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Please be advised that this information was generated on 2021-10-03 and may be subject to change. arXiv:0903.5477v2 [astro-ph.SR] 3 Apr 2009 iiu, Belgium. tifique, ehi tr, cas f aibe al Btp stars ders B-type early E-mail: * variable of class a stars, Cephei ß 2008 Received 2008. Accepted (2002) 1-15 000, Soc. stron. A R. Not. Mon. Cecer ulfe u od Ntoa d l Rcece Scien­ Recherche la de National Fonds au Qualifie Chercheur Flan­ J Research, Scientific for Fund the of Fellow Postdoctoral f Yang S. L. S. athias10, M P. Uytterhoeven13, K. Masuda9, S. Telting12, J. Lehmann8, H. Mkrtichian11, E. Krzesinski7, D. J. Kambe6, E. Ilyin5, I. odelling m ic seism and n C Aerts C. and 12 (DD) Lacertae (12 Lac) is one of the best observed observed best the of one is Lac) (12 Lacertae N IO T (DD) C U 12 D O R T IN 1 Handler4, G. at3, C De P. Zima1, W. Thoul2|, A. identification Briquet1!, M. ode Desmet1*, m M. observations, e th of spectroscopic y d stu ic ultisite m asteroseism n A KalShazcidObevtru Tëigr adstrwat, 78 atnug Gemany erm G Tautenburg, 7778 arte, Landessternw ustria A Thëringer ien, W , 1180 17, bservatorium arl-Schwarzschild-O K Türkenschanzstrasse 8 ien, W niversistët U ie, stronom A r fë t stitu In 4 Tksi Sine su As amu Ln Tksia 4 - 2 iiaai Nt, tn-h, tn-u, ouhma 7-11 Japan 779-0111, a Tokushim Itano-gun, Itano-cho, Nato, Kibigadani, 22 - Japan 45 Poland Cracow, 719-0232, Tokushima, 30-084 Okayama Land 2, u m ta su A Kamogata, Podchorazych any , Ul. erm G useum M , Observatory, University, Potsdam ical Science Belgium a Pedagogical D-14482 Astronom Liège, 16 Tokushim Cracow 9 4000 ational arte N Sternw Août, 6 Observatory, der du Observatory, n A Suhora Allée , t. 17, M 7 Potsdam Astrophysical Liège, de Okayama Institute 6 ’Université l de Astrophysical 5 Géophysique de et ’Astrophysique 3 d t stitu In 2 4 prmet fPyis n Atoo , iesiy f itra Vcoi, P, Canada 3P6, W 8 V C B Victoria, Victoria, of Ukraine Spain niverstity U Odessa, y, Palma, 65014 La 1v, de Astronom Cruz and arazlievskaya, Santa M Physics of 15 38700 ent 474, University, epartm D 14 ational Apartado N Odessa Telescope, of 13 Optical Observatory Nordic 12 ical Astronom 11 10 1 nsiut o too - evn Ceetjela 20, 01 evn Belgium Leuven, 3001 200D, elestijnenlaan C Leuven, U K - y stronom A of te stitu In nnlje trewah vn egë Rnla 3 18 Busl Belgium Brussel, 1180 3, Ringlaan België, van acht Sterrenw oninklijke K nttt d Asrßia e aais Cle i Lce sn, 80 L Lgn (F, Spain) , (TF Laguna La 38205 , s/n Lactea Via Calle Canarias, de stroßsica A de Instituto prmet fAtohsc, iest o imee, O o 91, 50 L imee, h Netherlands The egen, Nijm GL 6500 9010, Box PO egen, Nijm of niversity U Astrophysics, of ent epartm D N, NR, C, aps ars, M 62 H Fza, -60 Nc Cdx , France 2, Cedex Nice F-06108 Fizeau, H. 6525 UMR Valrose, Campus OCA, RS, CN UNS, [email protected] 1,15 respectively. O ur seismic modelling shows th a t f 2 is likely the radial first overtone and and overtone first radial the eter likely is 2 f param t ain a m th four overshooting shows frequency core the the ith modelling w t identify a seismic We th ur g-mode O spectroscopy. SPB-like respectively. (1,1), our the in = 1) detected (11,m is particular, as In before modes reported frequencies. -1 d 0.3428 bination com and ics tr: interiors stars: star Cephei 3 the for paign cam ultisite m spectroscopic a of results the present We for for S/N spectra gathered w ith 8 different telescopes in a tim e span of 11 m onths. In In onths. m 11 confirm of We span ents. e tim easurem m a in spectroscopic telescopes archival erous different num 8 of use ith w make we gathered addition spectra S/N T C A R T S B A K e y w ords: ords: w y e K 12 (DD) Lacertae. O ur study is based on more th an thousand high-resolution high high high-resolution thousand an th more on 10 based is study ur O Lacertae. (DD) 12 needn feunis eety icvrd rm htmer, s el s amon­ harm as well as etry, photom from discovered recently frequencies independent 1 f .794-1, = 5342d , 1 5.334224d- = 2 f , 1 5.178964d- = itd Api 2009 pril A 3 rinted P tr: ansqec - sas idvda: 2 a - sas oclain - - oscillations stars: - Lac 12 individual: stars: - main-sequence stars: l, 2)(l2,m ß i rcnl. hns o n nesv pooerc ulti­ m un­ photometric achieved intensive not an was to modes campaign, Thanks site observed recently. the til of tification order been has to star radial refer the We which low studied. during of grav­ century extensively to one for due pulsations known are been changes pressure-mode velocity and radial ity and light whose very detailed summ ary of the work accomplished in the past. the in accomplished work the of ary summ detailed very Sakv Hnlr 2005 Handler & (Stankov pe sa 1 Lacertae: 12 star ephei C M IAT (MN ept te ueos ale suis sf md iden­ mode safe studies, earlier numerous the Despite 00, 1, ) 10 ad (14 and (1,0) = 3) (13,m (0,0), = aov is lower th an 0.4 local pressure scale heights. scale pressure local 0.4 an th lower is e X 3 f tl fl v2.2) file style ade e a. (2006 al. et Handler = 5.066316d-1 and and 5.066316d-1 = ). The variability of 12 Lac has has Lac 12 of variability The ). ade e a. (2006 al. et Handler 14 uabgosy iden- unambiguously ) f4 .913-1, 1 5.490133d- = ,m4) = (2,1) (2,1) = fr a for )

2 M. Desmet et al.

0.75 ------1------1------1------1------4550 4551 4552 4553 4554 4555 Wavelength [A]

F igure 1. The average profiles of the Si III A4553 A line from each observatory. The dashed line shows the average profile from the BAO observatory. This profile deviates from all the others because the relatively small number of spectra implies that the beat cycle is not well covered (see Table 1).

Wavelength [A] tified the l-values of the five modes with the high­ Figure 2. The Si III line profiles of 1 night (JD 2452898) taken est photometric amplitudes. In addition, they found con­ with the McDonald telescope. The arrow indicates the flux scale straints on l for the six other independent modes de­ of one spectrum. tected in their dataset. Their l-identifications ruled out the assumption previously adopted for stellar modelling (e.g. Dziembowski & Jerzykiewicz 1999) that three of the strongest modes, almost equidistant, belong to the same over the northern continents. Table 1 summarizes the log­ multiplet. Indeed, these three modes are actually associated book of our spectroscopic data. The resolution (A/AA) of to three different values of the degree l. the instruments ranged from 30 000 to 80 000 and the aver­ Obviously, reliable empirical mode identification was in­ age S/N ratio near 4500 A between 180 and 380. dispensable before any attempt of in-depth seismic mod­ All data were subjected to the normal reduction pro­ elling of 12 Lac. To complement the photometric results, a cess, which consists of de-biasing, background subtraction, spectroscopic multisite campaign has also been devoted to flat-fielding and wavelength calibration . Finally, the helio­ the star. The additional constraints are presented in this pa­ centric corrections were computed, and all spectra were nor­ per. They mainly concern the identification of m-values for malized to the continuum by fitting a cubic spline function. the strongest modes and the derivation of the stellar equa­ For our study, we considered the Si III triplet around torial velocity. A detailed abundance analysis of 12 Lac was 4567 A because its characteristics simplify the modelling already presented in Morel et al. (2006), showing that the of the line-profile variations, which we use for mode iden­ abundances of all considered chemical elements are indistin­ tification. Indeed, these silicon lines are sufficiently strong guishable from the values found for OB dwarfs in the solar without being much affected by blending. Moreover, they neighbourhood. are dominated by temperature broadening, such that the Besides our line-profile study, we also describe a detailed intrinsic profile can simply be modelled with a gaussian. Fi­ stellar modelling based on all available observational results, nally, they are not too sensitive to temperature variations state-of-the-art numerical tools and up-to-date physical in­ (see De Ridder et al. 2002; Aerts & De Cat 2003), so that puts appropriate to model 0 Cephei stars, as explained in neglecting them remains justified. detail in Miglio (2007). Dziembowski & Pamyatnykh (2008) The similar average radial velocity of about -15 km s- 1 already computed models for 12 Lac, making some assump­ for 1990-1992, 2003-2004 and earlier measurements (e.g. tions and restrictions not supported by our study. In the Wilson 1953) definitely excludes the possibility of 12 Lac present paper, we discuss our conclusions which differ from being a spectroscopic binary. Besides the pulsation effect, those of Dziembowski & Pamyatnykh (2008). the different centroids are due to the different zero points of the different telescopes. Before extracting the pulsation in­ formation from the whole dataset, it is necessary to correct for this effect. The spectra were shifted in such a way that 2 OBSERVATIONS AND DATA REDUCTION the constant of a least-squares sine fit using the first three The data originate from a spectroscopic multisite campaign dominant modes is put to the same value for each observa­ dedicated to our studied star. In addition, we added the tory. 903 spectra of Mathias et al. (1994), which allowed us to In Fig. 1 we superimpose the average profiles of the Si III increase the time span and, thus, to achieve a better fre­ A4553 A line computed for each observatory separately, af­ quency accuracy. In total 1820 observations were gathered ter correction for the different zero points. We note that the using 9 different small- to medium-sized telescopes spread centroid of the lines are in good agreement but not exactly An asteroseismic study of the ß Cephei star 12 Lacertae 3

HJD -2452000

F igure 3. The densest part of the time series of the Si III radial velocities ((v1)) of 12 Lac derived from each spectrum taken during the dedicated multisite campaign (7 Aug. 2003 - 17 Nov. 2003).

Table 1. Log of our spectroscopic multisite campaign. The Julian Dates are given in days, AT denotes the time-span expressed in days, N is the number of spectra and S/N denotes the average signal-to-noise ratio for each observatory measured at the continuum between 4500 and 4551 A.

Observatory Long. Lat. Telescope Julian Date Data amount and quality Observer(s) (Name of the instrument; resolution) Begin End AT N S/N

2003-2004 -2450000 Apache Point Observatory -105° 49' +32°46' 3.5-m 2927 2928 2 93 300 JK (ARC; 31500) Bohyunsan Astronomical Observatory + 128° 58' +36°09' 1.9-m 3109 3109 1 27 180 DM (BOES; 30000) Dominion Astrophysical Observatory -123° 25' +48°31' 1.2-m 2858 2859 2 66 250 SY (45000) 2909 2911 3 42 McDonald Observatory -104° 01' +30°40' 2.1-m 2893 2901 9 210 380 GH (Sandiford; 60000) 2.7-m 3195 3198 4 31 350 GH Nordic Optical Telescope -17° 53' +28°45' 2.6-m 2947 2951 5 32 370 KU, JT, II (SOFIN; 80000) 2957 2960 4 47 Okayama Astrophysical Observatory +133° 35' +34° 34' 1.88-m 3201 3202 2 7 180 SM (HIDES; 685000) 3205 3205 1 6 3208 3210 3 17 Thüringer Landessternwarte Tautenburg + 11°42' +50°58' 2-m 2863 2869 7 103 240 HL (67000) 2886 2889 4 100 2895 2900 6 105 2914 2918 5 31 1990-1992 -2440000 Observatoire de Haute-Provence +5°42' +43° 55' 1.52-m 8135 8135 1 70 220 PM (AURELIE; 42000) 8224 8224 1 52 8227 8227 1 35 8845 8858 14 582 8884 8889 6 164

Total 82 1820

at the same position. Larger differences can be seen for the the McDonald telescope during only one night. In Fig. 3, we width and the depth of the lines. Such deviant average pro­ show the radial velocities for the densest part of the multi­ files are due to the limited time spread of the observations site campaign (7 Aug. 2003 - 17 Nov. 2003). A clear beating together with the multiperiodic character of the pulsations. pattern is seen. In the case of a similar spectroscopic multisite campaign devoted to v Eridani, A erts et al. (2004) observed similar differences, which are not caused by instrumental and/or signal-to-noise ratio effects. 3 FREQUENCY ANALYSIS Fig. 2 displays the large and complex radial-velocity We performed a frequency analysis on the first three ve­ variations of 12 Lac. The plotted spectra were taken with locity moments (v1), (v2) and (v3) (see Aerts et al. 1992, 4 M. Desmet et al.

Table 2. Frequencies and radial velocity amplitudes of the tional independent frequency (fp = 5.30912 d-1) was found first moment of the Si III A4553a line together with their S/N by the latter authors and not by us. An important result ratio (we refer to the text for explanation). Error estimates is that we clearly recover the low-frequency signal possibly (Montgomery & O’Donoghue 1999) for the independent frequen­ originating from a g-mode. Such a SPB-like oscillation does cies range from ±0.000002 d- 1 for f 1 to ±0.00002 d- 1 for f g. The not seem to be uncommon among 0 Cephei stars. For in­ error on the amplitude is 0.01 km s- 1. stance, it was also observed in 19 Mon (Balona et al. 2002), in v Eridani (Handler et al. 2004), in S Ceti (Aerts et al. ID Frequency Amplitude in (v1) S/N 2006) and in V1449 Aql (Briquet et al., in preparation). We [d- 1 ] [i‘Hz ] [km s 1 ] also point out that such hybrid SPB-0 Cep pulsators are the­ oretically predicted for hotter stars than previously thought fl 5.178964 59.941713 14.50 99.6 (Miglio et al. 2007 versus Pamyatnykh 1999). f 2 5.334224 61.738704 7.70 52.2 f3 5.066316 58.637917 6.26 42.7 The equivalent width (EW) of the silicon line of 12 f4 5.490133 63.543206 2.61 17.5 Lac clearly varies with the two dominant modes f 1 and f 2 . fg 0.342841 3.968067 1.34 7.1 The sum frequency f 1 + f 2 is even present. Fig. 5 represents h 4.241787 49.09475 0.91 6.7 phase diagrams for both frequencies. Such a strong EW vari­ f 6 5.218075 60.394387 0.84 5.8 ability of 9% is generally not observed in pulsating B-type f 7 6.702318 77.573125 0.62 4.4 stars. Indeed, typical relative EW peak-to-peak amplitudes f8 7.407162 85.731042 0.67 4.9 are of the order of a few percent (De Ridder et al. 2002; 5.84511 67.65184 0.79 5.2 f9 De Cat & Aerts 2002). However, for several 0 Cephei stars, 2fi 10.35814 119.88590 0.63 4.5 one clear sinusoidal variation is observed in their EW, with f 2 + f3 10.40056 120.37693 0.72 3.8 the same frequency as in the radial velocity. The known 10.51319 121.68044 2.59 19.7 f 1 + f 2 cases are targets pulsating multiperiodically with a high­ 2fl + f3 15.42400 178.51860 0.72 6.6 amplitude dominant radial mode, such as S Ceti (Aerts et al. fl + Ï2 + Ï3 15.57950 180.31838 0.71 6.4 1992), v Eridani (Aerts et al. 2004) and V1449 Aql (Briquet et al., in preparation). In the case of 12 Lac, we note that the EW signal is multiperiodic with a dominant (1,1)-mode, fol­ for a definition of the moments of a line profile) of lowed by a radial mode of lower amplitude (see next section). the Si III A4553 A line by means of the program Pe- For the four strongest modes, the frequency values derived riod04 (Lenz & Breger 2005). For some 0 Cephei stars from our spectroscopic measurements are exactly the same, (see Telting et al. 1997; Schrijvers et al. 2004; Briquet et al. within the errors, as the photometric ones by Handler et al. 2005) a two-dimensional frequency analysis on the spec­ (2006). For the modes with radial velocity amplitudes typ­ tral lines led to additional frequencies compared to a one­ ically lower than 1 kms-1, there are differences between dimensional frequency search in the moments. For 12 Lac values derived from both kinds of data, which might have 1D analysis on (v2) and (v3) did not reveal additional inde­ some influence on the mode identification outcome. Because pendent periodicities compared to those present in the first the number of observations as well as the time span of the moment. We also do not find additional frequencies in the datasets in the literature are much larger than the dedi­ 2D frequency analysis. The 2D analysis is more sensitive to cated spectroscopic ones, the literature frequency values are the detection of high-degree modes than a 1D analysis so expected to be more accurate and hence we adopt them. here we have already an indication that the frequencies are low-degree modes. In what follows we only describe our frequency analysis on (v1), which is the radial velocity from which the fitted 4 MODE IDENTIFICATION averaged radial velocity is subtracted. 4.1 The methods used The frequencies were determined with the standard method of prewhitening the data. In addition we made use We used two different and independent methods to identify of a procedure available in Period04 which allows to improve the modes, namely the moment method (Briquet & Aerts the detected frequency from non-linear least-squares fitting 2003) and of the Fourier parameter fit method (FPF with the maxima in the Fourier transform as starting val­ method, Zima 2006). ues. At each step of prewhitening we subtracted a theoretical The basic principles of both methods are the follow­ multi-sine fit with the amplitudes, phases and also optimized ing. With the moment method the wavenumbers (l,m) and frequencies that yielded the smallest residual variance. All the other continuous velocity parameters are determined in peaks exceeding an amplitude signal-to-noise ratio above 4 such a way that the theoretically computed first three mo­ in the Fourier periodogram (Breger et al. 1993, 1999) were ment variations of a line profile best fit the observed ones. retained. The noise level was calculated from the average In the FPF method, for every detected pulsation frequency, amplitude, computed from the residuals, in a 2 d - 1 interval the zero point, amplitude, and phase are computed for ev­ centered on the frequency of interest. ery wavelength bin across the profile by a multi-periodic The detected frequencies are listed in Table 2 and the least-square fit. Subsequently the zero-point profile, the am­ Fourier periodograms are shown in Fig. 4. Our frequency plitude and the phase values across the profile are fitted with analysis reaffirmes the presence of the already well-known theoretical values derived from synthetic line profiles. The 5 main frequencies (M athias et al. 1994) together with the first technique has the advantage to be less computation­ new independent signals and combination frequencies dis­ ally demanding, allowing to test a huge grid for (l, m) and covered by Handler et al. (2006). We note th a t one addi­ for the other parameters. Moreover, the modes are deter­ An asteroseismic study of the 3 Cephei star 12 Lacertae 5

Frequency [d "1]

Figure 4. Amplitude spectra of 12 Lac computed from the first moment of the Si iii A4553 A line. The uppermost panel shows the spectral window of the data. All subsequent panels show the periodograms in different stages of prewhitening. The significance limit in red is calculated according to 4 times the noise over a 2d -1 -range around the significant frequency.

mined simultaneously by fitting a multiperiodic signal tak­ Such a methodology already proved to be successful for two ing into account the coupling terms appearing in the second other 0 Cephei stars (Briquet et al. 2005; M azumdar et al. and third moments. For the second technique, the fitting is 2006). Mode identification results have been obtained with carried out by applying genetic optimization routines in a the software package FAMIAS1 (Zima 2008). large parameter space and mono-mode fits are used in order In what follows we present in detail our mode iden­ to speed up the computations. Once the best solutions are tification by the FPF method. The moment method gave constrained, a multi-mode fit is performed. The strength of compatible results, but did not give additional constraints the FPF method is its ability to estimate the significance on the (l, m) than the ones we present here. Both techniques of the derived mode parameters by means of a x 2 test (all found the same m-values for the three main modes, giving X2-values listed in the text are reduced x 2 values, see Zima us much confidence in our outcome. For the fourth mode, 2008). the moment method derived the sign of m in agreement with The FPF method is not optimally suited to study high­ the FPF method, which could further constrain its value. amplitude modes, but as described further, the applica­ tion of the FPF method for 12 Lac proved to be succesful. As shown by Zima et al. (2004) and by Zima (2006), both 4.2 Derivation of the wavenumbers mode identification techniques constrain the value of the az­ For the FPF method, a sophisticated theoretical formalism imuthal number m better than the degree I for low v sin i. for the modelling of the displacement field is used and in- Since the photometric data provided us with unambiguous l-values for the 5 main modes (Handler et al. 2006), a valid strategy is to make use of spectroscopy to identify the val­ 1 FAMIAS has been developed in the framework of the FP6 Euro­ ues of m, adopting the l-values obtained from photometry. pean Coordination Action HELAS - http://www.helas-eu.org/. 6 M. Desmet et al.

0.265 0.26 0.255 0.25 „ 0.245 g 0.24 LU 0.235 0.23 0.225 0.22 0.215 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Phase (5.179034 d-1) Velocity [km s 1] 0.025 ------F igure 6 . Observed zero-point profile (points) and best fit when taking into account the four main modes (solid line) by means of 0.02 a synthetic rotationally broadened profile (v sini = 36.7kms-1 , 0.015 W =15.3kms- 1 , u = 22.1 km s-1 ).

0.01

< 0.005

§ 0 ing our appropriate grid. Moreover v sin i will be overesti­ -0.005 mated. When taken into account the pulsational broaden­ -0.01 ing caused by the four main modes we acquire a v sin i of 36.7kms-1 (see Fig. ). A genetic optimization with the fol­ -0.015 6 lowing free parameters was afterwards adopted: l G [0,4] -0.02 ------with a step of 1, m G [—l, l] with a step of 1, the sur­ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 face velocity amplitude a G [10,110] kms- 1 with a step of Phase (5.334224 d-1) 1 km s-1 , the stellar inclination angle i G [5, 85] ° with a step Figure 5. Phase diagrams of the equivalent width of the Si III of 5 ° , v sin i G [20, 48] k m s- 1 w ith a step of 0.5 k m s- 1 and A4553 A line, for f (top panel), and for ƒ2 (bottom panel) after a G [10, 30] k m s- 1 with a step of 0.1 kms-1. We fixed f 2 as prewhitening with f . a radial mode and only fitted the continuous parameters for th a t mode. In Table 3, the 5 best solutions from a fit to the cludes, for instance, first-order effects of the Coriolis force Fourier parameters are listed for each frequency. The 95%- as well as temperature variations of the stellar atmosphere. confidence limit of x 2 is 1.15. The most probable solutions We refer to Zima (2006, 2008) for details on the computation are (l1,m 1) = (1 , 1), (l3 ,m 3) = (1 , 0) and (l4,m 4) = (2 , 1) of the synthetic line-profile variations. The fixed parameters (where a positive m-value denotes a prograde mode). We are the mass and the radius. When taking into account tem­ point out that the agreement between theory and observa­ perature effects, the effective temperature Teff and the loga­ tions is not perfect. More precisely, the observed asymmetry rithm of the gravity log g have also to be fixed. For 12 Lac, with respect to the centroid of the amplitudes across the line we adopted the values M = 13.5 M q (Handler et al. 2006), of the dominant modes is not completely reproduced by our R = 6.9 R q (Handler et al. 2006), Teff = 24500 ± 1000K model. Such an asymmetry can be caused by EW varia­ (Morel et al. 2006), logg = 3.65 ± 0.15 (Morel et al. 2006). tions of the intrinsic line profile due to local temperature We note that the mode identification results are robust when variations (Schrijvers & Telting 1999). We considered EW using different values for these parameters, within the errors. variations in our fitting procedure but we did not obtain an In a first step we identified the 3 main modes (f1>3,4) by improved solution compared to the case of a constant EW. fitting mono-mode Fourier parameters, which are computed In a second step we attempted to improve the solutions from synthetic line profiles evenly sampled over one pulsa­ by simultaneously fitting the four main modes and using all tion cycle (for more explanations see Zima 2006). Moreover, the original data points. This however was computationally we neglected the equivalent width variations of the spec­ not possible. To restrict the computation time we took a sub­ tral line due to temperature variations. In order to restrict set of our complete time series. Our subset resulted in 388 the parameter space, we estimate roughly the values of the spectra with a time span of 4 months. We also fixed the wave projected rotational velocity v sin i, the width of the intrin­ numbers according to our mono-mode fit described above. sic gaussian profile a, and the equivalent width W from a We succeeded in improving the surface velocity amplitudes least-squares fit of a rotationally broadened synthetic pro­ together with the inclination, v sin i and the width of the file to the zero-point profile. This gives only a crude estimate intrinsic profile. The x 2 values are lower than in the mono­ of these three parameters, because the pulsational broaden­ mode fit and we are able to reproduce part of the asymmetry ing is ignored in this fit, but it serves the purpose of defin- in the amplitude across the line as explained above (see Ta- An asteroseismic study of the ß Cephei star 12 Lacertae 7

Table 3. Mode parameters derived from the FPF mono-mode method. For each pulsation mode, the five best solutions are shown together with the second best (l,m ) identifications. For f 5 we list the different solutions for m. a is the surface velocity amplitude (kms- 1 ); i is the stellar inclination angle in degrees; v sin i is the projected rotational velocity, u is the width of the intrinsic profile, both expressed in km s- 1 . Due to spherical symmetry we have not indicated the inclination angle for radial modes. The 95% significance limit is at x 2 = 1.15.

f 1 = 5.178964 d-1 ƒ3 = 5.066316 d- 1 x 2 1 m tts % v sin % a x 2 l1 m ttß % v sin % a [km s- 1 ] [° 1 [km s- 1 ] [km s- 1 ] [km s- 1 ] [° 1 [km s- 1 ] [kms- 1 15.29 1 1 68.7 74.6 46.47 21.00 10.34 1 0 24.8 59.1 42.69 26.23 15.30 1 1 64.9 85.0 38.15 27.82 10.38 1 0 24.2 59.2 42.70 26.23 15.31 1 1 48.2 72.1 43.72 21.64 10.39 1 0 25.0 51.5 42.38 26.24 15.32 1 1 72.3 51.4 36.03 28.62 10.40 1 0 26.4 48.9 42.38 26.35 15.33 1 1 50.9 59.2 42.00 22.58 10.41 1 0 25.7 59.2 43.02 26.41 31.62 1 0 68.4 41.2 30.79 26.66 25.99 1 1 18.0 74.6 43.65 26.70 f 2 = 5.3324224 d -1 ƒ4 = 5.490133 d- 1 x 2 l1 m tts % v sin % a x 2 l1 m ttß % v sin % a [km s- 1 ] [° 1 [km s- 1 ] [km s- 1 ] [km s- 1 ] [° 1 [km s- 1 ] [km s- 1 12.36 0 0 41.6 - 43.65 25.17 7.73 2 1 26.2 15.3 40.15 27.23 12.83 0 0 37.0 - 36.66 28.11 7.76 2 1 26.2 12.7 40.16 27.17 13.07 0 0 42.6 - 43.33 25.00 7.77 2 1 26.2 12.7 40.79 26.76 13.53 0 0 33.4 - 40.47 26.41 7.79 2 1 27.1 12.7 40.79 27.11 13.68 0 0 33.3 - 44.60 24.70 7.83 2 1 24.2 12.8 42.38 26.35 43.68 2 2 23.7 82.4 41.76 26.26

Table 4. Mode parameters derived from the FPF method Table 5. Final results for the mode identifications of 12 Lac through a simultaneous fit of the four main modes. We used the from our spectroscopic analysis together with the results from same symbol conventions as in Table 3. The 95% significance limit the photometric amplitude ratios (Handler et al. 2006). is at x 2 = 1.31. ID Frequency l m [d- 1 ] Spectr. Phot. 2 f 1;(1, 1) 5 f 2;(0,0) 5 f 3;(1,0)> f4;(2,1) x 2 tts1 as2 as3 aS4 % v sin % a [kms- 1 ] [° ] [kms- 1 ] [kms- 1 ] f 1 5.178964 1 1 1 f 2 5.334224 0 0 0 7.26 91.6 54.9 24.5 29.6 43.7 36.7 22.1 f3 5.066316 1 1 0 7.34 91.6 59.3 26.9 28.8 46.2 37.0 21.6 f4 5.490133 2 2 1 7.51 98.4 57.4 18.0 28.8 43.7 35.1 22.0 0.342841 - 1,2,4 - 7.55 83.8 51.1 24.5 28.8 54.0 35.4 21.7 ƒ5 4.241787 - 2 0, 1, 2 7.64 90.6 50.4 26.9 28.8 43.7 37.1 22.1 f 6 5.218075 - 2,4 - f 7 6.702318 - 1 - f8 7.407162 - 1,2 - f9 5.84511 - 1,2 - /p 5.30912 - 1,2 - ble 4). Fig. 7 illustrates the best models for the amplitude and phase across the line profile for the four main modes. Finally, we tried to identify the other low-amplitude 4.3 Derivation of the surface equatorial rotational modes with the same methodology and by imposing the velocity and the inclination wavenumbers (l,m) deduced for the 4 main modes. Unfor­ tunately, we could not obtain additional conclusions. Our From mode identification we can also estimate the inclina­ failure is not very surprising since these modes have am­ tion, v sin i and the equatorial rotational velocity veq. By us­ plitudes lower or of the same order as the harmonics and ing x 2 as a weight we can construct histograms for i, v sin i combination frequencies also present in the signal while non­ and veq. In Fig. 9 we display these histograms. They are com­ linear terms are not taken into account in our line-profile puted by considering all the solutions of the FPF method modelling. Fig. 8 depicts the amplitude and phase across the with the correct wavenumbers of the four dominant modes line for these low amplitude modes. Despite the fact that the through a simultaneous fit, and by giving each parameter phase across the profile is well determined for these modes (ik, v sin ik and v£q,k) its appropriate weight Wk = x o /x k , and seems to point to axisymmetric or prograde modes, the where xO is the x 2-value for the best solution. By calculating x 2 values between the different options were not discrimi­ a weighted mean and standard deviation of the data, we get native so we do not use that information in the following. i = 48 ± 2 °, v sin i = 36 ± 2 k m s - 1 and veq = 49 ± 3kms-1. In Table 5, we summarize the final outcome of both the This leads to a surface rotational frequency between 0.11 and photometric (Handler et al. 2006) and spectroscopic mode 0.12 d - 1 for the appropriate stellar models in Table6 , which identifications. will be discussed further on. In the same way we constructed 8 M. Desmet et al.

f1=5.178964 d-1 f2=5.334224 d

(A (A c C (D(D (D ~(D "O .E "O .E "0 B "0 U U <1 < |

Velocity [km s ] Velocity [km s -1 ]

-1 f4=5.490133 d

O

Velocity [km s-1] Velocity [km s -1 ]

F igure 7. Observed Fourier parameters (black points with error bars) across the line profile together with the best fit for the four identified modes (full line). The amplitudes are expressed in units of continuum and the phases in 2n radians. An asteroseismic study of the 3 Cephei star 12 Lacertae 9

ffl= 0.342841 d f5=4.241787 d f6=5.218075 d 0.008 0.008 0.007 0.007 0.006 0.006 0.005 0.005 0.004 0.004 0.003 0.003 0.002 0.002 0.001 0.001 0 0 -0.001 -0.001

0.4

0.2

0

-0 .2

-0 .4

-0 .6

-0 .8

-1

-1 .2 - 4 0 0 Velocity [km s ] Velocity [km s-1] Velocity [km s-1]

f7=6.702318 d f8=7.407162 d f9=5.84511 d

-4 0 0 40 80 Velocity [km s-1] Velocity [km s-1] Velocity [km s-1]

F igure 8 . Observed Fourier parameters (black points with error bars) across the line for the detected pulsation frequencies (f9,5—9). The amplitudes are expressed in units of continuum and the phase in 2n radians. these histograms with the solutions from the multi-mode fit vective overshooting parameter. Such asteroseismic con­ of the moment method. However, the moment method did straints have recently been derived for a few 0 Cephei not have the ability to limit the range for i and, by implica­ stars: 16 Lac (Thoul et al. 2003), V836 Cen (Aerts et al. tion, leads to a less trustworthy estimate of veq. The reason 2003; D upret et al. 2004), v Eri (Pam yatnykh et al. 2004; for this is probably that the moment method is based on Ausseloos et al. 2004; Dziembowski & Pamyatnykh 2008), integrated quantities over the line profile, while in the FPF S Ceti (Aerts et al. 2006), 0 CM a (M azumdar et al. 2006), method we use all the information across the line profile 0 Oph (Briquet et al. 2007). Finally, Ausseloos (2005) and, as a whole, which is more sensitive to the position of the more recently, Dziembowski & Pamyatnykh (2008) already nodal lines across the stellar surface and thus to the stellar made a comparison with stellar models for 12 Lac. inclination.

5.1 Numerical tools and physical inputs 5 MODELLING In our work, we used the following numerical tools and In this section, we check if state-of-the-art stellar mod­ physical inputs. The stellar models for non-rotating stars els with standard physics can account for the observed were computed with the evolutionary code CLES (Code frequency spectrum together with the derived wavenum- Liegeois d’Evolution Stellaire, Scuflaire et al. 2008b). We bers (l, m) for our studied star. By doing so, we can used the 0PAL2001 equation of state (Rogers & Nayfonov constrain 12 Lac’s model parameters, which are the 2002; Caughlan & Fowler 1988), with nuclear reaction rates mass, the central hydrogen abundance and the core con­ from Formicola et al. (2007) for the 14N (p ,y) 15 O cross- h FF method. FPF the resentative of the star, the problem is obviously increased. increased. rep­ obviously is more are problem con­ we which the in If 0.016, star, discussed 0.016). As an the th of than smaller larger Z resentative of (typically values Z sider of values the large ith w also t u b of three-dimensional models e-dependent, tim hydrodynamical from derived values solar First, arguments. following the by justified perature is tem low the ith w 4.1 < of T log at tables completed are bles from ture oes Nyoo 2002 Nayfonov & Rogers excita­ This the the of ith eter. w param values encountered is realistic overshooting for problem core observed tion excited all and be not can metallicity opacities, modes OPAL the using ith w pulsation when confronted at, was th and Lac 12 of problem study seismic liminary Secondly, therein). ences clas­ the convection using of by Theory treated is Length Mixing transport sical Convective section. ve­ rotation projected inclination, the for Histograms 9. igure F % of the mode id. solutions are stable w ith OPAL opacities opacities OPAL ith w stable are the ith w agreement good in errors, the within are, Fe and S from derived star the of velocity rotational equatorial and locity 1958 (2006 2005 (Seaton 10 20 20

Our choice for the m etal m ixture and the opacities opacities the and ixture m etal m the for choice Our ). For the chemical composition, we used the solar mix­ solar the used we composition, chemical the For ). 0 5 0 5 0 5 0 5 70 65 60 55 50 45 40 35 30 ) found th a t the abundances of He, C, N, O, Mg, Al, Si, Al, Mg, O, N, C, He, of abundances the t a th found ) ------M. Desmet et al. et Desmet M. Fruo t l (2005 al. et Ferguson Apud t l (2005 al. et Asplund 1 ------cmptd o ti lte mxue Tee a­ ta These mixture. latter this for puted com ) Biut t l (2007 al. et Briquet sln t l (2005 al. et Asplund 1 ------nlnto (degrees) inclination Ausseloos cn e rdce t b unstable be to predicted be can ) i i(ms 1) s (km i sin v 1 ------Apud t l (2005 al. et Asplund Veqs-1) (km ). ). We used OP opacity tables tables opacity OP used We ). 1 ------

Rgr & geis 1992 Iglesias & (Rogers ) for 0 Oph, modes th at at th modes Oph, 0 for ) (2005 rvse t l (1993 al. et Grevesse 1 ------) m ixture, even for for even ixture, m ) ) performed a pre­ a performed ) (Bohm-Vitense 1 ------oe e al. et Morel ad refer­ and , r ) ;

of the oscillation up to l = 4 using a standard adiabatic adiabatic standard a using 4 = l degree a to ith w up g-modes and oscillation p- the low-order of of spectrum of case quency the in etal m modes and tables pulsation of stars. opacity B-type excitation the adopted on the of ixtures m of implication ­ modelling the ta the on of case opacity the to OP in occurs. refer of We use Lac. modes situation the if the check pulsation improves OPAL of to bles for wish one driving Cephei the the 0 consequently than We where typical larger a region 25% in is the at th in opacity is OP reason the The model opacities. OP ith w was determined. Moreover, other works corroborate the ne­ the 0.45 corroborate ~ works of value other higher Moreover, even an studied which all determined. for core for of ph was O [0.05-0.25] 0 range occurrence except the the ov in stars several a of showed ith w above) modelling (listed overshooting asteroseismic stars recent Cephei t a 0 th is a reason considered The also ith­ we w but models diffusion, stellar account into non-rotating taking We puted out Oph. com 0 of we e­ case param the simplicity, in stellar checked to derived carefully diffusion the a refer been affect are has not including effects It does layers, ters. these not or since ent surperficial However, very treatm the star. to the of limited photospheric interior the the alter to being can zone mechanisms convective a surface diffusion its being and thin, star reason The very the rotator following. and slow the is value relatively composition this solar adopting the also for to typical Lac 12 above, corresponds for (see choice found also metallicity errors a the to ithin w neighbourhood, solar corresponds the in value dwarfs B of chosen first The conclusions. the change not does 0.0089±0.0018. = Z spec­ metallicity a same ined the determ on ethod from m 3.65±0.1 another as = and g log Using tra and K K spectra. 500±1000 24 optical 23500±700 = Teff = obtained (2005 Teff Daszynska-Daszkiewicz provides & Niemczura Lac 3.4±0.4 12 = g log of etry code for non-rotating stellar models models stellar non-rotating for code ihr hnwht s eie fo spectra. from derived is hat w than higher Z compared surface the at smaller Z to leading abundances, = X range adopted reasonable a We in . X M for eter mass value param the different a and Z overshooting using but 0. 72 convective metallicity core the ov, the ,a X abundance 3.60±0.1. = g log and K 24000±1000 0.15. ± 3.65 = g log and K ­ 23600±1100 photom Geneva Ta- constraints. in observational summarized additional as characteristics le pulsation b the Besides non- linear the ith w modes checked we pulsation the selected, the are of ithin w modes excitation observed frequencies leads the the but fitting pulsation code models theoretical non-adiabatic a same an th the to faster much is which . Simi analysis ic Seism 5.2 by developed MAD code adiabatic 10 was which fit, the of precision adopted 5 For each stellar model, we calculated the theoretical fre­ theoretical the calculated we model, stellar each For We considered two values for Z, i.e. 0.010 and 0.015. 0.015. and 0.010 i.e. Z, for values two considered We Several values for a ov in the range [0.0-0.5] were tested. tested. were [0.0-0.5] range the ov in a for values Several hydrogen initial the by etrized param are models Stellar w as ue ef lgg n te ealct Z metallicity the and g log Teff, use also we , oe t l (2006 al. et Morel rqe e a. (2007 al. et Briquet Mgi e a. (2007 al. et Miglio Hnlr t l 2006 al. et (Handler ), Lefever oe e a. 2006 al. et Morel ) for a deeper discussion. For For discussion. deeper a for ) ) for a detailed discussion discussion detailed a for ) pe (2001 upret D

oe e a. (2006 al. et Morel Sulie t l 2008a al. et (Scuflaire (2008 ). From IUE spectra, spectra, IUE From ). oe e a. (2006 al. et Morel ddcd ef = Teff deduced ) -3 drvd ef = Teff derived ) d - 1 . Once the the Once . 1 - d ). The second second The ). ). ) also also ) 12 12 ), ) An asteroseismic study of the 3 Cephei star 12 Lacertae 11 cessity to include core overshooting in modelling of massive Table 6 . Physical parameters of the models that match f 2 (being B-type stars. For instance, Deupree (2000) derived a value the first overtone) and f3, with X = 0.72 and Z = 0.015. X c is of about 0.45 by means of 2D hydrodynamic simulations of the central hydrogen abundance. The age t is expressed in million zero-age main-sequence convective cores. We also mention years. that, with his study of 13 detached double-lined eclipsing binaries, Claret (2007) found that models with core over­ shooting of aov ~ 0.2 are needed to analyse ~10 M© stars. a 0v M (M0 ) Teff (K) log g Xc R (Re ) log (L/L0 ) r (My) The observed radial mode with f 2 = 5.334224 d -1 and 0.0 14.4 25600 3.70 0.13 8.8 4.48 11 the I = 1 zonal mode with f 3 = 5.066316 d- were selected 0.1 13.1 24500 3.68 0.15 8.6 4.38 13 as the first two frequencies to be confronted with the theo­ 0.2 12.0 23600 3.66 0.17 8.4 4.29 16 retically predicted frequencies. This was an obvious choice 0.3 11.0 22750 3.65 0.19 8.2 4.21 20 since these are zonal modes (m = 0) and thus we do not need 0.4 10.2 22000 3.64 0.21 8.0 4.13 23 any information about the equatorial frequency of rotation to fit them with the model frequencies. We considered two different scenarios: f 2 being either the radial fundamental mode or the first overtone. 3.3 The theoretical frequency spectrum with f 2 being the radial fundamental mode is globally compatible with the ob­ 3.4 served frequencies of 12 Lac. However, all such models fail 3.5 to reproduce the I = 1 mode with f 7 = 6.702318 d-1. If f 2 corresponds to a radial fundamental mode, f 7 can only be an I = 2 mode according to theoretical models, while the OCT «ov=0 4 observed photometric amplitude ratios for this mode point “ 3.7 “ ov=0 out an I = 1 (Handler et al. 2006). Unfortunately, we can­ not confirm this strong constraint since our spectroscopic 3.8 mode identification was not conclusive for this mode. How­ 3.9 ever, stellar models with f 2 corresponding to the radial first overtone can account for a I = 1 mode with f7. We conse­ 4 quently favour a radial order n = 2 for f2. An additional 4.5 4.45 4.4 4.35 4.3 4.25 strong argument to favour f 2 as the radial first overtone lo9 Teff is the following. The Ledoux rotational splitting is defined Figure 10. The error box represents the position of 12 Lac in by fLed0ux = m/Snifrot, where ¡3ni is a structure constant the log(Teg ) — logg diagram. The positions of the models which depending on the stellar model, and originates from the fre­ fit exactly f 2 (being the first overtone) and f 3 are also shown for quency splitting defined as different a 0v values.

— CR 0 (r) d r fm = fo + mprdfvot = V0 + rn K ('/’) —----- —, (1) Jo 2n R 5.3 Radial order identifications where fm is the cyclic frequency of a mode of azimuthal- The frequency f 1 = 5.178964 d -1 can only belong to the p 1 order m, fi(r) is the angular velocity, and K (r) is the ro­ triplet. The frequency splitting f 1 — f 3 is 0.1126 d - 1 . Ac­ tational kernel (for more information, see Scuflaire et al. cording to the models, ¡3ni = 0.5 for this mode. This yields 2008a). W ith f 2 as fundamental, the difference between the an averaged frequency of rotation fr0t = (f 1 — f 3)/m 0 ni. frequency f 5 = 4.241787 d -1 and the nearest quadrupole This averaged rotation frequency leads to a rotational ve­ mode (at ~3.8d-1) is too large to be explained by the locity twice as high as the surface value estimated from our Ledoux rotational splitting and thus f 5 cannot be fitted. spectroscopic observations (with corresponding velocities of W ith f 2 as first overtone, f 5 is easily identified as the g2 100 km s-1 versus 49kms-1). We thus confirm that 12 Lac mode with (I, m) = (2,1) as shown below (see also Fig. 11). must rotate more rapidly in its inner parts than at the sur­ W ith f 2 being the first overtone, f 3 is identified as p1. face, as already emphasized by Dziembowski & Pamyatnykh The couple (X, Z) being fixed, the fitting of two fre­ (2008). quencies suffices to derive the stellar age and stellar mass As can be seen in Fig. 11, the (I = 1, p3) mode can for adopted a 0v. The positions in the log(Teff) — log g dia­ account for f 7 = 6.702318 d-1. The match is not perfect for gram of the models matching f 2 and f 3 for different values all models in Table 6 . This is probably due to the fact that of a0v are shown in Fig.10. The corresponding parameters there can occur small shifts due to rotation. On the other are listed in Table 6 for Z = 0.015. The mass decreases when hand a change in metallicity will also cause a small shift the core overshooting parameter increases. We also note the of the theoretical frequencies (see Sect. 5.5). The frequency perfect agreement between the log g of the models and the f 8 = 7.407162 d -1 and f 9 = 5.84511 d -1 can only be the value deduced from spectroscopic observations. Moreover, (I = 2 , p2) mode and the (I = 1 , p2) mode, respectively. the best agreement in Teff corresponds to models with a 0v Indeed, no theoretical frequency of an I = 1 zonal mode between 0.0 and 0.4 and a mass between 10.0 and 14.4 M©. and an I = 2 zonal mode is in the vicinity of f 8 and f 9, Note that a decrease in Z implies an increase in mass and, respectively. We also note that an l = 0 radial-mode solution considering a Z of 0.010 instead of 0.015 increases the mass for f 8 is excluded (see Table 5). by 0.5 M© at most. The frequency fp = 5.30912 d -1 found in photometry 12 M. Desmet et al.

Table 7. The I, n and m identifications according to our mode Table 8 . Observed photometric amplitude ratios taken from identification and modelling with the radial mode as first over­ Handler et al. (2006). We also list our best fitting theoretical am­ tone. We use the usual convention that negative n-values denote plitude ratios. They originate from the model with a ov = 0.4 g-modes (see e.g. Unno et al. 1989). given in Table 6.

ID Frequency I n m ID Frequency v/u V /u [d- 1 ] ' [d- 1 ] ' obs. theory obs. theory

fi 5.178964 1 1 1 f 2 5.334224 0.529(7) 0.64 0.456(7) 0.56 f2 5.334224 0 2 0 f 3 5.066316 0.716(8) 0.73 0 .686(8) 0.67 f3 5.066316 1 1 0 h 5.490133 0.82(1) 0.83 0.78(1) 0.78 f4 5.490133 2 0 1 f 5 4.241787 2 -2 1 f6 5.218075 4 -2 or -1 f 7 6.702318 1 3 5.4 Comparison with photometry 7.407162 2 2 f8 We used our best models listed in Table 6 to compare 5.84511 1 2 f9 our results with the photometric mode identification of fp 5.30912 2 0 0 Handler et al. (2006). We computed theoretical photomet­ ric amplitude ratios for our three main identified pulsation modes (f2, f3, f4; f 1 belongs to f3) following D upret et al. and the frequency f 4 = 5.490133 d -1 were both identified as (2003). We computed the amplitude ratios for Stromgren u quadrupole modes and can be accounted for by the (l = 2 , ƒ) and v, and for Johnson V. For this computation we must de­ mode. Indeed, the theoretical Ledoux frequency splitting as­ termine the nonadiabatic pulsation mode parameters and suming the averaged internal rotation frequency value frot fg, where corresponds to the local effective temperature derived from ƒ — f 3 ranges in [0.186-0.190] d -1 for differ­ variation and fg corresponds to the local effective gravity ent overshooting values, which corresponds to the difference variation, both at the level of the photosphere. We used the between fp and f4. The frequency fp thus corresponds to non-adiabatic code MAD (Dupret 2001) to compute these (l,m ) = (2 , 0) and f 4 to (l,m) = (2 , 1) in accordance with fT,g parameters for our best seismic models of 12 Lac. the m —identification for f4. Note that the theoretical fre­ We list the observed and theoretical photometric am­ quency value for the (l = 2 , ƒ) mode is in best agreement plitudes for the three main modes in Table 8 . We refer to for a ov = 0.0 and is 5.287 d - 1 . Fig. 4 of Handler et al. (2006) for the results of the photo­ = 4.241787 d -1 being a prograde l = 2 mode, is metric mode identification. For all the models, we observed uniquely identified as the (l = 2, g2) mode. The corre­ that the amplitude ratios decrease with decreasing mass. For sponding theoretical zonal mode has a frequency between f 2 we find amplitude ratios for v/u ranging from 0.68 until 4.048 and 4.070 d -1 with a Ledoux splitting for that mode 0.64, and for V /u ranging from 0.51 until 0.56. Thus, we en­ ranging between 0.183 and 0.185 d -1 deduced from frot and counter exactly the same problem as Handler et al. (2006), Pni = 0.80. In addition, we thus conclude that m = 1 for 1.e., we could not find an perfect agreement for the observed this mode. ratios of f2. For f 3 we find amplitude ratios for v/u ranging = 5.218075 d -1 may be either identified as (l = from 0.84 until 0.73, and for V/u ranging from 0.74 until 4, g2) or (l = 4, g1) (see also Fig. 11). The models give 0.67. Our models give amplitude ratios for f 4 for v/u rang­ Pni = 0.97 for this mode. In the first case, the zonal fre­ ing from 0.90 until 0.83, and for V/u ranging from 0.84 until quency ranges between 4.38 and 4.55 d -1 with a Ledoux 0.78. Our conclusion is that we can reproduce the observed splitting of 0.21-0.22 d -1 deduced from f.ot. This corre­ photometric amplitude ratios for the two main modes (f3 sponds to an m-value of 3 or 4. In the second case, the zonal and f4) from our best seismic models. Moreover, from this a frequency ranges between 5.57 and 5.70 d -1 with a Ledoux posteriori consistency check, we favour a the more evolved splitting of 0.20-0.21 d-1. This corresponds to an m-value seismic models, i.e., those with lower mass and higher core of -2 . overshoot (see Tables 6 and 8). Finally, nothing can be concluded for fg = 0.342841 d -1 because of the dense theoretical frequency spectrum in the 5.5 Discussion low-frequency range and because of the lack of observed con­ straints on the wavenumbers of this mode. We additionally We have shown that adopting n = 2 for the note that we cannot definitely exclude this frequency to be radial mode cannot be excluded as was done by linked with the frequency of rotation. For instance, fg/3 Dziembowski & Pamyatnykh (2008), and, in fact, explains is compatible with our observed surface equatorial rotation all observed frequencies according to their photometric and frequency. As explained in Briquet et al. (2001, 2004), one spectroscopic mode identification except for fg. The iden­ way to discriminate between the pulsation and rotation in­ tifications of the radial order together with the additional terpretation is to compare the variability of the considered l identifications are summarized in Table 7. A perfect frequency in lines of different chemical elements. Unfortu­ agreement in frequency values is not achieved. One reason nately, we could not achieve such a test because lines other might be that slightly different values for (X, Z) than those than Si lines are at our disposal only for two nights of ob­ adopted could lead to a better agreement. Another one may servation. This time span is not long enough to recover such be that small frequency shifts due to rotation may occur as a low frequency value. already pointed out by Dziembowski & Pamyatnykh (2008), An asteroseismic study of the 3 Cephei star 12 Lacertae 13

F igure 11. Comparison between the theoretical frequencies (m = 0, dots) and the observed frequencies (full and dotted lines) labeled with their frequency identifications as listed in Table 5. The full lines correspond to the two axisymmetric modes f 2 and f 3. The top and bottom panels correspond to f 2 as the radial first overtone and the radial fundamental, respectively. The theoretical models are calculated for X = 0.72, Z = 0.015 and «oV = 0.0.

although the surface equatorial rotation frequency is only assumption of non-rigid rotation is based on the fact that a few percent of the measured oscillation frequencies. We f 5 cannot be fitted because the closest l = 2 mode is too far found evidence for some degree of differential rotation from to be explained by the splitting due to a uniform rotation. the observed splitting between f 1 and f 3 . Using the Ledoux We recall that, on the contrary, our models reproduce the splitting, all the other detected frequencies can be well fit­ l = 1 for ƒ7 and the Ledoux splitting for /rot can explain ƒ5 ted with one and the same averaged rotation frequency /ot as the (l, m,n) = (2 , 1, —2 ) mode. As additional argument inside the star, where /rot is derived from the model fitting to non-rigid rotation, Dziembowski & Pamyatnykh (2008) of f 1 and f 3 . In the absence of the observations of multiplets stated that their model with uniform rotation has no iden­ we do not have enough constraints to derive the internal ro­ tification for f 6 . We can explain it as the (l = 4, g2 ) mode tation profile as defined in Equation (1) and, consequently, or the (l = 4, g 1) mode in agreement with the photometric to determine its effects on the frequencies and frequency l-identification. Finally, even with a non-uniform rotation, splittings. they cannot reproduce f 5 appropriately while they used this frequency value to exclude rigid rotation. Dziembowski & Pamyatnykh (2008) partly included ro­ tation effects in their modelling and proposed a model with Until now, the only convincing proof of non-rigid ro­ the angular rotation velocity more than 4 times higher in the tation has been achieved for V836 Cen (Aerts et al. 2003; centre of the star compared to the surface. They imposed Dupret et al. 2004) and for v Eri (Pamyatnykh et al. 2004; that the radial mode is the fundamental without consid­ Dziembowski & Pamyatnykh 2008). The only other star ering alternative identifications. Consequently, they could with two observed multiplets is 6 Oph for which the observed not account for an l = 1 for ƒ7. As our results are the rotational splittings cannot rule out a rigid rotation model. same than theirs under the assumption of the radial fun­ For 12 Lac, we are able to fit the observed frequency spec­ damental mode, we refer to their paper for the properties of trum using non-rotating stellar models and Ledoux split­ the resulting seismic model in this case without repeating tings based on one consistent value /ot for the averaged in­ it here (see in particular their Fig. 7 and our Fig. 11). Their ternal rotation frequency derived from the splitting between 14 M. Desmet et al.

ƒ and f3. The deviation between the rotation frequency tion of linear oscillation modes. The most significant out­ derived from fitting the splitting fi — f 3 and from the ob­ come of our modelling is the fact that there is a strong pref­ served surface rotation frequency points to some degree of erence that the observed radial mode (f2) is the radial first non-rigidity. W ith only two components of one and the same overtone. If f 2 is taken as the radial fundamental, the fre­ multiplet observed, we are not able to quantify this state­ quency spectrum of 12 Lac cannot be reproduced satisfac­ ment. torily. We do point out that the theory we used, as well as Besides our basic stellar modelling, we also checked if Dziembowski & Pamyatnykh (2008), does not include non­ non-adiabatic computations are able to reproduce the ex­ linear effects, while these may be present in 12 Lac (e.g. citation of the observed frequencies. Indeed, recent works Mathias et al. 1992). It remains to be studied how these revealed shortcomings in the details of the excitation mech­ would effect mode identification and seismic modelling, but anism for these massive B-type pulsating stars. For instance, given the very moderate effect and the limit to only very few for v Eri (Pam yatnykh et al. 2004; Ausseloos et al. 2004), third-order combination frequencies, we do not expect this the observed lowest g-mode and highest p-mode frequen­ to alter our conclusions, just as for the case of v Eri, where cies were not predicted to be excited by standard models. a linear theory was also used (Pamyatnykh et al. 2004). The same conclusion is found for 12 Lac. Neither the fre­ The complete frequency spectrum of 12 Lac, except the quency fg nor the highest p-mode frequencies (f 7 and f 8) low frequency, can be fully identified. Our seismic modelling are excited by current models, even with OP opacity ta­ also revealed an excitation problem. Indeed, the range of fre­ bles. Since 1991, the agreement between the non-adiabatic quencies theoretically excited is not large enough compared computations for 0 Cephei models and the observations to the observations. This might reflect the fact that opaci­ has improved with subsequent update of opacities (see e.g. ties are still underestimated in the region where the driving Miglio et al. 2007; Miglio 2007). Consequently, a most prob­ of pulsation modes occurs. This conclusion is valid for both able explanation for the encountered excitation problem is the OPAL and OP opacities. that current opacities are still underestimated in the region Finally, our best-fit models indicate that the overshoot­ where the driving of pulsation modes occurs. We refer to ing parameter aov has to be lower than 0.4 local pressure Dziembowski & Pamyatnykh (2008) for an additional dis­ scale heights to get the best agreement with physical param­ cussion on this matter. eters derived from spectroscopic observations. A more re­ fined seismic modelling requires the computation of a much denser grid of models and will be done in future work. 6 CONCLUSIONS Our study was based on 1820 ground-based, high-resolution, high-S/N, multisite spectroscopic measurements spread over ACKNOWLEDGMENTS 14 years. As pointed out in Handler et al. (2006) this effort MD, MB, WZ and CA acknowledge financial support from was necessary to identify the previously unknown azimuthal the Research Council of Leuven University under grant order of the pulsation modes and thus to be able to perform GOA/2008/04. GH has been supported by the Austrian a detailed seismic modelling of the star. We used the Si III Fonds zur Forderung der wissenschaftlichen Forschung un­ 4553 A line to derive the pulsation characteristics of 12 Lac. der grants R12-N02 and P18339-N08. Part of this work was In total we find 10 independent frequencies which were also based on observations made with the Nordic Optical Tele­ detected in photometry. Worth mentioning is that we also scope, operated on the island of La Palma jointly by Den­ clearly recover the low-frequency signal. One of our aims mark, Finland, Iceland, Norway, and Sweden, in the Spanish was to provide a unique identification of as many modes Observatorio del Roque de los Muchachos of the Instituto as possible for 12 Lac. The important result of combining de Astrofisica de Canarias. our spectroscopic results with the ones from the intensive photometric campaign (Handler et al. 2006) is the unique identification of both land m of the four highest-amplitude modes. REFERENCES With two state-of-the-art methods, the moment method and the FPF method, we were able to identify the azimuthal Aerts C., De Cat P., 2003, Space Science Reviews, 105, 453 order of the four main modes. We could also give a constraint Aerts C., De Cat P., Handler G., Heiter U., Balona L. A., on the azimuthal order of the fifth frequency. Two main Krzesinski J., Mathias P., Lehmann H., Ilyin I., De Rid­ frequencies were identified as axisymmetric modes. One of der J., Dreizler S., Bruch A., Traulsen I., Hoffmann A., which has an lvalue of 1 (f3), the other one is a radial James D., Romero-Colmenero E., Maas T., Groenewegen mode (f2). The two other main frequencies are identified M. A. T., Telting J. H., Uytterhoeven K., Koen C., Cot­ as (li,m i) = (1,1) and (l4 ,m 4) = (2,1). The conclusion for trell P. L., Bentley J., Wright D. J., Cuypers J., 2004, the fifth frequency is that the azimuthal order is likely to be MNRAS, 347, 463 positive. In addition, the FPF method could also constrain A erts C., de Pauw M., Waelkens C., 1992, A&A, 266, 294 the inclination, i = 48 ± 2 °, and the surface equatorial ro­ Aerts C., Marchenko S. V., Matthews J. M., Kuschnig R., tational velocity, veq =49 ± 2km s- i . Guenther D. B., Moffat A. F. J., Rucinski S. M., Sasselov The definite identification of four of the observed modes D., Walker G. A. H., Weiss W. W., 2006, ApJ, 642, 470 together with some constraints of the wavenumbers (either Aerts C., Thoul A., Daszynska J., Scuflaire R., Waelkens l-values or sign of m) on the other modes allowed us to C., Dupret M. A., Niemczura E., Noels A., 2003, Science, carry out a detailed seismic modelling, under the assump­ 300, 1926 An asteroseismic study of the ß Cephei star 12 Lacertae 15

Asplund M., Grevesse N., Sauval A. J., 2005, in Astro­ E., Sareyan J.-P., Parrao L., Lorenz D., Zsuffa D., Drum­ nomical Society of the Pacific Conference Series, Vol. 336, mond R., Daszynska-Daszkiewicz J., Verhoelst T., De Cosmic Abundances as Records of Stellar Evolution and Ridder J., Acke B., Bourge P.-O., Movchan A. I., Gar­ Nucleosynthesis, Barnes III T. G., Bash F. N., eds., pp. rido R., Papará M., Sahin T., Antoci V., Udovichenko 25-+ S. N., Csorba K., Crowe R., Berkey B., Stewart S., Terry Ausseloos M., 2005, PhD thesis, KULeuven, Belgium D., Mkrtichian D. E., Aerts C., 2006, MNRAS, 365, 327 Ausseloos M., Scuflaire R., Thoul A., Aerts C., 2004, MN- Handler G., Shobbrook R. R., Jerzykiewicz M., Krisciu- RAS, 355, 352 nas K., Tshenye T., Rodríguez E., Costa V., Zhou A.-Y., Balona L. A., James D. J., Motsoasele P., Nombexeza B., Medupe R., Phorah W. M., Garrido R., Amado P. J., Pa- R am nath A., van Dyk J., 2002, MNRAS, 333, 952 paro M., Zsuffa D., Ramokgali L., Crowe R., Purves N., Böhm-Vitense E., 1958, Zeitschrift fur Astrophysik, 46, 108 Avila R., Knight R., Brassfield E., Kilmartin P. M., Cot­ Breger M., Handler G., Garrido R., Audard N., Zima W., trell P. L., 2004, MNRAS, 347, 454 Paparo M., Beichbuchner F., Zhi-Ping L., Shi-Yang J., Lefever K., 2008, PhD thesis, KULeuven, Belgium Zong-Li L., Ai-Ying Z., Pikall H., Stankov A., Guzik J. A., Lenz P., Breger M., 2005, Communications in Asteroseis- Sperl M., Krzesinski J., Ogloza W., Pajdosz G., Zola S., mology, 146, 53 Thomassen T., Solheim J.-E., Serkowitsch E., Reegen P., Mathias P., Aerts C., Gillet D., Waelkens C., 1994, A&A, Rum pf T., Schmalwieser A., Montgomery M. H., 1999, 289, 875 A&A, 349, 225 Mathias P., Gillet D., Crowe R., 1992, A&A, 257, 681 Breger M., Stich J., Garrido R., Martin B., Jiang S. Y., Li Mazumdar A., Briquet M., Desmet M., Aerts C., 2006, Z. P., Hube D. P., Ostermann W., Paparo M., Scheck M., A&A, 459, 589 1993, A&A, 271, 482 Miglio A., 2007, PhD thesis, Universite de Liege, Belgium Briquet M., A erts C., 2003, A&A, 398, 687 Miglio A., Montalban J., Dupret M.-A., 2007, MNRAS, Briquet M., Aerts C., Löftinger T., De Cat P., Piskunov 375, L21 N. E., Scuflaire R., 2004, A&A, 413, 273 Montgomery M. H., O’Donoghue D., 1999, Delta Scuti Star Briquet M., De Cat P., Aerts C., Scuflaire R., 2001, A&A, Newsletter, 13, 28 380, 177 Morel T., Butler K., Aerts C., Neiner C., Briquet M., 2006, Briquet M., Lefever K., Uytterhoeven K., Aerts C., 2005, A&A, 457, 651 MNRAS, 362, 619 Niemczura E., Daszynska-Daszkiewicz J., 2005, A&A, 433, Briquet M., Morel T., Thoul A., Scuflaire R., Miglio A., 659 Montalbán J., Dupret M.-A., Aerts C., 2007, MNRAS, Pamyatnykh A. A., 1999, Acta Astronomica, 49, 119 381, 1482 Pamyatnykh A. A., Handler G., Dziembowski W. A., 2004, Caughlan G. R., Fowler W. A., 1988, Atomic Data and MNRAS, 350, 1022 Nuclear D ata Tables, 40, 283 Rogers F. J., Iglesias C. A., 1992, ApJS, 79, 507 Claret A., 2007, A&A, 475, 1019 Rogers F. J., Nayfonov A., 2002, A pJ, 576, 1064 De Cat P., A erts C., 2002, A&A, 393, 965 Schrijvers C., Telting J. H., 1999, A&A, 342, 453 De Ridder J., Dupret M.-A., Neuforge C., Aerts C., 2002, Schrijvers C., Telting J. H., Aerts C., 2004, A&A, 416, 1069 A&A, 385, 572 Scuflaire R., Montalban J., Theado S., Bourge P.-O., Miglio Deupree R. G., 2000, ApJ, 543, 395 A., Godart M., Thoul A., Noels A., 2008a, Ap&SS, 316, Dupret M. A., 2001, A&A, 366, 166 149 Dupret M.-A., De Ridder J., De Cat P., Aerts C., Scuflaire Scuflaire R., Theado S., Montalban J., Miglio A., Bourge R., Noels A., Thoul A., 2003, A&A, 398, 677 P.-O., Godart M., Thoul A., Noels A., 2008b, Ap&SS, Dupret M.-A., Thoul A., Scuflaire R., Daszynska- 316, 83 Daszkiewicz J., Aerts C., Bourge P.-O., Waelkens C., Seaton M. J., 2005, MNRAS, 362, L1 Noels A., 2004, A&A, 415, 251 Stankov A., Handler G., 2005, VizieR Online Data Catalog, Dziembowski W. A., Jerzykiewicz M., 1999, A&A, 341, 480 215, 80193 Dziembowski W. A., Pamyatnykh A. A., 2008, MNRAS, Telting J. H., Aerts C., Mathias P., 1997, A&A, 322, 493 385, 2061 Thoul A., Aerts C., Dupret M. A., Scuflaire R., Korotin Ferguson J. W., Alexander D. R., Allard F., Barman S. A., Egorova I. A., Andrievsky S. M., Lehmann H., Bri­ T., Bodnarik J. G., Hauschildt P. H., Heffner-Wong A., quet M., De Ridder J., Noels A., 2003, A&A, 406, 287 Tam anai A., 2005, A pJ, 623, 585 Unno W., Osaki Y., Ando H., Saio H., Shibahashi H., 1989, Formicola A., Imbriani G., Costantini H., Angulo C., Be- Nonradial oscillations of stars. Nonradial oscillations of mmerer D., Bonetti R., Broggini C., Corvisiero P., Cruz stars, Tokyo: University of Tokyo Press, 1989, 2nd ed. J., Descouvemont P., Fulop Z., Gervino G., Guglielmetti Wilson R. E., 1953, Carnegie Institute Washington A., Gustavino C., Gyurky G., Jesus A. P., Junker M., D.C. Publication, 0 Lemut A., Menegazzo R., Prati P., Roca V., Rolfs C., Ro­ Zima W., 2006, A&A, 455, 227 mano M., Rossi Alvarez C., Schumann F., Somorjai E., — , 2008, Communications in Asteroseismology, 155, 1 Straniero O., Strieder F., Terrasi F., Trautvetter H. P., Zima W., Kolenberg K., Briquet M., Breger M., 2004, Com­ Vomiero A., Zavatar elli S., 2007, Physics Letters B, 591, munications in Asteroseismology, 144, 5 61 Grevesse N., Noels A., Sauval A. J., 1993, A&A, 271, 587 This paper has been typeset from a TßX/ LT eX file prepared Handler G., Jerzykiewicz M., Rodríguez E., Uytterhoeven by the author. K., Amado P. J., DorokhovaT. N., Dorokhov N. I., Poretti