Modelling the light variability of the Ap star epsilon Ursae Majoris
D. Shulyak, J. Krticka, Z. Mikulasek, O. Kochukhov, T. Luftinger
aa r X i v : . [ a s t r o - ph . S R ] S e p Astronomy&Astrophysicsmanuscript no. epsilon-uHa c (cid:13)
ESO 2018November 20, 2018
Modelling the light variability of the Ap star ε Ursae Majoris
D. Shulyak , J. Krtiˇcka , Z. Mikul´aˇsek , , O. Kochukhov , and T. L ¨uftinger Institute of Astrophysics, Georg-August-University, Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany Department of Theoretical Physics and Astrophysics, Masaryk University, Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic Observatory and Planetarium of J. Palisa, VˇSB – Technical University, Ostrava, Czech Republic Department of Physics and Astronomy, Uppsala University, Box 516, 751 20, Uppsala, Sweden Institut f¨ur Astronomie, Universit¨at Wien, T¨urkenschanzstraße 17, 1180 Wien, AustriaReceived / Accepted
ABSTRACT
Aims.
We simulate the light variability of the Ap star ε UMa using the observed surface distributions of Fe, Cr, Ca, Mn, Mg, Sr, andTi obtained with the help of the Doppler imaging technique.
Methods.
Using all photometric data available, we specified light variations of ε UMa modulated by its rotation from far UV to IR.We employed the LL models stellar model atmosphere code to predict the light variability in di ff erent photometric systems. Results.
The rotational period of ε UMa is refined to 5 . d a -parameter. Finally, we show thatthe abundance spots of considered elements cannot explain the observed variabilities in near UV and β index, which probably haveother causes. Conclusions.
The inhomogeneous surface distribution of chemical elements can explain most of the observed light variability of theA-type CP star ε UMa.
Key words. stars: chemically peculiar – stars: variables: general – stars: atmospheres – stars: individual: ε UMa
1. Introduction
Chemically peculiar (CP) stars are main-sequence A- and B-typestars that have a number of characteristic photometric and spec-troscopic anomalies. Slow rotation (and, consequently, weakmeridional currents), globally structured magnetic fields, andweak convection allow a build up of the prominent abundanceanomalies and inhomogeneities as a result of particle di ff usiondriven by the radiation field (first proposed by Michaud 1970).These inhomogeneities are frequently seen as abundance spotson stellar surfaces (see, for example, Khokhlova et al. 2000;Kochukhov et al. 2004; Lehmann et al. 2007), and as clouds ofchemical elements concentrated at di ff erent heights in stellaratmospheres (Ryabchikova et al. 2002, 2008; Kochukhov et al.2006; Shulyak et al. 2009, and references therein).Among the defining characteristics of CP stars, their lightvariability in di ff erent photometric systems still awaits quanti-tative explanation. That this variability is seen in independentsystems, simultaneously in narrow- and broad-band bands, andin anti-phase in far-UV and optical spectral regions suggeststhat the global flux redistribution caused by the phase-dependentabsorption is the probable cause of the observed phenomena(Molnar 1973, 1975). If so, there may be a point (or points oreven regions, see Mikul´aˇsek et al. 2007b) in the spectra wherethe flux remains almost unchanged, and vary more or less in anti-phase on both sides from it. This region, called a “null wave-length region”, has been noted and confirmed by a number of Send o ff print requests to : D. Shulyak,e-mail: [email protected] studies (see, for example, Leckrone 1974; Molnar 1973; Jamar1977; Sokolov 2000, 2006). However, that the truly “null wave-length region” in stars with more complex light curves may notexist (Sokolov 2006, 2010). The connection between the abun-dance spots and flux redistribution has emerged as a natural ex-planation of the rotationally modulated light curves of CP stars,but the details of this relation still have to be determined on asolid theoretical basis.With the appearance of detailed abundance Doppler im-ages (DI) of the stellar surfaces (e.g., L¨uftinger et al. 2003;Kochukhov et al. 2004; Lehmann et al. 2007) and advancedcomputers it became possible to numerically explore the role ofuneven distributions of chemical elements as a source of the lightvariability. First, Krtiˇcka et al. (2007) succeeded in reproducingthe light variability of a hot ( T e ff = T e ff =
1. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
In this paper, we continue to present results of the light-curvemodelling of CP stars, concentrating on the brightest A-type CPstar ε UMa (HD 112185, HR 4905). This star has been studiedextensively in the past and many good photometric observationsare available, including the data obtained by space photometryexperiments. The surface distributions of seven elements (Ca,Cr, Fe, Mg, Mn, Sr, Ti) were derived via the DI technique byL¨uftinger et al. (2003), providing the necessary basis for theo-retical modelling of the photometric variability.
2. Observations
The CP star ε UMa is a well-known spectral, magnetic, and pho-tometric variable star. The Ca ii K line and profuse lines of Cr ii vary oppositely in strength with a period of 5 . d ii minimum chosen to be the phase 0.00. Farultraviolet OAO-2 spectrometer observations of ε UMa indi-cate strong light variations in the anti-phase with optical ones(Molnar 1975; Mallama & Molnar 1977).Spectropolarimetry of ε UMa has detected its veryweak variable magnetic field whose maximum coin-cides with the optical light maximum (Borra & Lanstreet1980; Bohlender & Landstreet 1990; Donati & Semel 1990;Wade et al. 2000).Until now, almost all papers dealing with variations in thelight of ε UMa have used the Guthnick (1931) ephemeris
HJD (Ca ii min . int . ) = . + . d × E , (1)where E is the epoch.In this paper, we use light curves observed in di ff erent photo-metric systems. Str¨omgrem uvby measurements were taken fromPyper & Adelman (1985), and extended by 10-color medium-band Shemakha photometry with filters similar to standard uvby filters (Musielok et al. 1980). B and V observations closeto the Johnson standard were performed by Provin (1953).Unfortunately, no reliable data has been obtained in the broad-band U filter, the available UBV observations of Srivastava(1989) been ususable due to their excessive scatter. The sameconcerns available data in the B T and V T passbands from Tychocatalog, while Hipparcos H P photometric measurements (ESA1997) are well calibrated and were used here in the light curvemodelling.We also used photometric data derived by Molnar (1975) andMallama & Molnar (1977) using spectrometry of ε UMa per-formed by the ultraviolet satellite OAO-2 (more about the projectis given in Code et al. 1970). We also attempted to use spec-troscopic observations obtained by the International UltravioletExplorer (IUE) and Copernicus space missions available via theMultimission Archive at STScl . Finally, we used a dataset ofphotometric observations carried out by the Wide Field InfraredExplorer (WIRE) obtained during June-July 2000, although wehad to rejected several parts strongly a ff ected by instrumentale ff ects and instabilities (see Retter et al. 2004, for details). http://archive.stsci.edu/
3. Methods
To perform the model atmosphere calculations, we used themost recent version of the LL models (Shulyak et al. 2004) stel-lar model atmosphere code. For all calculations, local thermody-namical equilibrium (LTE) and plane-parallel geometry were as-sumed. The VALD database (Piskunov et al. 1995; Kupka et al.1999) was used as a main source of the atomic line data for com-putation of the line absorption coe ffi cient. The VALD compila-tion contains information about 66 × atomic transitions. Mostof them come from the latest theoretical calculations performedby R. Kurucz .The global magnetic field of the star is very weak (of polarintensity on the order of a few hundred Gauss (Donati & Semel1990; Bohlender & Landstreet 1990)), which allows us to ignoreits possible e ff ect on the variability of the outgoing flux. In previous attempts to simulate the light curves of HD 37776(Krtiˇcka et al. 2007) and HR 7224 (Krtiˇcka et al. 2009), the fol-lowing approach has been taken:1. The construction of a grid of model atmospheres for a num-ber of abundances combinations that cover the parameterspace provided by DI maps.2. Interpolation of specific intensities (or fluxes) from the gridonto a combination of abundances for all meshes visible ata given phase of rotation. The resulting flux is then obtainedby surface integration of specific intensities (or fluxes withassumed limb-darkening law).This procedure is applied to every rotational phase and wave-length interval occupied by a given photometric filter. Althoughsimple, this approach becomes challenging when modelling thelight curves of ε UMa. Indeed, in both cases of HD 37776 andHR 7224, only two elements were mapped (He and Si for theformer; Fe and Si for the latter). To produce a smooth lightcurve, it was enough to compute a grid containing tens of modelswith di ff erent sets of abundances. Obviously, with the increas-ing number of mapped elements, the number of models neededto perform abundance interpolation can become comparable oreven larger than the number of surface elements originally usedin DI calculations.Seven mapped elements of ε UMa would require time-consuming computations. We instead use a slightly di ff erent ap-proach. To reduce computational expenses, we decrease the res-olution of DI maps by interpolating them from the original gridof longitudes and latitudes onto a more sparse one. Then, for ev-ery pair of new latitude and longitude, we compute a model at-mosphere with individual abundances that represent a given sur-face mesh. Next, for every model (i.e. surface mesh), we com-pute the light curves in a particular photometric filter with thehelp of modified computer codes taken from Kurucz (1993) andextended by Hipparcos photometry with passbands from Bessel(2000). The expected magnitude at a given phase c phase is ob-tained by surface integration of individual fluxes F i from all vis-ible meshes c phase = − . U Z visible surface F i cos θ i u ( θ i ) dS , (2) http://kurucz.harvard.edu
2. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
Table 1.
Stellar parameters of ε UMa. E ff ective temperature T e ff g (cgs) 3 . i ◦ Abundance ranges from DI mapsCa [ − . , − . − . , − . − . , − . − . , − . − . , − . − . , − . − . , − . where θ is the angle between the normal to the surface elementand the line of sight, u ( θ ) describes the limb darkening for agiven filter, which we assume to be a quadratic law, i.e., u ( θ ) = − a (1 − cos θ ) − b (1 − cos θ ) , (3)and U is given by U = Z visible surface u ( θ i ) cos θ i dS . (4)We adopted the same limb darkening coe ffi cients regardless ofthe true surface elements. In particular, the quadratic coe ffi cients a and b in each color were taken from the tables of Claret (2000)for the solar abundance model. For the Hipparcos H P filter, thesecoe ffi cients were taken to be the average between the Johnson B and V filters. Since there is no definite filter curve for the WIREstar tracker, limb darkening coe ffi cients for Johnson V filter wereused instead (H. Bruntt, private communication). Because ofthis uncertainty, the WIRE photometry is presented in this pa-per mainly for illustrative purposes.The original resolution of DI maps of ε UMa is 33 latitudeand 68 longitude points equidistantly spaced between [0 , , no visible di ff erences in light curves when computingall 2244 models at the original resolution, and 450 models fromthe reduced grid (see Sect. discussion for more details). Thus, inthe remaining light curve modelling presented in this work wealways used a lower number of 450 surface elements than the2244 original.Table 1 summarizes the basic stellar parameters adopted inthe present study taken from L¨uftinger et al. (2003). The abun-dances are given relative to the total number ot atoms, i.e. ε el = log ( N el / N total ).
4. Results
Observational data used in the following analyzes of rotation-ally modulated variations of ε UMa were obtained in the past 70years during which the star has revolved more than five thousandtimes. Although we do not know the true uncertainty in the de-termination of the canonical rotational period P = . d ± . d f m a x Fig. 1.
The dependence of the primary maximum phases derivedfrom Eq. 7 on time. ∇ – BV photometry (Provin 1953), full cir-cle – UV photometry from OAO-2 satellite (Molnar 1975), ^ –10-color medium-band photometry (Musielok & Madej 1988), (cid:3) – Hipparcos photometry (ESA 1997), △ ubvy photometryof Pyper & Adelman (1985), the oblique line represents theGuthnick’s ephemeris from Eq. (1), and the star on it the phaseposition to which are referred the spectroscopic maps we used. Table 2.
Maximum phases according to individual sources ofphotometric data. ∆ ϕ is the di ff erence between new phases andphases evaluated according former ephemeris, see Eq. (8). Year max. phase ∆ ϕ N source1952.3 -0.004(10) 0.006 36 Provin (1953)1971.0 -0.000(3) 0.024 38 Molnar (1975)1975.7 -0.001(5) 0.028 179 Musielok et al. (1980)1982.1 0.012(7) 0.035 39 Pyper & Adelman (1985)1991.4 -0.006(6) 0.044 175 ESA (1997)1995.2 0.047 L¨uftinger et al. (2003) This uncertainty manifests itself in the uncertainty of 0 . p ε UMa rotational pe-riod. For this purpose, we used all available reliable photo-metric data, especially the 36 BV measurements of Provin(1953), 38 synthetic magnitudes derived from spectrometricdata in 191 nm by Molnar (1973), 179 individual observa-tions in the ten-color Shemakha medium-band system taken byMusielok et al. (1980), 80 Str¨omgreen uvby measurements ob-tained by Pyper & Adelman (1985), andthe 314 B T V T , and 122 H P measurements derived from data of the Hipparcos satellite(ESA 1997). Thus, we had 769 individual measurements cover-ing the period of 1952–1993 (see Table 2); unfortunately mea-surements from the present time were absent. Measurementstaken in various narrow- and broad-band filters were divided into10 groups according to their e ff ective wavelengths - see Table 3.We also used a very extensive data-set of WIRE photometry(Retter et al. 2004), which was unfortunately unreliable in sev-eral ways:1. We have been unable to interconnect the timing of its mea-surements with standard HJD timing.2. Because of their large scatter and instabilities, we omittedthe first part of measurements and then one day of measure-ments.
3. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris m agn i t ude Fig. 2.
Observed light curves constructed from original datain several passdands, whose e ff ective wavelengths in nm aregiven on the right. For clarity, individual LCs are shifted intheir magnitudes. Full circles – UV photometry from OAO-2satellite (Molnar 1975), ^ – 10-color medium-band photometry(Musielok et al. 1980), △ ubvy photometry of Pyper & Adelman(1985), ∇ − BV (Provin 1953), (cid:3) − Hipparcos photometry (ESA1997), and small full circles relate to broad-band photometry ofWIRE (Retter et al. 2004) - each point is the median of 1600individual time-consecutive measurements.3. We found that consecutive measurements are not indepen-dent. We therefore aggregated WIRE measurements intogroups of about 1600 members. Altogether we used 195 nor-mal points with standard uncertainty of 0.55 mmag.4. We identified significant trends in the observations, butwhich could be represented well by cubic polynomials andremoved.The double-waved light curves are more or less similar (seeFig. 2), di ff ering only in their e ff ective amplitudes A c (for a def-inition, see Mikul´aˇsek et al. 2007a), where the subscript c de-notes the photometric passband. Light-curve magnitudes canthen be expressed as m c j ( t ) ≃ m c j + A c F ( ϑ ) , (5)where m c j ( t ) is the magnitude in color c observed by the j -thobserver, m c j is the mean magnitude, which can be variable overthe long term. This was the case for WIRE measurements, wherewe had to assume a cubic trend. Table 3.
Dependence of the e ff ective amplitude of light varia-tions on wavelength. N denotes the number of measurements. λ e ff [nm] A e ff [mag] N e ff ective filter191 0.1232(25) 38 Molnar350 -0.0336(22) 38 u , U
360 -0.0325(33) 18 P
410 -0.0147(27) 39 v , X
440 -0.0190(22) 175 B , B T
465 -0.0218(18) 38 b , Y
510 -0.0220(15) 140 H p , Z
550 -0.0209(01) 407 V , y , V T , V , WIRE
635 -0.0230(33) 37 HR , S
740 -0.0205(28) 34 MR , DR λ eff [nm] e ff e c t i v e a m p li t ude [ m ag ] Fig. 3. E ff ective amplitudes of observed light curves taken invarious e ff ective wavelengths. Squares correspond to the ampli-tudes of LCs taken in individual photometric filters, full circlesare values adopted from Mallama & Molnar (1977), and crossesare e ff ective amplitudes of LCs modelled in this work.Function F ( ϑ ) is the simplest normalised periodic functionthat represents the observed photometric variations of ε UMa indetail. The phase of maximum brightness is defined to be 0.00,and the e ff ective amplitude is defined to be 1.0. The function, be-ing the sum of three terms, is described by two dimensionless pa-rameters β and β , where β quantifies the di ff erence in heightsof the primary and the secondary maxima and β expresses anyasymmetry in the light curve F ( ϑ, β , β ) = q − β − β cos(2 π ϑ ) + β cos(4 π ϑ ) + β (cid:20) √ sin(2 π ϑ ) − √ sin(4 π ϑ ) (cid:21) , (6)where ϑ is the phase function. The O-C diagram (see Fig. 1 andTable 2) indicates that the function is linear; thus, we assumed itto have the form ϑ = ( t − M ) / P , HJD (max I) = M + P × E , (7)where P is the period of the linear fit and M is the HJD time ofthe primary maximum nearest the weighted centre of all observa-tions except WIRE ones. The time of basic primary maximum inWIRE timing is then M . All 39 model parameters were com-puted simultaneously by a weighted non-linear LSM regressionapplied to the complete observational material.
4. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
We found these parameters to be P = . d , M = . , and M = . d ff ective am-plitudes A c are given in Table 3, β = . β = − . ϕ Guth can be transformed to new phases ac-cording to Eq. (7), ϕ new by means of the relation ϕ new = ϕ Guth + . × − ( t − = ϕ Guth + . × − ( T − . , (8)where t denotes the time in JD, T the time in years, and theirrespectivefractions.Figure 3 represents the dependence of e ff ective amplitudesof observed light curves on their e ff ective wavelengths. Valuessummarized in Table 3 were calculated using UV e ff ective am-plitudes derived by Mallama & Molnar (1977) from spectromet-ric observations of the OAO-2 satellite. This figure clearly illus-trate the dependence of the e ff ective amplitude on wavelength.Mallama & Molnar (1977) predicted that the ”zero point” shouldoccur at a wavelength of 345 nm. However, the ”zero point” (ifany) appears to occur at a slightly shorter wavelength.Applying weighted LSM analysis to 28 measurements of thee ff ective magnetic field published in Borra & Lanstreet (1980),Bohlender & Landstreet (1990), Donati & Semel (1990), andWade et al. (2000), we concluded that the observed sinusoidalvariations can be satisfactorily reproduced by the simple modelof centered magnetic dipole inclined by the angle β to the ro-tational axis (see Fig. 4). The north magnetic pole then passesthe central meridian at the phase ϕ = − . β and the inclination of the rotational axis with respect to the ob-server i , to be tan( β ) = . i ). Assuming (together withL¨uftinger et al. 2003) that i = ◦ , we found that β = ◦ (6 ◦ ),thus the north magnetic pole is located close to the center of theleading photometric spot on ε UMa, which is unlikely to be acoincidence.The magnetic field of ε UMa is the weakest measured mag-netic field in magnetic CP stars which would be hardly de-tectable in any more rapidly rotating and fainter star. Its max-imum (polar) surface magnetic field is weaker than 400 Gauss.Thus we could neglect its influence in our computation.
Before presenting the results of the light curve modelling, we at-tempt to verify model predictions about the impact of individualelements on overall energy distribution. Figure 5 illustrates thesynthetic energy distributions for models computed with maxi-mum, minimum, and mean abundances of every element fromDI maps. This exercise illustrates the maximum possible energyredistribution e ff ect caused by enhanced chemistry inside a spot.From Fig. 5, one can note two general features. First, there areonly two elements that have (on average) the strongest e ff ect onradiative energy balance: Fe and Cr. This is mainly because theyhave the largest number of strong lines in the UV and thus ef-fectively block radiation at those frequencies. Absorbed energyis then redistributed at optical wavelengths. This is illustratedby the positive delta’s of uvby passbands shown on every plot.For Cr, the di ff erence in the y -band between models computed −0.2 0 0.2 0.4 0.6 0.8 1 1.2−100−50050100150 rotational phase B e ff [ G ] Fig. 4. E ff ective magnetic field and its uncertainty in gaussversus rotational phase. Note that the field is extraordinarilyweak and the phase of its maximum coincides exactly withthe maximum of brightness. Measurements taken form: ◦ –Borra & Lanstreet (1980), ^ – Bohlender & Landstreet (1990), △ – Donati & Semel (1990), (cid:3) – Wade et al. (2000). Areas of in-dividual markers correspond to the weight of the particular mea-surement, which is inversely proportional to the squares of theiruncertainties.with mean and maximum Cr abundance is almost zero due to thestrong Cr absorption features around 5500Å that keep fluxes al-most unchanged for the highest abundance of [Cr] = − .
56 dex.This never happens for Fe, which has a smooth flux excess in uvby bands. Both elements contribute significantly to the 5200Åflux depression frequently found in CP stars (see Kupka et al.2003, and references therein).Models with enhanced Mg, Mn, and Sr cause only marginalor small changes in photometric filters. Interestingly and in con-trast to any other element, overabundant Sr leads to a noticeableflux deficiency in the v parameter because of some very strongSr ii λ , , uby passbands exhibit a farlower sensitivity to abundance changes (the same applies to the v filter in the Ca enhanced model because of the Ca ii H & K lines,but with much smaller amplitude). This e ff ect is even compara-ble to that of enhanced Fe. However, the mean abundance of Sris only − . ffi cult to expect any notice-able impact of it on the resulting light curve. This is investigatedin more detail later in this work. Finally, the contributions ofCa and Ti to the magnitudes of Str¨omgren photometry are verysmall.Although the relative roles of di ff erent elements can be esti-mated by the aforementioned simulations, to obtain quantitativeresults it is still necessary to carry out accurate modelling of lightcurves by taking into account the individual contributions of allspotted elements. Using DI maps of individual elements, we verified the relativecontribution of each of them to the total light variation. Theresult is presented in Fig. 6. We note that WIRE data weresmoothed over 200 points to provide more or less a reasonableview, yet some instrumental e ff ects are still visible.As expected from the previous paragraph, among the sevenmapped elements there are only two that strongly contribute to
5. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
Fig. 6. Di ff erential e ff ect of inhomogeneous surface distributions of individual elements on light variations of ε UMa. Fe – fullthick, Cr – dashed, Ca – dash-dotted, Mg – dash-dot-dotted, Mn – long dashed, Sr – dotted, Ti – thin dashed.
Left panel: (cid:3) –Pyper & Adelman (1985), ^ – Musielok et al. (1980). Right panel: ^ – Provin (1953), △ – Retter et al. (2004), (cid:3) – ESA (1997). See online version for color figures. the amplitude of light curves in all presented photometric pa-rameters: Fe and Cr. Next follows Mn, whose contribution canonly be recognized by a keen eye (mostly in b and V filters).The impact of the remaining elements is negligible. The cumu-lative impact of all seven elements on the light curves of ε UMais presented in Fig. 7 (thick solid line). A very good agreementin terms of both the shape and amplitude of the light variabilityis found for almost all photometric bands (see Fig. 3). The onlyexceptions are passbands shortward of the Balmer jump, namelyStr¨omgren’s u , Shemakha U , and P for which the observedamplitude is approximately two times higher than the predictedone (see Fig. 3). In principle, this may be a signature of someother elements that were not mapped in L¨uftinger et al. (2003),but still contribute to the light curve. Interestingly, a poorer fit tothe u passband than to others was also reported by Krtiˇcka et al.(2009) (see their Fig. 10). Nevertheless, the amplitudes in otherpassbands are well reproduced in both the present work andthose cited above. Taking into account the large scatter in theobserved points in general for the ubvy system, it is di ffi cult toexplain the observed discrepancy in u -band in terms of impact ofsome missing element, at least without additional DI mapping, ifever possible. Finally, Hipparcos photometry exhibits significantscatter with a peak amplitude that is a little larger than the onetheoretically predicted.The forms of the simulated LCs agree with the observed onesvery well, all of the modelled curves being systematically shiftedwith respect to the observed light curves by + . p As can be seen from Fig. 5, the main source of light variability isthe energy redistribution from UV to visual due to the enhancedabundances of certain elements. Thus, the fluxes in these twospectral regions should display an anti-phase behaviour. Thishas been experimentally already reported for some CP stars (seeSokolov 2000, 2006, for CU Vir and 56 Ari, respectively). Lightcurve modelling of HD 37776 and HR 7224 has also success-fully predicted this characteristic signature of stellar spots.For ε UMa, we again had a problem with the lack of well-calibrated phase-resolved UV observations that could be used tostudy the e ff ect of flux redistribution. The IUE archive mostlycontains spectra obtained with a small aperture. This means thatsome of the stellar flux was missed during individual exposures,which is important to note in our study of the true variations inthe continuum flux. Unfortunately, the only two spectra obtainedwith a large aperture (LWR06920RL and SWP07944RL) wereobtained at the same rotational phase ( ϕ = . −
6. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
Fig. 7.
Same as in Fig. 6, but taking into account surface distributions of all elements simultaneously. Theoretical predictions areshown for linear (dash-dot-dotted) and quadratic limb darkening laws based on the assumptions of [M / H] =+ . / H] = . / H] = − . − λ ff er-ent wavelength and, even worse, at di ff erent phases. We werethus unable to choose a unique spectral region that had been ob-served at a su ffi ciently large number of rotational phases (thecase would be a narrow region around 1200Å, but only 6 phasesare available and with large errors). Nevertheless, to confirm theanti-phase behavior of the UV flux relative to the visual, we per-formed an exercise similar to those presented in Fig. 5 of Molnar(1975), which illustrates the light curve of the OAO-2 λ ≤ λ ≤ FWHM of the OAO-2 S3F1 filter (see Meade1999). The theoretical magnitudes are shown in Fig. 8. The V -filter observations by Provin (1953) are also plotted. The oppo-site variations in the optical and UV fluxes with respect to eachother can be clearly seen. The double wave behavior of the sim-ulated OAO-2 curve is in excellent agreement with those pre-sented in Molnar (1975), yet the theoretical amplitude is smallerby a factor of two. We emphasize that there is a systematical de-viation between the predicted and observed magnitudes of theOAO-2 photometry, as illustrated in Fig. 3. Theoretical compu-tations display a large scatter blueward of log λ = .
4. At thesame time, some predicted points do follow the observed lineartrend of the dependence of amplitude on wavelength in UV re-gion. This is the case for filters at log λ = . , . , .
5. We note,however, that we did not attempt a detailed quantitative com-parison between the observed and modelled light curves or their amplitudes (also because the original OAO-2 filter data, as wellas the filter values, are not available online), but only confirmedthe anti-phase variations in UV and optical fluxes from the mod-elling point of view. a -system The photometric parameter a is often used as a measure of thepeculiarity in CP stars. It was introduced by Maitzen (1976) andcorresponds to the well-known flux depression at 5200Å fre-quently seen in spectra of CP stars. Using model atmosphere cal-culations, Kochukhov et al. (2005) noted and Khan & Shulyak(2007) later confirmed (on the basis of more detailed computa-tions) that Fe is the main contributor to the 5200Å depression inthe temperature range of CP stars. In addition, at low tempera-tures Si and Cr are also important. For HR 7224, Krtiˇcka et al.(2009) also identified the dominant role of Fe relative to Si. Wecarried out the same investigation and illustrate in Fig. 9 the im-pact of di ff erent elements on the variation in the a -index (notethat, since the a -index is positive, the negative delta value in theplot correspond to the higher a at corresponding phases). Wethus confirm the major role of Fe, although, the contribution ofCr is very important too. Moreover, at phase ϕ = . a -index appears to be small, on theorder of a few mmag. A similar weak variation was reported forthe hotter star HR 7224 (Krtiˇcka et al. 2009).
7. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris
Fig. 5. E ff ect of individual elements on the synthetic energydistribution. The di ff erence in fluxes in Str¨omgren uvby pass-bands between models computed with mean abundances andmaximum abundance of the particular is presented on each plot.Fluxes were convolved with FWHM =
20Å Gaussian for betterview. The passbands (corresponding to the width of individualfilter at half-maximum) of uvby filters are indicated by grayedboxes.
Fig. 8.
Simulated light curves in the region of Copernicus OAO-2 λ V -band photometry by Provin (1953).The simulated light curve has been shifted along the y-axis for aclear visualization. Fig. 9.
Impact of di ff erent elements on the light curve of the pe-culiar a parameter. Fe – full thick, Cr – dashed, Ca – dash-dotted,Mg – dash-dot-dotted, Mn – long dashed, Sr – dotted, Ti – thindashed. Cumulative impact of all elements is shown by the thickfull thick line with highest amplitude. We found a very small variation in the β index on the stan-dard system defined by Crawford & Mander (1966) related tothe Balmer H β line, with a peak amplitude of ≈ . ff ecting mainly profilesof hydrogen lines, as long as other photometric parametersare well fitted. For example, the rotational modulation of hy-drogen lines may be caused by a non-zero magnetic pres-sure in the stellar atmosphere, as reported by Shulyak et al.(2010, 2007). An overview of other possible mechanisms canbe found in Valyavin et al. (2004) and we refer the interestedreader to this work for more details. Additional observationsof the β index from Musielok et al. (1980) have large errorsand have significant scatter preventing the direct comparisonwith Musielok & Madej (1988). Last but not least, the data fromMusielok & Madej (1988) contains only three points at the max-imum of the β light curve and a definite outlier at ϕ = .
7, whichshould be interpreted with caution. We thus conclude that, if true,
8. Shulyak et al.: Modelling the light variability of the Ap star ε Ursae Majoris the high amplitude of the observed variability in the β index can-not be attributed to abundance anomalies alone.
5. Discussion ε UMa is the third CP star for which a model-based light-curveanalysis predicts excellent agreement between theory and ex-periment, based on the assumption of the inhomogeneous sur-face distribution of chemical elements derived by DI techniques.The same analysis of two other and hotter Bp stars HD 37776(Krtiˇcka et al. 2007) and HR 7224 (Krtiˇcka et al. 2009), but us-ing slightly di ff erent methods, has also uniquely found that abun-dance spots may be a dominating mechanism of light variabilityin CP stars.The maximum di ff erences between the predicted and ob-served light curves are seen in Str¨omgren u and Hipparcos H P bands, as illustrated in Fig. 7. At the same time, other photo-metric parameters are fitted well enough to exclude any otherelement significantly influencing the light curves. For instance, BV parameters from Provin (1953) probably provide the clos-est fit, again, within the error bars of the observations (see threeconsecutive points at phase 1 .
0, for instance). These error barsare larger for Str¨omgren passbands. Krtiˇcka et al. (2009) find thesame larger discrepancy in u but for the hotter star HR 7224. Thisensures that the impact of any missed element is less significant,yet not negligible. In principle, taking into account the relativeimportance of di ff erent elements described in Khan & Shulyak(2007), Si may represent such a missed element, but it could notbe mapped in L¨uftinger et al. (2003). The calibration of the u -band may also be problematic. Nevertheless, the remaining vby passbands can be fitted much better without the need to involveother elements in the modelling.As described above, to reduce computational time we de-creased the resolution of the original DI maps from 2244 sur-face elements to 450. The characteristic di ff erence in computedmagnitudes between these two sets of maps was found to be ∆ c ≈ × − mag, which is well below the detection limit.The choice of the limb darkening model used to reconstructthe light curves in di ff erent photometric bands can also influencethe amplitude of variations. In our study, we used a quadraticlaw with coe ffi cients for every photometric band (except thosedescribed above in the paper) taken from the solar abundancemodel. However, neither metallicity nor the choice of limb-darkening law seem to significantly a ff ect the light curves, norcan they explain the discrepancy in u . This is illustrated in Fig. 7where, as an example, we overplot the predictions of the lin-ear and quadratic limb darkening laws and the solar abundancemodel, as well as for the quadratic law with [ M / H ] = + . M / H ] = − . M / H ] = + .
6. Conclusions
We have successfully simulated the light variability of Ap star ε UMa caused by the inhomogeneous surface distribution ofchemical elements detected by the DI maps of L¨uftinger et al.(2003). A very close agreement in both the amplitude and shapeof the light variability has been found for di ff erent and inde-pendent photometric systems. Our main conclusions are sum-marized below: – The presence of abundance spots on the stellar surface is themajor contributor to the light variability. We did not intro- duce any free parameters to improve the agreement betweentheory and observations. – The light variability is due to the flux redistribution from theUV to visual region, which is initiated by the presence ofabundance spots. – We support the conclusion of Khan & Shulyak (2007) thatnumerous lines of Fe and Cr are the main contributors to thewell-known depression around 5200Å. – The variation in the theoretical β index is very weak andapproximately one order of magnitude smaller than the ob-served one. We thus conclude that inhomogeneous distribu-tion of chemical elements alone cannot explain the observedlight curve in the β index. – The strong contribution of Fe and Cr to the flux variationsallows us to conclude that, as for the hotter Bp stars thathave been analysed so far (HD 37776, T e ff = T e ff = ε UMa. This mustof course be tested in stars with values of T e ff lower than T e ff = ε UMa.
Acknowledgements.
We would like to express our gratitude to Drs. TimothyBedding and Hans Bruntt for kindly providing us with data from WIREmission. This work was supported by the following grants: DeutscheForschungsgemeinschaft (DFG) Research Grant RE1664 / ffi ce of Space Science via grantNNX09AF08G and by other grants and contracts. References
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