnetic spot.To our knowledge, fossil, slowly decaying global magnetic fields are organized in a nearlyforce-free poloidal configuration . In contrast, if the suggested vertical orientation of themagnetic field lines in the spot is correct, the uncompensated “magnetic pressure” of sucha localized field may dominate against the tension causing a strong impact on the pressure-temperature balance in the photosphere of the degenerate. This may produce a temperaturedifference between the strong-field area and other parts of the star’s surface. As a resultwe may expect rotationally-modulated photometric variability of WD 1953-011 (as observedby Wade et al. (2003) and Brinkworth et al. (2005)). For these reasons (and by analogy tosunspots) such fields might be unstable if not supported by other dynamical processes suchas differential rotation, and may therefore exhibit secular drift with respect to the stellarrotation axis. According to the basic properties of the Maxwell stress tensor (see, for instance, Parker (1979)) themagnetic field B creates in the atmospheric plasma an isotropic pressure B π and tension B i B j π directed alongthe magnetic lines of force. While neighboring lines of force of a magnetic field try to expand due to themagnetic pressure, tension tends to compensate for this effect. The force-free configuration is possible onlywhen the gradient of the magnetic pressure is fully compensated by the tension forces.
19 –In the above context we note that significant photometric variability of WD 1953-011 hasbeen established (Wade et al. 2003; Brinkworth et al. 2005). Also, remarkably, the authorshad encountered problems analyzing the periodicity of the variable differential flux. Whenindividual epochs of their photometric data are phased according to the rotation period ofabout 1.45 days, the resultant folded lightcurves are smooth and approximately sinusoidal.However, they had difficulty obtaining an acceptable fit to all epochs of photometric data con-sidered simultaneously (Wade et al. 2003). Besides, periodograms (Brinkworth et al. 2005)obtained separately for their 7 individual observing runs indicate a significant spread in theperiod distribution . The individual peaks are stochastically distributed around a rotationperiod P ≈ d .
45 from P ≈ d .
415 to P ≈ d .
48, also suggesting a probable phase shift fromepoch to epoch with characteristic times from tens to hundred of days. Combining all thedata, they establish their version ot the rotation period P = 1 . d ± REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
22 –Table 1: Spectral and spectropolarimetric observations of WD 1953-011: column 1 is theJulian Date of the midpoint of the observation, column 2 is the exposure time, and column 3reports the telescope used in the observations (
AAT indicates the high resolution spec-troscopy presented by Maxted et al. (2000)). For data obtained with the VLT the exposuretimes are presented for the three consecutive
I, V / Q / U modes of observations (in thiscase the midpoint corresponds to observations of the Stokes I, V parameters).JD
Exp (sec) Stokes Telescope2450676.955 600 I AAT2451391.948 600 I AAT2451391.955 600 I AAT2451391.962 600 I AAT2451392.059 1800 I AAT2451392.957 1800 I AAT2451393.066 1800 I AAT2451393.106 1800 I AAT2451393.943 1200 I AAT2451393.958 1200 I AAT2451393.973 1200 I AAT2451393.988 1200 I AAT2451394.003 1200 I AAT2452048.801 840/1200/1200
I, V /Q/U
VLT2452048.893 840/1200/1200
I, V /Q/U
VLT2452076.671 840/1200/1200
I, V /Q/U
VLT2452076.883 840/1200/1200
I, V /Q/U
VLT2452078.722 840/1200/1200
I, V /Q/U
VLT2452078.879 840/1200/1200
I, V /Q/U
VLT2452079.672 840/1200/1200
I, V /Q/U
VLT2452079.892 840/1200/1200
I, V /Q/U
VLT2452087.621 840/1200/1200
I, V /Q/U
VLT2452087.670 840/1200/1200
I, V /Q/U
VLT2452087.722 840/1200/1200
I, V /Q/U
VLT2452087.768 840/1200/1200
I, V /Q/U
VLT2452505.290 3600
I, V
BTA2452505.327 3600
I, V
BTA2452505.360 3600
I, V
BTA2452505.397 3600
I, V
BTA 23 –Table 2: Determinations of the mean modulus | B G | of the weak-field component. Column 1is the Julian Date, Col. 2 and Col. 3 are the equivalent widths of the H α core ( EW core ) andassociated error bar, Col. 4 and Col. 5 are the inferred field strength | B G | and its error bar σ (kG), Col. 6 is the telescope used. Uncertainties at the measured equivalent widths arecalculated as a noise fraction of the flux (due to Poisson noise) in the total flux under theline profile. The mean field modulus and its uncertainty obtained from the high-resolutionspectroscopy with the AAT are taken from Maxted et al. (2000). Uncertainties at thecalibrated field strengths (observations with the
VLT and
BTA ) result from regressionerrors in the EW core – | B G | relationship shown in Fig. 4.JD EW core σ | B G | ( kG ) σ ( kG ) Telescope2450676.955 1.040 0.008 91 5 AAT2451391.948 1.073 0.008 93 4 AAT2451391.955 1.105 0.016 100 4 AAT2451391.962 1.045 0.008 93 4 AAT2451392.059 1.068 0.012 93 2 AAT2451392.957 0.905 0.012 83 1 AAT2451393.066 0.943 0.012 80 2 AAT2451393.106 0.933 0.012 83 1 AAT2451393.947 1.008 0.020 92 3 AAT2451393.958 1.013 0.016 87 2 AAT2451393.973 0.945 0.016 84 2 AAT2451393.988 0.935 0.020 84 2 AAT2451394.003 0.981 0.020 83 2 AAT2452048.801 0.901 0.012 80 3 VLT2452048.893 0.917 0.012 81 3 VLT2452076.671 1.012 0.012 89 3 VLT2452076.883 1.057 0.016 93 3 VLT2452078.722 1.013 0.016 89 3 VLT2452078.879 0.939 0.012 83 3 VLT2452079.672 1.052 0.016 93 3 VLT2452079.892 1.102 0.012 97 3 VLT2452087.621 0.952 0.016 82 3 VLT2452087.670 0.939 0.016 83 3 VLT2452087.722 0.911 0.016 78 3 VLT2452087.768 0.921 0.016 79 3 VLT2452505.290 0.978 0.020 84 4 BTA2452505.327 1.000 0.020 88 4 BTA2452505.360 1.028 0.020 91 4 BTA2452505.397 1.034 0.020 91 4 BTA 24 –Table 3: Determinations of the mean modulus | B spot | of the strong-field component. Column 1is the Julian Date, Col. 2 and Col. 3 are relative area of the spot on the disc S (in percentof the full disc area) and associated error bar obtained as a noise fraction of the flux in thetotal flux under the strong-field satellite features. Col. 4 and Col. 5 are the magnetic fieldstrength | B spot | and its error bar σ (kG) obtained as uncertainty in the determination of thesatellite positions deblended by Gaussians. Col. 6 is the telescope used.)JD S (%) σ (%) | B spot | ( kG ) σ ( kG ) Telescope2450676.955 13.1 0.6 521 40 AAT2451391.948 9.5 0.6 513 30 AAT2451391.955 12.0 1.2 495 30 AAT2451391.962 11.7 0.6 494 30 AAT2451392.059 12.0 0.6 527 15 AAT2451392.957 0.9 0.9 invisible AAT2451393.066 0.6 0.9 invisible AAT2451393.106 0.7 0.9 invisible AAT2451393.947 5.4 1.5 invisible AAT2451393.958 2.4 1.2 invisible AAT2451393.973 0.6 1.2 invisible AAT2451393.988 0.7 1.5 invisible AAT2451394.003 0.7 1.5 invisible AAT2452048.801 0.6 0.9 invisible VLT2452048.893 0.6 0.9 invisible VLT2452076.883 7.8 1.2 520 15 VLT2452078.622 3.2 1.2 invisible VLT2452078.879 0.5 1.2 invisible VLT2452079.672 8.4 1.2 529 30 VLT2452079.892 12.3 0.9 511 15 VLT2452087.621 0.6 1.2 invisible VLT2452087.670 0.6 1.2 invisible VLT2452087.722 0.7 1.2 invisible VLT2452087.768 0.6 1.2 invisible VLT2452505.290 10.4 1.2 500 35 BTA2452505.327 12.3 1.2 492 40 BTA2452505.360 12.3 1.2 524 35 BTA2452505.397 10.8 1.5 502 45 BTA 25 –Table 4: Determinations of the longitudinal magnetic field of the low- and high-field compo-nents of WD 1953-011. Column 1 is the Julian Date, Col. 2 and Col. 3 are the longitudinalfield of the low-field component and associated error bar, Col. 4 and Col. 5 are the deducedlongitudinal magnetic field of the strong-field component and its error bar (“no” means“below the detection level”), Col. 6 is the telescope used.)JD B lG ( kG ) σ ( kG ) B sl ( kG ) σ ( kG ) OBS2452048.801 -41.5 1.5 no VLT2452048.893 -39.6 1.6 no VLT2452076.883 -41.0 1.6 430 70 VLT2452078.722 -42.9 1.8 VLT2452078.879 -42.2 1.7 no VLT2452079.672 -41.9 1.6 360 60 VLT2452079.892 -46.8 1.7 460 60 VLT2452087.621 -41.5 1.6 no VLT2452087.670 -39.8 1.7 no VLT2452087.722 -40.1 1.7 no VLT2452087.768 -40.1 1.5 no VLT2452505.290 -46.2 2.3 440 80 BTA2452505.327 -44.8 2.3 450 80 BTA2452505.360 -45.0 2.5 no BTA2452505.397 -42.0 2.7 no BTA 26 –Table 5: Phase-resolved observable magnetic quantities of the weak-field component ofWD 1953-011. The first column is rotation phase φ obtained according to the magneticephemeris characterized by the rotational period of 1.448 days found here; the second andthird columns are the mean field modulus or “surface magnetic field” | B G | and associatederror bar; the fourth and fifth columns are the longitudinal field B Gl and its error bar. φ | B G | σ B lG σ ( kG ) ( kG ) ( kG ) ( kG )0.05 81.5 4 -40.5 1.30.15 82 20.25 89 30.35 94 3 -44 1.50.45 93 20.55 97 3 -47 20.65 89 30.75 86 4 -42.2 20.85 83 1 -41 10.95 78 2 -40 1 27 –Fig. 1.— Stokes IQU V H α profile timeseries of WD1953-011, obtained using the FORS1spectropolarimeter at the ESO VLT. Phases correspond to the magnetic ephemeris obtainedin this paper and are expressed in part per mil at right. The thin lines represent theobservations obtained at phase 0, and are reproduced to emphasise the variability of theStokes profiles. 28 –Fig. 2.— Stokes V at the H α line obtained with the VLT (the solid line) and the BTA (thedashed line). The spectra correspond to phases of the maximum visible strong-field Zeemansatellite features at the H α wings ( φ ≈ . φ ≈ . α profiles obtained at the VLT, AAT and BTA. High resolution spectra areconvolved to the spectral resolution of the FORSE1 and UAGS. The solid lines illustrateprofiles at two extreme rotation phases at which the spot component is most clearly seen(the shallowest profile) and where the spot component is absent (the deepest profile).Fig. 4.— The relationship EW core – | B G | obtained from the convolved high-resolutionspectra for which | B G | values are estimated by Maxted et al. (2000). The dotted line is alinear fit of the relationship. 30 –Fig. 5.— An example of the model technique as described in sect. 5. The thick solid line isthe Stokes V observed spectrum containing strong circular polarization from the strong-fieldcomponent; the dotted and dashed lines are modeled Stokes V spectra of the weak- andstrong-field components respectively; the thin solid line is their sum.Fig. 6.— Power spectrum of the magnetic field variations in WD 1953-011. 31 –Fig. 7.— The phase curve of the equivalent widths at the H α core phased with the 1.4480-dayperiod.Fig. 8.— The magnetic phase curves of WD 1953-011 and their fits with the 1.448-dayperiod. The upper plot illustrates variation of the field modulus of the weak-field component;the lower plot is variation of the its longitudinal magnetic field. 32 –Fig. 9.— Observations and modeling of mean longitudinal field (bottom panel) and meanfield modulus, or surface field (top panel). The dashed lines show a fit obtained with dipolemodel. The solid lines show the best-fit obtained by means of a dipole + quadrupole model. 33 –Fig. 10.— Phase-resolved projection S of the strong-field area: open circles illustrate thedata from the AAT, filled circles and triangles are the observations with the VLT and BTArespectively. All the data have been phased according to the magnetic ephemeris obtainedin Sect. 6. 34 –Fig. 11.— Model fits (red lines) of the observed (black lines) Stokes IV QU spectra obtainedin observations with the VLT. The corresponding tomographic portraits of the star’s mag-netosphere and rotational phase (marked as