The Β Cep/SPB Star 12 Lacertae 3 of Metallicity: Z = 0.015 and Z = 0.010

arXiv:1304.4049v1 [astro-ph.SR] 15 Apr 2013 -yeplaosi hc yial o re / oe are the termed modes were p/g variables order These low observed. impor- earl typically in most which g-modes in the high-order pulsators of of B-type One discovery 2010). the al. was al. et results et tant (Baglin (Koch mis- CoRoT Kepler space-based 2003), or the al. 2006) from et (Walker as MOST well multi- like as sions the objects fo such from organized of the data campaigns several for spectroscopic to and obtained thanks photometric site been mainly Since have pulsators, dominate. results B-type M-shell intriguing iron many the then, within transitions cause ‡ † ⋆ in ntehayeeetin Ilsa ta.19,Seaton bump opacity 1992, metal 2 al. the around et revealed transi- opacities (Iglesias of new ions number These element 1993). huge heavy a the of in included re-computation tions after which possible data, been opacity has the Dziem- Gautschy and & 1993) 1993, old Moskalik Pamyatnykh Saio years 1992, & & al. 20 Dziembowski et 1992, only Kiriakidis bowski is 1992, years. al. phenomenon many et this for the (Cox persisted of in instability had explanation pulsation stars The of sequence origin main the B-type about puzzle The INTRODUCTION 1

.Daszyńska-DaszkiewiczJ. modelling seismic complex and identification The o.Nt .Ato.Soc. Astron. R. Not. Mon. c 1 ntttAtooizyUiesttWo asi Wroc Wroc Uniwersytet lalawski, Astronomiczny Instytut -al [email protected] E-mail: [email protected] E-mail: -al [email protected] E-mail: RAS · β 10 5 e/P tr1 aete xeddmode extended Lacertae: 12 star Cep/SPB ,wihi aldthe called is which K, 000 1– , ?? 2Lc–aoi aa opacities data: atomic – Lac 12 of values words: theoretical Key and empirical the of concordance the osrc e fsimcmdl hc ttoplainlfrequenc pulsational two fit which modes models the seismic to of set a construct htteO alsaepeerdada nraeo pcte nthe in opacities of increase an and preferred are enhancemen tables opacity OP and the mixture chemical that data, opacity spheres, eoiyof and velocity five a first (the frequencies of values empirical esi oesrpoueas h frequency the also reproduce models variations, seismic flux bolometric the is nldn h ffcso oain eso httedmnn f dominant the that show we rotation, of effects the Including cies. ra r rsne.Uigdt ntemliclu htmtya degree, photometry mode multi-colour the the constrain or on determine data we Using variations, presented. are ertae ieyapure a likely eut fmd dnicto n esi oeln fthe of modelling seismic and identification mode of Results ABSTRACT )Pitd2 ue21 M L (MN 2018 June 21 Printed () Z 1 − ⋆ .Szewczuk W. , rFe or β e/P stars Cep/SPB ℓ V − ℓ tr:erytp tr:oclain tr:rtto stars: – rotation stars: – oscillations stars: – early-type stars: rot oeadtelwfrequency, low the and mode 1 = up be- bump, 0 = ≈ f , ,Poland w, 5k/.I h etse,w hc h ffcso oe atmo- model of effects the check we step, next the In km/s. 75 oevr twspsil ofidamdlfitn h i 2Lac 12 six the fitting model a find to possible was it Moreover, . p 1 y r and ℓ 1 ezkeize l 05,1 aete(ade ta.2006; al. or et 2009) (Handler al. Lacertae et 12 Desmet 2005), al. et Jerzykiewicz rhbi ustr n e fte eedtce pto up detected were them of few e.g., a now, and pulsators hybrid or nmdl ihmse agrta bu 7 about than excited larger g-modes masses high-order with the get models sti to in that, is of problem Despite common 2009). & 2005, a Grevesse al. 1993, et Asplund Noels rede- 1998, & Sauval been (Grevesse 1996, has times Seaton mixture several 1996, chemical termined Rogers solar ta- the & computation Moreover, opacity (Iglesias 2005). in revised data added atomic first these been the of have of improvements release many the bles Since 2009). al. noeve ftesa.Scin3i eoe omd identifi- mode to devoted is 3 give Section we star. 2, the Section of In overview Lacertae. an 12 pulsator B-type hybrid the 2004). opacit Dziembowski the the & for Handler of e.g., increase called, artificial was coefficient an and ,,blamed” account are inconsistencies into data the opacity take the adequately Usually, physics. cannot stellar in or understand of something instability still not with is do problem there the that indicates hand, interior g-modes other high-order the whole On the star. almost order a scan high of to and modes potentially p/g allow order g-modes low both of Excitation ities. † ν 1 = f A o ohfeunissmlaeul.Sm fthese of Some simultaneously. frequencies both for , n .Walczak P. and ,ol fterttoa pitn a acltdfor calculated was splitting rotational the if only ), nti ae eraayetefeunysetu of spectrum frequency the reanalyse we paper this In possibil- as well as challenges give pulsators hybrid The , A g T 1 E tl l v2.2) file style X n erdc lotecmlxapiueof amplitude complex the also reproduce and ν rdn Hnlre l 04 et ta.2004, al. et Aerts 2004, al. et (Handler Eridani ν A samode a as , ν A sadpl ergaemd.We mode. retrograde dipole a is , γ eai(ade 09 ade et Handler 2009, (Handler Pegasi 1 ‡ ℓ o l ustoa frequen- pulsational all for , f . ℓ β 1 = ν e/B tr1 Lac- 12 star Cep/SBP r tr(Pamyatnykh, star Eri requency, , .Orrslsshow results Our t. g drda velocity radial nd e corresponding ies 13 Z M rg or − ⊙ upspoils bump . individual: ν 14 1 n its and , smost is , for we ll y 2 Daszyńska-Daszkiewicz, Szewczuk & Walczak12 Lac OPAL AGSS09 AO=0

cation, which we made both within the zero-rotation approx- 4.50

n=2 imation and including some eﬀects of rotation for frequencies Z=0.010

Z=0.015 which can be mostly affected by it. In Section 4, we present n=1 results of seismic modelling which we made in two steps, i.e., 1) fitting frequencies and 2) fitting the nonadiabatic 4.25 f−parameter. We study the effects of model atmospheres,

opacity data, chemical mixture and opacity enhancement. 13M Conclusions are summarised in Section 5. logL/L

4.00 12M

2 THE VARIABLE STAR: 12 LACERTAE 11M 12 Lacertae (DD Lac, HR 8640, HD 214993) is a well-known

10M pulsating star of B2III spectral type. Studies of the variabil- 3.75 ity of the star have begun in the early twentieth century

4.47 4.44 4.41 4.38 4.35 when Adams (1912) discovered its radial velocity variations. logT eff The period of these changes amounted to P = 0d.193089 as determined by Young (1915). Figure 1. The HR diagram with a position of the observational In the following years 12 Lac had been extensively stud- error box of 12 Lac as determined by Handler et al. (2006). The ied by many authors, e. g. Young (1919), Christie (1926). evolutionary tracks were computed for two values of metallicity, Z =0.010, 0.015, the hydrogen abundance of X = 0.7, initial However, the physical interpretation of the radial veloc- rotational velocity of Vrot = 50 km/s, no overshooting from the ity changes was incorrect; they thought that 12 Lac is a convective core and the OPAL tables. Lines labeled as n = 1, binary system. Stebbins (1917) and Guthnick (1919) re- n = 2 and an asterisk will be discussed latter in the text. ported the light variations with periods of 0d.193 and 0d.1936, respectively, which agreed well with the spectroscopic value of Young (1915). Fath (1938) explained a for 12 Lac were organized. Handler et al. (2006) reported shape of the light curve as an interference between two results on the photometric campaign; they confirmed five close periods of the order of 0d.193. In his further stud- from six already known frequencies and added six new ies, Fath (1947) found three periods; the well-known pri- independent modes, including the low frequency typical for mary 0d.19308902, the secondary 0d.164850 and the third Slowly Pulsating B-type stars (SPB). The frequency values d 0 .1316. However, these periods have not been confirmed determined from these data are: ν1=5.179034, ν2=5.066346, by de Jager (1953), who determined the new value of ν3=5.490167, ν4=5.334357, ν5=4.24062, ν6=7.40705, d the primary period 0 .1930883 and found the secondary ν7=5.30912, ν8=5.2162, ν9=6.7023, ν10=5.8341 and d 0 .197367. In 1956, one of the first worldwide observing νA=0.35529 c/d. Based on the colour photometry campaigns was organized called The International Lacer- Handler et al. (2006) provided the spherical degrees ℓ tae weeks (de Jager 1963). The analysis of these observa- for the five strongest frequency peaks and constraints on tions by Barning (1963) led to a discovery of four different ℓ for the low-amplitude frequencies. Then, Desmet et al. periods; two already known: 0d.19308883, 0d.197358, and (2009) presented results on the spectroscopic campaign. two new ones: 0d.182127, 25d.85. Jerzykiewicz (1978) chose They confirmed 10 independent frequencies discovered from a more homogeneous sample of yellow passband data from photometry. The important result was that in their spec- d this campaign and detected six periods: P1 = 0 .1930754, troscopy the SPB-type frequency has been also detected. d d d P2 = 0 .19737314, P3 = 0 .18214713, P4 = 0 .1874527, Thus, 12 Lac is another known hybrid pulsator in which d d P5 = 0 .0951112 and P6 = 0 .23582180. the β Cep and SPB type modes are excited simultaneously. Sato (1979) examined observational data of 12 Lac to From spectroscopy, Desmet et al. (2009) arrived at a unique test stability of pulsational periods. Next, secular varia- identification of the mode degrees, ℓ, and azimuthal orders, tions of the period were searched by Ciurla (1987), but no m, for four pulsational frequencies. variations larger than 0.08 s per century were found. Af- Dziembowski & Pamyatnykh (2008), based on results ter the great effort made during The International Lacer- from the 2003 world-wide campaign, made seismic modelling tae weeks more photometric data were gathered and anal- of 12 Lac and showed that the rotation rate increases sig- ysed by Jarzebowski et al. (1979) and Jerzykiewicz et al. nificantly towards the star centre. They addressed also a (1984). Smith (1980) examined SiIII λ 4567 line profile problem with excitation of low frequency mode νA. variations. His findings were generally in agreement with In Fig. 1, we show the observational error box of 12 Lac previous results of Jerzykiewicz (1978). The spectroscopic in the HR diagram. The star is in an advanced phase of the analysis based on high spectral and time resolution ob- core hydrogen burning. We adopted the effective tempera- servations had been carried out by Mathias et al. (1994). ture, log Teff = 4.375 ± 0.018 and luminosity, log L/L⊙ = Their frequency analysis revealed a presence of 6 photo- 4.18 ± 0.16 from Handler et al. (2006). In the literature var- metric frequencies in the radial velocity curve. Subsequently ious determinations of metallicity are given. Niemczura & Dziembowski & Jerzykiewicz (1999) investigated the possi- Daszyńska-Daszkiewicz (2005) obtained [m/H]≈−0.2 ± 0.1 bility of existence of the equidistant triplet, ν1, ν4, ν3 (corre- from the ultraviolet IUE spectra, which corresponds to sponding to periods P1, P4 and P3) and performed seismic Z ≈ 0.012 ± 0.003, whereas Morel et al. (2006) derived modelling. Z = 0.0089 ± 0.0018 from the high resolution optical spec- In the years 2003-2004, the two multisite campaigns tra. Therefore, we show evolutionary tracks for two values

c RAS, MNRAS 000, 1–?? The β Cep/SPB star 12 Lacertae 3 of metallicity: Z = 0.015 and Z = 0.010. Computations comes from models of stellar atmospheres and these are pass- were performed using the Warsaw-New Jersey evolutionary band flux derivatives over effective temperature and grav- x code (Pamyatnykh et al. 1998), with the OPAL equation of ity as well as limb darkening, included in the terms D1,ℓ x state (Rogers & Nayfonov 2002), the OPAL opacity tables and D3,ℓ, respectively. Here, we rely on two grids of stel- (Iglesias & Rogers 1996) and the latest heavy element mix- lar atmosphere models: LTE (Kurucz 2004) and non-LTE ture by Asplund et al. (2009), hereafter AGSS09. For lower (Lanz & Hubeny 2007) and consider different values of the temperatures, the opacity data were supplemented with the atmospheric metallicity, [m/H], and the microturbulent ve- Ferguson tables (Ferguson at al. 2005; Serenelli at al. 2009). locity, ξt. The second input is connected with pulsational We assumed the hydrogen abundance of X = 0.7, initial ro- properties and this is the so-called nonadiabatic complex tational velocity of Vrot = 50 km/s and no overshooting parameter f describing a ratio of the amplitude of the radia- from the convective core. Rotational velocity was chosen of tive flux perturbation to the radial displacement at the pho- the same magnitude as in Desmet et al. (2009) who derived tosphere level (e.g. Daszyńska-Daszkiewicz, Dziembowski, Vrot sin i = 36 ± 2 km/s and Vrot = 49 ± 3 km/s. It means Pamyatnykh 2003).The f-parameter is incorporated in the x that 12 Lac rotates at a speed of about 10% of its break-up temperature term, D1,ℓ. crit GM velocity (Vrot ≈ p R ≈ 480 km/s). A comprehensive overview of possible sources of uncer- tainties in the theoretical values of the photometric amplitudes and phases of early B-type pulsators has been studied in details by Daszyńska-Daszkiewicz & Szewczuk (2011). 3 MODE IDENTIFICATION The value of f for a given mode can be either ob- To identify pulsational modes detected in 12 Lac, we tained from the linear nonadiabatic computations of stellar will use the most popular observables, i.e., the ampli- pulsations (Cugier, Dziembowski, Pamyatnykh 1994) or de- tudes and phases of the photometric and radial velocity termined from observations (Daszyńska-Daszkiewicz et al. variations. Here, we rely on the Strömgren uvy photom- 2003, 2005). Here, we applied the two approaches. etry (Handler et al. 2006) and the radial velocity changes To identify the degree ℓ of pulsational modes of 12 Lac (Desmet et al. 2009) determined as the first moment of the using the theoretical f-parameter, we compared the theo- SiIII 4552.6A˚ line. We performed mode identification in two retical and observational values of the photometric ampli- steps. Firstly, we determined the degree ℓ ignoring all effects tude ratios and phase differences in various pairs of pass- of rotation on pulsation. This is a commonly used approach bands. To this aim, we considered models from the error and in the case of 12 Lac, which rotates at a speed of about box corresponding to masses from 10 to 12M⊙ and used 50 km/s, seems to be justified for the most frequencies. How- the nonadiabatic pulsational code of Dziembowski (1977). ever, this value of rotation can already couple pulsational To check how our identification of ℓ is robust we considered modes and affects properties of slow modes such as the fre- two values of metallicity, Z = 0.015 and 0.010, two chemi- quency νA detected in 12 Lac. Therefore, in the second step, cal mixtures, AGSS09 and GN93 (Grevesse & Noels (1993)), we checked whether the dominant frequency, ν1, can be a ro- two sources of the opacity tables, OPAL (Iglesias & Rogers tational coupled mode and apply traditional approximation 1996) and OP (Seaton 2005), and two sets of stellar model to identify the angular numbers of the low frequency, νA. atmospheres, LTE and non-LTE. The hydrogen abundance was fixed at X = 0.7 and no overshooting from the convective core was allowed. We considered the values of ℓ from 0 3.1 The zero-rotation approximation up to 6 and the frequency range appropriate to the observational values. If all effects of rotation on stellar pulsation are neglected Using the theoretical values of f, we arrived at the fol- then identification of the mode degree, ℓ, from photometric lowing identification. The dominant mode ν1 as well as ν2 amplitudes and phases are independent of the inclination an- are certainly dipoles (ℓ = 1). Identification of ℓ for ν3 and ν5 gle, i, intrinsic mode amplitude, ε, and azimuthal order, m. is also unique; they are quadruple modes (ℓ = 2). There is This is also a disadvantage of disregarding rotation because also no doubt about identification of ℓ for ν4; certainly this the full geometry of mode cannot be determined. Within the is a radial mode (ℓ = 0). Identification of ℓ for the remaining zero-rotation approximation, the complex amplitude of the frequencies is ambiguous: ν6 is most likely a dipole, but it light variation in the x passband in the framework of linear may be also ℓ = 2 or 3; ν7 can be ℓ = 1, 2, 3, 5; for ν8 we theory can be written as (e.g., Daszyńska-Daszkiewicz et al. were not able to associate any values of ℓ; ν9 is either ℓ = 3 2002) or 1 and ν10 can be ℓ = 1 or 2 or 3. Finally for the SPB like x m x x x mode, νA, we got ℓ = 1 or 6, but because of the strong aver- A (i)= −1.086εY (i, 0)b (D + D2,ℓ + D ), (1) ℓ ℓ 1,ℓ 3,ℓ aging effects which increase with the harmonic degree, the m x where Yℓ is the spherical harmonic, bℓ is disc averaging ℓ = 6 identification is much less probable. These identifica- x x factor, and D1,ℓ, D2,ℓ and D3,ℓ are the temperature, geo- tions are independent of the metallicity, opacity data, mix- metrical and pressure effects, respectively. Expressions for ture and atmosphere models. An influence of the different these quantities can be found in Daszyńska-Daszkiewicz et stellar parameters (within the error box) is also negligible. al. (2002). In the second approach the mode degree ℓ is determined The radial velocity variations due to pulsations are de- simultaneously with the empirical values of f by fitting the fined as the first moment of the well isolated spectral line theoretical values of the photometric amplitudes and phases and the formulae is given in Dziembowski (1977). to the observational counterparts (Daszyńska-Daszkiewicz To compute the theoretical values of the photometric et al. 2003, 2005). To this end we considered stellar parame- amplitudes and phases two inputs are needed. The first one ters from the center and edges of the error box. We checked

c RAS, MNRAS 000, 1–?? 4 Daszy´nska-Daszkiewicz, Szewczuk & Walczak

Table 1. The most probable identification of ℓ for the 12 Lac fering by 2 (Soufi, Goupil & Dziembowski, 1998). The two frequencies obtained from two methods described in the text. In modes are close if the difference between their frequencies is the last column, we put the values of m determined by Desmet of the order of the rotational frequency, i.e., |ν1 − ν2|≈ νrot. et al. (2009). Then, the complex photometric amplitude due to a pulsational mode coupled by rotation is given as a sum of the frequency AV the most probable ℓ m photometric amplitudes of modes contributing to the cou- [c/d] [mmag] theoret. f empir. f Desmet2009 pling (Daszyńska-Daszkiewicz et al. 2002)

coup ν1=5.179034 38.1 1 1 1 A λ (i)= Pk akAλ,k(i), (2) ν2=5.066346 16.0 1 1 0 ν3=5.490167 11.1 2 2 1 where A(i) is given in Eq. 1. The coefficients ak results from ν4=5.334357 10.0 0 0 0 linear perturbation theory of rotating stars and describe con- ν5=4.24062 3.6 2 2 0,1,2 tributions of the ℓk-modes to the coupled mode. ν =7.40705 2.0 1,2,3 1,2 – 6 According to Desmet et al. (2009), the surface equato- ν =5.30912 2.0 1,2,3,5 – – 7 rial rotational velocity of 12 Lac amounts to about 50 km/s. ν8=5.2162 1.3 ? 2 – Adopting the stellar radius of about 7 R⊙ we get a surface ν9=6.7023 1.3 3,1 3,1 – ν10=5.8341 1.3 1,2,3 1,2 – rotational frequency of about νrot = 0.14 c/d. As has been νA=0.35529 5.0 1,6 1,4 – shown by Daszyńska-Daszkiewicz et al. (2002), already rotation of this order can couple modes. The dominant mode of 12 Lac, which has been identified also the effects of adopted model atmospheres, i.e., effects as the ℓ = 1 mode, has the photometric amplitude much of the microturbulent velocity, ξt, metallicity, [m/H], as well higher than the other ones; its value of AV is about 2.5 as the non-LTE effects. All the above parameters do not larger than for ν2. Because usually, it is the radial mode change identification of ℓ although a quality of the fit ob- which dominates, we checked whether ν1 could be tained with the LTE and non-LTE models can differ signif- the ℓ = 0 mode coupled with the ℓ = 2 mode by icantly. Usually, with the non-LTE atmospheres we got the rotation. Pulsational models which satisfy the rotational better goodness of the fit. From this approach, we found a coupling conditions for the frequency ν1 can be found in a unique determination of ℓ for the six frequencies. For the first very wide range of stellar parameters within the 12 Lac error five modes with the largest amplitudes results are consistent box. with the determinations obtained with the theoretical values In Fig. 2, we show an example of a mode coupling be- of the f−parameter. In the case of some other frequencies tween the ℓ = 0,p1 and ℓ = 2,g1 modes with frequencies we succeed in getting better constraints on ℓ, i.e., ν6 and before the coupling of about 5.17 and 5.15 c/d, respec- ν10 can be ℓ = 1 or 2 and ℓ = 3 is excluded, ν8 has been tively. The parameters of the model are: M = 9.43M⊙, identified uniquely as a quadruple. The degree of ν7 is inde- log Teff = 4.3411 and log L/L⊙ = 3.975. The value of terminable because this frequency has not been detected in Vrot sin i = 36 km/s was kept constant and the values of Vrot spectroscopy. The slow mode νA is now ℓ = 1 or 4. are indicated at each point. The pure modes with ℓ =0, 1, In Table 1, we list frequencies of 12 Lac and the most 2 are marked as big dots. The observational position for the probable identification of the degree ℓ from the two ap- frequency ν1 is shown as an asterisk. As we can see the best proaches. The last column contains the values of m as deter- agreement is obtained if ν1 is considered as the pure ℓ = 1 mined by Desmet et al. (2009). As we can see, identifications mode. The same conclusion was reached for other models of of ℓ from the two methods are consistent. In the case of νA, 12 Lac and other pairs of modes. Therefore, in our further the common identification is ℓ = 1 and this is the most likely analysis we assume that ν1 is the pure ℓ = 1 mode. degree. Moreover, the visibility of modes with ℓ = 4 and 6 In the case of high-order g-modes, whose frequencies are is much lower. Our identifications of ℓ agree with those ob- of the order of the rotational frequency, incorporation of the tained by Handler et al. (2006). Coriolis force is crucial. One way to account for these effects From the second approach, besides the degree ℓ, we get of rotation is the so-called traditional approximation (e. g. the empirical value of f, which has a great asteroseismic Lee & Saio 1987, Townsend 2003). In the case of the low potential, and the intrinsic mode amplitude multiplied by frequency of 12 Lac, νA = 0.35529 c/d, the spin parameter, the aspect factor,ε ˜ = εY (i, 0). These quantities will be dis- s = 2νrot/ν, amounts to 0.8-1.0 depending on the model. cussed in subsequent sections. At such value of s, the photometric amplitudes and phases can be changed. The effects of rotation on photometric observables for high-order g-modes excited in the SPB star 3.2 Including effects of rotation models were studied by Townsend (2003). The formulas for the light and radial velocity variations, we use here, were Rotation causes that the 2ℓ + 1 degeneracy is lifted what given by Daszyńska-Daszkiewicz, Dziembowski & Pamyat- allows to determine the full geometry of pulsational modes, nykh (2007). The semi-analytical form of the formula for the i.e., the spherical harmonic degree, ℓ, and azimuthal order, complex amplitude of the light variations due to pulsations m. It also complicates mode identification because the pho- in a high-order g-mode is as follows: tometric amplitude ratios and phase differences become de- pendent on the inclination angle and rotational velocity. Atrad λ (i)= Pj γℓj Aλ,ℓj (i), (3) In the case of low order p- and g-modes, the most important effects of rotation is mode coupling occurring for where γℓj are coefficients of the expansion of the Hough func- close frequency modes with the same values of m and ℓ dif- tion into the series of the Legendre functions (see Daszyńska-

c RAS, MNRAS 000, 1–?? The β Cep/SPB star 12 Lacertae 5

3 3

'=0

'=2

60 50

36 2 2 y

50 /A u

60-90

A =1

1 1

40 90

-1.5 -1.0 -0.5 0.0 -1.5 -1.0 -0.5 0.0

- [rad] - [rad]

u y u y

Figure 2. The diagnostic diagrams with Str¨omgren passbands u and y, showing positions for coupled ℓ = 0,p1 and ℓ = 2,g1 modes excited in the model with M = 9.43M⊙, log Teﬀ = 4.3411 and log L/L⊙ = 3.975. Big dots indicate positions of the pure modes with ℓ = 0, 1, 2. The numbers at small dots are the values of the rotational velocity in km/s. The values of the inclination angle result from the condition Vrot sin i = 36 km/s. An asterisk denotes an observational position of ν1 with errors.

inclination inclination

67.767.7 38.1 27.527.5 21.7 18.018.0 15.3 13.413.4 67.767.7 38.1 27.527.5 21.7 18.018.0 15.3 13.413.4

l=1 m=0

l=1 m=+1

2.0

l=1 m=-1

l=2 m=0

l=4 m=-1

1.5

l=1 m=0

2 l=1 m=+1 ) [km/s]

l=1 m=-1 rad

l=2 m=0

l=3 m=-1 A(V

1.0

l=4 m=-1

0.5

0.0

40 80 120 160

V [km/s] V [km/s]

r ot r ot

2 Figure 3. Left panel: the values of χ for the νA frequency as a function of the rotational velocity. Only unstable modes in the model with M = 9.91M⊙, log Teﬀ = 4.3546 and log L/L⊙ = 4.039 were included. The corresponding values of the inclination angle result from a condition Vrot sin i = 36 km/s and are indicated at the top X axis. Right panel: the resulting amplitudes of the radial velocity variations calculated for the optimal values of ε. The observational range of A(Vrad) is marked as horizontal lines.

Daszkiewicz, Dziembowski & Pamyatnykh, 2007) and A(i) discriminant includes the best value of the intrinsic mode is given in Eq. 1. amplitude, ε, to ﬁt photometric amplitudes and phases in all passbands simultaneously. The value of χ2 is a function In order to determine the angular numbers of νA, we apply the same discriminant, χ2, as proposed by Daszy´nska- of both the inclination angle and rotational velocity. Daszkiewicz, Dziembowski & Pamyatnykh (2008). The χ2 In the left panel of Fig. 3, we plot the dependence of

c RAS, MNRAS 000, 1–?? 6 Daszy´nska-Daszkiewicz, Szewczuk & Walczak

12 Lac OP nF AGSS09 X=0.7 Vrot=50

χ2 on the rotational velocity for unstable modes with fre- 4

n=1 quencies close to νA. Again the value of Vrot sin i = 36 km/s was kept constant. We show an example for model with the 3 following parameters: M = 9.91M⊙, log Teﬀ = 4.3546 and 2 2 log L/L⊙ = 4.039. Only modes which give χ < 6 were f

I shown. As we can see, all dipole modes are acceptable. For

1 Z=0.013 Z=0.015 a possible discrimination of the azimuthal order, m, in the Z=0.01

M, T right panel of Fig. 3, we plotted the amplitude of the radial eff 0 velocity variations obtained for the best value of ε as a func-

tion of Vrot. The horizontal lines mark the observed value of -1 A(Vrad) with errors. We can see that only the retrogarde

mode (ℓ = 1, m = −1) has the radial velocity amplitude -2 within the observational range. This identiﬁcation does not

-11 -10 -9 -8 -7

12 Lac OP nF AGSS09 X=0.7 Vrot=50 change if other models of 12 Lac are considered. Thus, we f R assume that νA is a dipole retrograde mode. 4

n=2 =0.0

Z=0.015 Z=0.013

3.3 Radial orders of pulsational modes M, T

eff

Z=0.01

Our assignment of the radial order, n, has been I done within the zero-rotation approximation. For 1 non-axisymmetric frequencies with the identiﬁed azimuthal order, m, we estimated the centroid fre- 0 quency, νnℓ, from the following equation

-1

νnℓm = νnℓ + (1 − Cnℓ)mνrot, (4)

-2 where Cn,ℓ is the Ledoux constant and νrot is the

-11 -10 -9 -8 -7 rotational frequency. f In the HR diagram (Fig. 1), we plotted the lines of con- R stant period P = 0.18746 d, corresponding to the radial Figure 4. Comparison of the empirical (boxes) and theoretical mode ν4, assuming that ν4 is the fundamental (n = 1) or first (lines with symbols) values of f for the radial mode, ν4, on the overtone mode (n = 2). Two values of metallicity, Z = 0.01 complex plane. The upper and bottom panels correspond to the and 0.015, were considered. Both lines are within the error hypothesis of the fundamental and first overtone radial mode, re- box, but the n = 2 line is cut near the middle, for the lower spectively. The theoretical values of the f-parameter were calculated with the OP tables for three values of metallicity, Z = 0.01, metallicity models. It is caused by a lack of the main se- 0.015 and 0.02. All models reproduce the frequency ν4 and are quence models reproducing ν4 with masses lower than about inside the error box of 12 Lac. The thin, black line in the bot- 11 M⊙ for Z = 0.01. tom panel indicates the instability border. Models to the right In order to identify the radial order, n, of ν4, we com- of this line are unstable. In the upper panel, all models are un- pared the empirical and theoretical values of the nonadia- stable. The arrows indicates the increasing masses and effective batic f-parameter (Daszyńska-Daszkiewicz, Dziembowski & temperatures. Pamyatnykh, 2003, 2005). We chose models which are located on lines of constant period and are inside the observational error box of 12 Lac. For these models, we computed the theoretical values of f for the frequency ν4 and com- also much less likely. The effect of the overshooting pa- pare them with the empirical counterparts. The results are rameter, αov , on identification of n is completely neg- presented in Fig. 4, where the real part of the f-parameter, ligible. Thus, we adopted that ν4 is the radial fundamental fR, is plotted against its imaginary part, fI. The empirical mode in our further analysis. Our identification of n for the values are marked as a box and the theoretical ones as lines frequency ν4 confirms that of Dziembowski & Pamyatnykh with symbols corresponding to the three values of metallic- (2008) ity: Z =0.010, 0.013, 0.015. All models were computed with Following Desmet et al. (2009), we assumed that the the OP opacities and no overshooting from the convective dipole mode ν2 is axisymmetric (m = 0). A survey of pul- core. In the upper panel of Fig. 4, we assumed that ν4 is sational models with different masses, M, metallicities, Z, the fundamental mode, while in the bottom panel - the first and core overshooting parameters, αov, showed, that if ν4 overtone. is the radial fundamental mode, then ν2 can be only the g1 As we can see, the agreement is achieved if ν4 is the mode. To account for the rotational splitting of ν1 and ν2, radial fundamental mode. Moreover, all these models are which belong to the ℓ = 1, g1 triplet, Dziembowski& Pamy- unstable, while in the case of the first overtone, only models atnykh (2008) suggested a nonuniform rotation with an in- to the right of the thin, black line labeled as η = 0.0 are un- crease of its rate between the envelope and core. Therefore, stable. We need quite high metallicity and large masses to we considered rotational splitting according to the two val- make ν4 unstable as a first overtone. With the OPAL data, ues of the rotational velocity: 50 and 80 km/s. The values of identification of n is not so unambiguous but a requirement the Ledoux constant were appropriate for models of 12 Lac. of instability and lower values of Z make the first overtone Identification of n for the first five modes is consistent for

c RAS, MNRAS 000, 1–?? 12 Lacertae, X=0.7, OPAL, nF, AGSS09, Vrot=50 The β Cep/SPB star 12 Lacertae 7

l ogT the chemical mixture (AGSS09 and the one derived for 12 l ogT

=0.0, =0, p OPAL

1 Lac by Morel et al. (2006)). 0.5

ef f

ef f = 4.357

= 4.375 =0.0, =1, g

1 4.1 Fitting frequencies

0.4 8.8M Fitting two, well-identiﬁed frequencies, ν4 and ν2, for an as-

9.0M

l o g T ov sumed hydrogen abundance, X = 0.7, allow to relate the

9.2M

0.3 ef f mass, M, and the overshooting parameter, αov, of seismic

= 4 .3 9 3

9.4M models with their metallicity, Z. These relations are presented in Fig. 5, where the overshooting parameter is plotted 9.6M

l ogL / L

0.2 versus metallicity. Models with constant masses are marked

9.8M

l ogL / L with black, thin lines. Thick, solid line indicates instability

l ogL / L

10.0M

= 4.18 border for the radial fundamental mode, while thick, dotted

= 4.02 10.2M 1 0.1 line - for the dipole g mode. The instability borders of these

= 4.34 10.4M two modes almost overlap and cover a very narrow range of

10.6M metallicity, i.e., Z ∈ (0.0105−0.0115). Models to the right of

11.0M 12.2M 11.8M 11.4M these lines are unstable. We marked also the lines of constant 0.0

0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 eﬀective temperature and luminosity. The showed values of

Z log Teﬀ and log L/L⊙ correspond to the center and edges of

Figure 5. The overshooting parameter, αov, as a function of the observational error box. metallicity, Z, for seismic models of 12 Lac found from fitting As we can see, there are a lot of seismic models of 12 the frequencies ν4 and ν2 (the modes ℓ = 0, p1 and ℓ = 1, g1, Lac fitting two frequencies, ν4 and ν2. An accuracy of this respectively) for the hydrogen abundance of X = 0.7 and the fitting amounts to 10−6 c/d. In general, models with a given OPAL opacities. There are plotted also lines of constant mass value of the core overshooting have smaller masses, if they (thin, black), effective temperature (black), luminosity (grey) and have larger metallicity. On the other hand, models with con- instability borders for the radial p1 mode (thick, solid) and dipole stant mass have increasing metallicity with decreasing over- g mode (thick, dotted). Models to the right of these lines are 1 shooting. These results are similar to those obtained for γ unstable. The grey area indicates models inside the error box. Peg (Walczak & Daszyńska-Daszkiewicz 2010, Walczak et al. 2013). The number of these seismic models can be limited if we take into account only unstable models and inside the

these two values of Vrot. The ν1 dipole mode was identified as observational error box (grey area). This slightly narrow the the prograde mode (m = 1) and it is the second component metallicity parameter, Z, which should be from about 0.010 of the g1 triplet. The ν3 quadruple mode is the prograde up to 0.024. Unfortunately, we are not able to constrain the mode (m = 1) and it is a component of the g1 quintuplet. overshooting parameter, αov. There are seismic models with For the ν5 quadruple mode, Desmet et al. (2009) were able no overshooting from the convective core as well as with to derive only that m > 0 what for most models fit to the αov ∼ 0.5. g2 quintuplet. In the next step, we will check whether these models re- As we have shown in Section 3, νA is the dipole retro- produce other well identified frequencies, i.e., ν3 = 5.490167 grade mode (ℓ = 1, m = −1). Identification of the radial c/d - a quadruple prograde mode (ℓ = 2, m = 1), ν5 = order n for low frequency modes is very sensitive to the 4.24062 c/d - a quadruple prograde mode (ℓ = 2, m> 0), adopted value of the rotational velocity. Assuming Vrot=50 and νA = 0.35529 c/d - the dipole retrograde mode (ℓ = km/s we got that νA is a g15 or g16 mode and for Vrot=80 1, m = −1). The centroid values of these frequencies were km/s - g13 or g14 are allowed. calculated according to Eq. (12) for two values of the rota- The remaining frequencies of 12 Lac do not have un- tional velocity, 50 and 80 km/s. Lines of models fitting addi- ambiguous identifications of the angular numbers so their tionally one of these three frequencies are applied in Fig. 6. radial orders cannot be determined as well. The left and right panels include models computed with the rotational splitting for Vrot = 50 and 80 km/s, respectively. The accuracy of the fitting of these three frequencies ranges from 10−5 to 5 · 10−4 c/d. As we can see, models computed with Vrot = 50 km/s 4 COMPLEX ASTEROSEISMOLOGY reproduce ν3, ν5 and νA as a g16 mode, only for high val- Our seismic modeling can be divided into two parts. First, ues of the overshooting parameter and quite high metal- we computed pulsational models which fit two well identi- licites. Models fitting νA as a g15 mode are nearly on fied axisymmetric pulsational frequencies. In the case of 12 the horizontal line and give constraints on overshooting: Lac, these are ν4 and ν2, identified as the radial fundamental αov ∈ (0.15, 0.31). Here, ν5 can be only the quadruple sec- and dipole g1 modes, respectively. Then, the theoretical val- toral mode (ℓ = 2, m = +2). In the case of models com- ues of the complex nonadiabatic parameter f corresponding puted with Vrot = 80 km/s, we obtained lower values of the to these two frequencies were compared with the empiri- overshooting parameter. The frequency νA has now lower cal counterparts. We termed this approach complex seismic radial order: n = 13 for αov ∈ (0.05, 0.24) and n = 14 for model. Here, we construct complex seismic models for two αov ∈ (0.4, 0.5) The frequency ν5 can be either m = +2 for sets of the opacity data (OPAL and OP), two sets of the αov ∈ (0.1, 0.38) or m = +1 for αov ≈ 0.5 and Z > 0.018. model atmospheres (LTE and non-LTE), and two sets of It is worth to notice that models fitting additionally

c RAS, MNRAS 000, 1–?? niestab. od Z=0.0104 12 Lacertae, X=0.7, OPAL, nF, AGSS09, Vrot=50 12 Lacertae, X=0.7, OPAL, nF, AGSS09, Vrot=50 -niestabilny, niestab. od Z=0.0109

3 8 Daszy´nska-Daszkiewicz, Szewczuk & Walczak 3 5

niestab. od Z= 0.00918. - stabilne stabilny rozczepienie dla Vrot=50 km/s rozczepienie dla Vrot=80 km/s

5 A

, = 1,

A OPAL V =80 km/s , g , , 3 m = 2, rot OPAL = -1 5 16 = 2,

0.5 0.5 g g , , m 1 m 1 = 1 V =50 km/s , 5 = 1

rot

= 2,

8.6M 8.6M , = 1,

A g g , m = -1 , 14 2

m 8.8M 8.8M 0.4 0.4

= 2

9.0M 9.0M

ov ov

= 2, 5

9.2M , 9.2M m

2 = 2

3 , =0.0, =0, p 0.3 0.3

= 2,

9.4M 9.4M

=0.0, =1, g

9.6M 9.6M

1 ,

=0.0, =0, p

0.2 0.2 = 1

9.8M 9.8M ,

A = 1, =0.0, =1, g

g 1

, m 15 = -1

10.0M 10.0M

10.2M 10.2M

0.1 0.1

, = 1, g , m = -1

10.4M 13 10.4M

10.6M 10.6M

11.0M 11.0M 12.2M 11.8M 11.4M 12.2M 11.8M 11.4M

0.0 0.0

0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027

Z Z

Figure 6. The same as in Fig. 5 but we added lines of models ﬁtting the third frequency: ν3 or ν5 or νA. The left and right panels correspond to models computed with the rotational splitting for Vrot = 50 and 80 km/s, respectively. The grey area indicates the observational error box.

LTE f 1 12 Lacertae, X=0.7, OPAL, nF, AGSS09, Vrot=50 12 Lacertae, X=0.7, OP, nF, AGSS09, Vrot=50 niestab. od Z=0.00915 =2 km/s

5 t

niestab od Z=0.00906, , =1, g niestab. od Z=0.0235 =2 km/s

3 A 14 rozczepienie dla Vrot=80 km/s t

OP OPAL V =80 km/s f( )

, V =80 km/s A rot 5 = 2,

rot 0.5 0.5 LTE

LTE g

, m 1 f( ) = 1 f( )

f A

( 8.6M

8.6M , , = 1, = 1 , A A g + 4 g , , m m = -1 = -1 1 4 14 8.8M

f( ) 8.8M 0.4 0.4 A

) 2

= 2 , 9.0M 5

9.0M g

ov , m ov 2 = 2

= 2, 5 9.2M

, 9.2M m

2 = 2

3 ,

0.3 0.3

=0.0, =0, p 9.4M = 2, f( + ) 9.4M 1

4 2

, =0.0, =1, g 9.6M 3

9.6M 1 1 , =0.0, =0, p = 2 ,

9.8M

0.2 = 1 0.2 g 9.8M

, =0.0, =1, g 1

1 , 10.0M

A = 1 , = 1

10.0M g

13 m 10.2M = -1

10.2M

0.1 0.1

10.4M

, = 1, g , m = -1

13 10.4M

10.6M

12.2M 11.0M 11.8M 11.0M 11.4M 12.2M 11.8M 11.4M

0.0 0.0

0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027

Z Z

Figure 7. The same as in the right panel of Fig. 6, but we marked models ﬁtting the empirical values of the nonadiabatic f-parameter (hatched areas) of both the radial fundamental mode and dipole g1 mode (labeled as f(ν4 + ν2). In the left panel, we used the OPAL data and in the right panel the OP opacities. In both cases we assumed the AGSS09 mixture and the rotational splitting for Vrot = 80 km/s. The LTE model atmospheres with the microturbulent velocity of ξt = 2 km/s were used.

the frequency νA are usually on the nearly horizontal lines, As we can see from the right panel of Fig. 6, there are a i.e., they have nearly constant overshooting regardless of few intersections of two lines which means that we have mod- the metallicity. Thus, the high-order g-modes could be po- els fitting four frequencies simultaneously: (ν4,ν2,ν3,νA) or tentially good indicators of αov but proper identification of (ν4,ν2,ν5,νA) or (ν4,ν2,ν3,ν5). n is crucial. A similar behaviour can be seen in the case of models fitting additionally ν5 except for the small values of metallicity, Z < 0.01. On the other hand the line of 4.2 Fitting the f-parameters models fitting additionally the frequency ν3 corresponding Each pulsational frequency is associated with the non- to (ℓ = 2,m = +1,g1), has the largest slope, i.e., the largest adiabatic complex parameter f which gives the ratio of the dependence of αov on Z. In general, models fitting pressure bolometric flux changes to the radial displacement. This modes form the slopped lines on the the αov vs. Z plane whereas models fitting gravity and mixed modes - nearly quantity is very sensitive to properties of subphotospheric horizontal lines. layers where mode driving occurs. In the case of B-type pulsators, f shows a strong dependence on metallicity and

c RAS, MNRAS 000, 1–?? 12 Lacertae, X=0.7, OPAL, nF, AGSS09, Vrot=50 NLTE NLTE f 1 12 Lacertae,=2 X=0.7, km/s OP, nF, AGSS09, Vrot=50 f 1 The β Cep/SPB star 12 Lacertae 9 t

=2 km/s

=0.0, =0, p OP OPAL =0.0, =0, p

1 1

0.5 0.5

non-LTE non-LTE

=0.0, =1, g =0.0, =1, g

1 1

(

+ 8.8M

4 0.4 8.8M 0.4

)

9.0M

9.0M ov

9.2M

9.2M f

( 0.3 0.3

+ 9.4M 4 9.4M

( ) 2 I 9.6M

9.6M

+ 4

0.2 0.2 9.8M

) 9.8M 2

10.0M

10.0M f f ( + )

4 2 ( 10.2M R I

10.2M

+ 0.1 0.1 4 f ( + )

10.4M

4 2

R ) 10.4M 2

10.6M

11.0M 12.2M 11.0M 12.2M 11.8M 11.4M 11.8M 11.4M

0.0 0.0

0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027

Z Z

Figure 8. The same as in Fig. 7, but we used the non-LTE model atmospheres in order to derived the empirical values of the f-parameter. There are also shown models which ﬁt separately the real (vertical bars) and imaginary (horizontal bars) parts of f.

opacity data. Thus, a comparison of the theoretical and em- peratures and luminosities. Instability borders for the ra- pirical values of f may yield constraints on these quanti- dial fundamental and dipole g1 modes are slightly larger for ties (Daszyńska-Daszkiewicz, Dziembowski & Pamyatnykh, the OP tables. The positions of lines representing models 2005). fitting additionally ν5 or νA are quite similar, the differ- In a set of seismic models fitting the frequencies ν4 and ence is that in the case of the OP models, there is no mode ν2, we tried to find those which reproduce also both values of ℓ = 2,m = +1,g2 for ν5. The biggest change is connected the f-parameter within the observational errors. Such mod- with ν3. The line of the OP models fitting the frequency ν3 els exist and are marked as hatched areas in Fig. 7 on the is shifted toward much larger metallicities. Z vs. αov planes and are labeled as f(ν4 + ν2). The left The empirical values of the f-parameter depend slightly and right panels correspond to models computed with the on the model atmospheres, so we decided to check this ef- OPAL and OP data, respectively. To derive the empirical fect. In Fig. 8, we show models fitting the empirical values values of f, we used the LTE model atmospheres with the of f for the ν4 and ν2 frequencies derived with the non-LTE microturbulent velocity of ξt = 2 km/s (Kurucz 2004). models of stellar atmospheres (Lanz & Hubeny, 2007). In As we can see, the hatched regions are quite small, what the left and right panels we used the OPAL and OP opacity substantially reduces the number of models. The OPAL seis- tables, respectively. Additionally, we marked models fitting mic models are slightly outside the observational error box separately the real, fR, (vertical bars) and imaginary, fI , of 12 Lac. They are cooler and less luminous. They have also (horizontal bars) parts of the f-parameter. As we can see, a high overshooting parameter, αov ∼ 0.35 − 0.55. The OP the overshooting parameter of models fitting fI decreases seismic models have smaller metallicity and overshooting. with increasing metallicity. In the case of models fitting the In this case, the area f(ν4 + ν2) is larger and entirely inside real part of f, the dependence of αov on Z is different. First, the error box, suggesting a preference for the OP tables. for low values of Z, the overshooting parameter increases In the case of the high-order g-mode, νA, all OPAL mod- rapidly with metallicity and then, for larger Z, it starts to els fitting this frequency reproduce also the empirical values decrease. With the LTE atmospheres, the shape of the re- of f. The line (ℓ = 1, g14) crosses the area f(ν4 + ν2) for gions fitting the real and imaginary part of f is similar; the αov ≈ 0.5 and Z ≈ 0.014. Also almost all OP models fitting main difference is in positions of the regions. The theoreti- the frequency νA reproduce its value of f; except for models cal values of f for ν4 and ν2 fit the empirical counterparts with the core overshooting of αov ≈ 0.45 and metallicity of derived with the non-LTE atmospheres for slightly larger Z ≈ 0.020 − 0.025. In this case the line (ℓ = 1, g13) crosses metallicity and lower core overshooting. the area f(ν4 + ν2) for αov ≈ 0.2 and Z ≈ 0.011. Thus with both opacity OP tables, we get seismic models reproducing the three frequencies ν4, ν2, νA and their f-parameters 4.3 Effects of the chemical mixture and the simultaneously, but only the OP models are located inside opacity enhancement the observational error box and have ,,reasonable” values of the core overshooting. In the case of ν3 and ν5, no seismic To evaluate the effect of the adopted chemical mixture, model reproduces their values of f in the allowed range of we performed our seismic modeling assuming the photo- parameters, neither with the OPAL and OP data. spheric chemical element abundances of 12 Lac as derived Comparing the OPAL and OP seismic models, we can by Morel et al. (2006). In Fig. 9, we show the results on the conclude that for a given Z and αov, models calculated with αov vs. Z plane. The left and right panels correspond to these two opacity tables have similar masses, effective tem- computations with the OPAL and OP data, respectively. In

c RAS, MNRAS 000, 1–?? 12 Lacertae, X=0.764, OP, nF, AGSS09, mix12Lac, Vrot=50 f 1 NLTE 12 Lacertae, X=0.764, OPAL, nF, AGSS09, Vrot=50, mix-12Lac 10 Daszy´nska-Daszkiewicz, Szewczuk &f Walczak LTE/NLTE, 1

rozczepienie dla 80 , =1, g niestab. od Z=0.01915 niestab do Z=0.0107, -niestab niestab od Z=0.0102 - stabilne niestab od Z=0.00906 A 14 3 5

5 A

5 OPAL = 2, OP

0.5 0.5 g

, mix 12Lac mix 12Lac m 2

= 1

, = 1, g

, m = -1 ,

= 1, g 14

, m = -1 9.2M

14 9.2M

0.4 0.4

LTE

9.4M

9.4M , = 2,

ov ov g 5 , m f = 2

2 ( , 9.6M = 2, 5 9.6M g +

4 , m

2 = 2 0.3 0.3

9.8M ) 2 9.8M

LTE =0, =0, p

, 3 1

10.0M

NLTE = 2, 10.0M

=0, =1, g

10.2M 1 g

0.2 0.2 =0.0, =0, p 10.2M , 1

1 m

10.4M

3 , = 1

10.4M =0.0, =1, g ,

A = 2, f( + ) 1 10.6M = 1, g

4 2

10.6M

g ,

A 10.8M , m = -1 13 = 1, g 0.1 0.1 1 ,

10.8M m

NLTE

, m = -1 11.0M = 1 13

11.0M

11.2M

12.8M 12.4M 12.8M 12.4M 12.0M 11.6M 12.0M 11.6M 11.2M 11.4M

0.0 0.0

0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027

Z Z

Figure 9. The same as in Fig. 5 but for models computed for the 12 Lac mixture. We marked models reproducing the empirical values of the nonadiabatic f-parameter of both ν4 and ν2, obtained with the LTE (horizontal bars) and non-LTE (vertical bars) atmospheres. In the left panel we used the OPAL data and in the right panel the OP tables.

both cases, the empirical values of f were determined adopt- termining the empirical values of f for ν2 and ν4 (see ing both the LTE (horizontal bars) and non-LTE (vertical Sect. 3.1 for the description of the method). bars) model atmospheres. In the next step, we were trying to find a model fitting In a comparison with computations performed with the first five frequencies, ν1, ν2, ν4, ν3, ν5, as well as νA. We AGSS09 and X = 0.7, now the models fitting additionally did it by setting the value of the rotational velocity which the f-parameters of ν4 and ν2 have considerably lower values gives an adequate splitting for ν1, ν3, ν5, νA. We obtained of the core overshooting parameter, while their metallicity Vrot =74.6 km/s what is consistent with the value suggested parameter was not changed much. The mean value of αov by Dziembowski & Pamyatnykh (2008) for the inner part of for models fitting the f-parameters with the 12 Lac mixture the star. In the case of these four mode a significant contri- is approximately 0.13 lower. The lower overshooting occurs bution to the kinetic energy density comes from the g-mode for both OP and OPAL opacities as well as for the LTE and propagating zone and their splitting is mainly determined non-LTE atmospheres. The lines of constant masses and a by the inner layers which presumably rotate faster. This position of a grey area corresponding to the error box were rotational rate agrees also with the value of Vrot obtained also moved towards lower overshooting. These changes are from identification of the angular numbers of the g-mode caused mainly by a higher hydrogen abundance, X = 0.764, frequency, νA, (cf. the right panel of Fig. 3). In Fig. 10, we which we adopted after Morel et al. (2006). Abundances of compare the observed frequency spectrum of 12 Lac with others elements had no significant influence on the results. the theoretical one corresponding to the model reproducing As we can see lines of models fitting additionally one of the the six frequencies. The model has the following parameters: three frequencies ν3, ν5, νA, also follow the general trend, M = 10.28M⊙, log Teff = 4.347 and log L/L⊙ = 4.00, and i.e., a shift towards slightly lower values of αov. In the case was computed for the metallicity Z = 0.0115, overshooting of the OPAL data, models fitting the f-parameters are still parameter αov = 0.39, OPAL tables and the 12 Lac mixture. mostly outside the error box. Amongst seismic models com- We called it MODEL-6ν and it is marked as an asterisk in puted with the OP tables, there are those that fit also the the HR diagram (Fig. 1). The theoretical p-modes were split frequency νA and its values of the f-parameter. Thus, also according to the rotational rate of Vrot = 50 km/s which with the 12 Lac mixture, there exist seismic models fitting corresponds to the outer layers. these three frequencies and corresponding values of the f- As we can see in Fig. 10, the dipole prograde mode parameter simultaneously. reproduces very well the dominant frequency, ν1. The fre- To summarize, we constructed seismic models quency ν6, identified as ℓ = 1 or 2 is quite close to the with two sets of opacity tables, two sets of model centroid of ℓ = 1, p2 and to ℓ = 3, p0 with m = +2, +3. The atmospheres, two values of microturbulent velocity frequency ν9 could be ℓ = 3,m = −3, p0 or ℓ = 2,m = 0, and two mixtures of chemical elements. Now, the p0 but ℓ = 2 was excluded in our identification for this fre- question is which case is ,,the best”. To pre-answer quency. The value of ν7 is not far from ℓ = 2,m = 0, g1 this question, in Table 2, we listed the models with and for ν8, identified as ℓ = 2, the closest counterpart is 2 the minimum values of χ for each case. These mod- ℓ = 3,m = +1, g2. The frequency ν10, identified as ℓ = 1 or els reproduce the centroid frequencies, ν2 and ν4, and 2, is very close to ℓ = 1,m = +1, p1. their empirical values of the f-parameter. The val- In Table 3, we compare the theoretical values of the f- ues of χ2 were obtained from fitting the photometric parameter for MODEL-6ν with the empirical counterparts and radial velocity amplitudes and phases when de- derived with the LTE model atmospheres. The first and sec-

c RAS, MNRAS 000, 1–?? The β Cep/SPB star 12 Lacertae 11

Table 2. A list of the ,,best” seismic models for each considered case. The basic stellar parameters are given in columns 2 − 6. The last 2 two columns contains the values of χ as determined from ﬁtting the photometric and radial velocity amplitudes and phases for ν2 and ν4, respectively.

2 2 Model Z αov M [M⊙] log Teﬀ log L/L⊙ χ (ν2) χ (ν4)

OPAL LTE ξt = 8 0.0135 0.45 9.3027 4.3463 3.9711 2.00 2.76 OPAL LTE ξt = 2 0.0135 0.40 9.5440 4.3513 3.9973 3.54 4.05 OPAL NLTE ξt = 2 0.0160 0.30 9.8440 4.3501 4.0003 2.77 3.45

OPAL mix12Lac LTE 0.0100 0.41 10.289 4.3522 4.0217 3.49 3.84 OPAL mix12Lac NLTE 0.0125 0.25 10.922 4.3549 4.0469 4.54 6.58

OP LTE ξt = 2 0.0115 0.32 10.152 4.3693 4.0846 8.95 5.80 OP NLTE ξt = 2 0.0130 0.24 10.462 4.3704 4.0962 6.20 5.32

OP mix12Lac LTE 0.0100 0.21 11.494 4.3733 4.1322 10.98 6.46

AO M Time[My] Teff lgL lgg R/Rs Xc Mc/M Mov/M Vrot OP mix12Lac NLTE 0.0115 0.12 11.899 4.3749OP AL4.1464 mix12Lac, 7.62 Vrot=50 5.11 km/s, roz=74.65 km/s

0.38500 10.2835009 22.307545 4.346531 3.9994 3.7864 6.7705 0.325785 0.2255 0.3121 46.924

Z=0.0115, X=0.764

7,4 1,8 A 5 9

10 6

p g p m = -3 -2 -1 0 1 2 3

1 1

3 g

p m = -2 -1 0 1 2 g 1

1 2 g

m = -1 0 1

g g p

14 1 1 1

p p p

2 3 1

0.2 0.6 4 5 6 7 8

[c/d]

Figure 10. Comparison of the observed frequencies of 12 Lac (vertical lines) with theoretical ones (symbols) corresponding to MODEL- 6ν ﬁtting six frequencies: ν1, ν2, ν3, ν4, ν5 and νA. The value of the rotational splitting was computed at Vrot = 50 km/s for p-mode frequencies and at Vrot = 74.6 km/s for g-mode frequencies. The mode degrees ℓ = 0, 1, 2, 3 were considered.

ond lines for each frequency correspond to the empirical and usually slightly lower. In the first column of Table 3, we list theoretical values, respectively. There is no ν7 because it was the empirical values of the intrinsic mode amplitude, ε, mul- m not found in the radial velocity variations. For the frequen- tiplied by the spherical harmonic, |ε˜| = |εYℓ (i, 0)|, where i cies ν6, ν9 and ν10, we gave the two values of f obtained for is the inclination angle. mode degrees, ℓ, as listed in Table 1. The differences in f re- For the radial mode, ν4, the value of |ε˜| is equal to |ε|, sulting from adopting different model atmospheres are small; i.e., to the root mean square of the the relative change of the f-parameter derived with the non-LTE atmospheres is the stellar radius, δr/R, over the surface. Because we have

c RAS, MNRAS 000, 1–?? 12 Daszy´nska-Daszkiewicz, Szewczuk & Walczak

m tulated an increase of stellar opacities near the Z-bump Table 3. The empirical values of |ε˜| = |ε|Yℓ (i, 0), the empirical and theoretical values of the f-parameter, the instability, η, and (Pamyatnykh, Handler & Dziembowski 2004; Miglio et. al 2 the value of χE for all detected frequencies of 12 Lac. We used 2007; Dziembowski & Pamyatnykh 2008). This should have the stellar model fitting six frequencies: ν1, ν2, ν3, ν4, ν5 and νA solved the problem with instability of some modes as well (MODEL-6ν). The empirical values of f were derived with the as improve fitting frequencies. We decided to test the arti- LTE model atmospheres. ficial opacity enhancement near this opacity bump. To this aim, we increased the opacity coefficient in the OPAL tables mode |ε˜| fR fI η by 50% in the metal bump. The obtained seismic models fit- empir 0.01431(62) -8.41(37) -1.24(38) — ting two frequencies, ν4 and ν2, were not changed much. The ν ,ℓ = 1 1 teor —— -7.97 −0.11 0.05 masses, effective temperatures and luminosities were similar. As one could expect, the biggest difference is related to the empir 0.00660(37) -7.69(44) -0.20(44) — ν ,ℓ = 1 instability. With higher opacities, all our models have un- 2 teor —— -7.77 −0.23 0.053 stable modes ℓ =0, p1 and ℓ = 1, g1, even for Z = 0.007. It empir 0.00429(41) -10.7(1.3) -0.6(1.3) — turned out, that models calculated with the modified OPAL ν ,ℓ = 2 3 teor —— -8.62 0.23 0.041 table mimic those computed with the original OP data. It is connected with the internal differences between the two empir 0.00631(22) -8.23(33) -0.31(33) — ν ,ℓ = 0 opacity tables; the OP data provide larger opacity coefficient 4 teor —— -8.17 0.06 0.044 near the Z-bump. empir 0.00187(14) -4.53(75) -4.96(75) — However, with the modified opacities we did not man- ν5,ℓ = 2 teor —— -6.52 −0.90 0.087 age to find models fitting the empirical values of the f- empir 0.00029(6) -20.6(4.0) 4.4(4.0) — parameter, neither for ν4 nor for ν2. The increase of the opac- ν ,ℓ = 1 6 teor —— -11.59 3.21 -0.148 ity coefficient changed significantly the real part f whereas empir 0.00046(6) -23.5(3.1) 4.2(3.1) — the imaginary part of f was almost insensitive to the opacity ν ,ℓ = 2 6 teor —— -11.71 3.21 -0.153 increase. Although the increase of the opacity parameter can solve problems with instabilities, a disagreement between empir 0.00163(30) 5.1(1.9) 4.7(1.9) — ν ,ℓ = 2 the empirical and theoretical values of f argues against this 8 teor —— -8.17 −0.07 0.052 change. The same result was obtained for γ Pegasi (Walczak empir 0.00070(12) -6.0(1.3) 3.3(1.3) — & Daszyńska-Daszkiewicz 2010, Walczak et al. 2013). ν ,ℓ = 1 9 teor —— -10.53 1.96 -0.056 empir 0.00300(15) -13.3(1.8) 23.2(1.8) — ν ,ℓ = 3 9 teor —— -10.78 1.91 -0.059 5 CONCLUSIONS empir 0.00056(10) -7.2(1.3) -0.9(1.4) — ν ,ℓ = 1 10 teor —— -9.11 0.68 0.02 The aim of the paper was to construct complex seismic mod- empir 0.00074(17) -4.4(2.2) -3.0(2.2) — els of the β Cep/SPB star 12 Lacertae, i.e., models which fit ν10,ℓ = 2 teor —— -9.20 0.66 0.023 simultaneously pulsational frequencies and the correspond- empir 0.00053(25) 26.3(12.5) -2.9(12.0) — ing values of the nonadiabatic parameter f. As usually, we ν ,ℓ = 1 A teor —— 18.13 5.37 -0.631 began with the mode identification from the multi-colour photometry and radial velocity data. Unambiguous determination of ℓ was possible for the six frequencies of 12 Lac; for the remaining ones usually two possibilities were ob- identification of the mode degree, ℓ, and azimuthal order, tained. Moreover, for the dominant mode, ν1, and the high- m, for ν1, ν2, ν3, ν5 and νA as well as the inclination angle, order g-mode, νA, the effects of rotation on identification o i ≈ 48 , (Desmet et al. 2009), we were able to derive the of ℓ were checked. As a result we got that ν1 is the pure relative change of the stellar radius also for these modes. (non-rotational coupled) dipole mode and νA is the dipole As a result we obtained that |ε| is equal to: 1.57% for ν1, retrograde mode. 0.57% for ν2, 0.32% for ν3, 0.63% for ν4, 0.25% for ν5 (as- We found plenty of models which reproduce the two suming that ℓ = 2,m = 2) and 0.06% for νA. The amplitude centroid frequencies ν4 and ν2, identified as ℓ = 0, p1 and of the relative radius changes caused by radial pulsations ℓ = 1, g1, respectively, and their empirical values of the f- of 12 Lac is similar to that of other β Cep stars. For ex- parameter simultaneously. These models are inside the ob- ample, |ε| of the radial mode for θ Oph is equal to 0.14% servational error box only if the OP opacity tables were used. (Daszyńska-Daszkiewicz & Walczak 2009), for ν Eri - 1.72% Models computet with the OPAL data had too small effec- (Daszyńska-Daszkiewicz & Walczak 2010) and for γ Peg - tive temperatures and luminosities. The next argument for 0.26% (Walczak & Daszyńska-Daszkiewicz 2010). The last the OP tables is that with this data, there exist models column contains the value of the instability parameter η. which fit additionally the high-order g-mode frequency, νA, Some of these modes are stable (η < 0). The most nega- identified as ℓ = 1,m = −1, g13, and its values of f for tive value is for the high-order g-mode, νA, and amounts to quite low values of the core overshooting, αov ≈ 0.2. With −0.63, what asserts the well-known problem of excitation of the OPAL data such models exist only for αov ≈ 0.5. Thus, high-order g-modes in the massive main sequence pulsators. we managed to find models which reproduce the three frequencies and their f-parameters simultaneously. This is the The last issue we would like to examine is the effect first star for which such a good concordance was achieved. of the opacity enhancement. The reason is that for 12 Lac, Moreover, we showed that if one assumes the rotational ve- as well as for some other β Cephei stars, there was pos- locity of the order of 75 km/s for the rotational splitting of

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c RAS, MNRAS 000, 1–??