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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2) ν The two hybrid B-type pulsators: Eridani and 12 Lacertae W. A. Dziembowski1,2⋆ and A. A. Pamyatnykh1,3 8 0 0 1 Copernicus Astronomical Center, Bartycka18, 00-716 Warsaw, Poland 2 2 Warsaw UniversityObservatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland n 3 Institute of Astronomy, Russian Academy of Sciences, PyatnitskayaStr. 48, 109017 Moscow, Russia a J 6 Accepted 000.Received2007June000;inoriginalform2007June000 1 ] ABSTRACT h The rich oscillation spectra determined for the two stars, ν Eridani and 12 Lacertae, p present an interesting challenge to stellar modelling. The stars are hybrid objects - o showinganumberofmodesatfrequenciestypicalforβ Cepstarsbutalsoonemodeat r frequencytypicalforSPBstars.Weconstructseismicmodelsofthesestarsconsidering t s uncertainties in opacity and element distribution. We also present estimate of the a interior rotation rate and address the matter of mode excitation. [ We useboth the OPandOPAL opacitydataandfindsignificantdifference inthe 1 results.Uncertaintyinthesedataremainsamajorobstacleinprecisemodellingofthe v objects and, in particular, in estimating the overshooting distance. We find evidence 1 for significant rotation rate increase between envelope and core in the two stars. 5 Instability of low-frequency g-modes was found in seismic models of ν Eri built 4 with the OP data, but at frequencies higher than those measured in the star. No 2 such instability was found in models of 12 Lac. We do not have yet a satisfactory . 1 explanation for low frequency modes. Some enhancement of opacity in the driving 0 zone is requiredbut we arguethat it cannotbe achievedby the ironaccumulation,as 8 it has been proposed. 0 : Key words: stars:variables:other– stars:early-type– stars:oscillations– stars:in- v dividual:ν Eridani–stars:individual:12Lacertae–stars:convection–stars:rotation i X – stars: opacity r a 1 INTRODUCTION induced mixing. A comprehensive survey of properties of thesevariableswaspublishedbyStankov&Handler(2005). Transport of chemical elements is still not well under- stood aspect of stellar interior physics. In the case of the PulsationencounteredinβCepstarsfrequentlyconsists upper main sequence there are uncertainties regarding the in excitation of a number of modes, which differ in their extent of mixing of the nuclear reaction products beyond probing properties. They are found most often in evolved the convective core, known as the overshooting problem, objects, wherelow ordernonradial modes havemixed char- as well as, regarding the survival of element stratification acter: acoustic-type in the envelope and gravity-type in in outer layers caused by selective radiation pressure and the deep radiative interior. Frequencies of such modes are diffusion. Closely related is another difficult problem of the very sensitive to the extent of overshooting. Their rota- angularmomentumtransportbecausethereisaroleofrota- tional splitting is a source of information about the deep tioninelementmixing.Theuppermainsequencepulsators, interior rotation rate. Pulsation is found both in very slow β Cephei stars, are potential source of constraints on mod- (<10kms−1)andveryrapid(>200kms−1)rotatingstars. ellingthetransportprocessesmassivestars.Recently,Miglio Thus,thereisaprospectfordisentanglingtheeffectofrota- etal.(2007b)andMontalbánetal.(2007)studiedsensitivity tioninelementmixing.Firstlimitsonconvectiveovershoot- of the oscillation frequencies in B stars to the rotationally ing and some measures of differential rotation have been already derived from data on the β Cep stars HD 129929 (Aerts et al. 2003), ν Eri (Pamyatnykh et al. 2004 (PHD), andAusseloossetal.2004).Unfortunately,bothobjectsare ⋆ E-mail:[email protected] veryslow rotators. To disentangle therole of rotation in el- 2 W. A. Dziembowski and A. A. Pamyatnykh ement mixing, we need corresponding constraints for more tion spectrum, in addition to four, which have been known rapidly rotating objects. for years. Furthermore, data from these campaigns allowed It seems that we understand quite well the driving ef- to determine (or at least constrain) the angular degrees for fect responsible for mode excitation in β Cep stars. The most of the modes detected in this star. A schematic oscil- effect arises due to the metal (mainly iron) opacity bump lation spectrum for this star is shown in the upper panel at temperature near 200 000 K. Yet, the seismic models of of Fig.1. The angular degrees, ℓ, are adopted after De Rid- ν Eri, which reproduce exactly the measured frequencies of der et al. (2004). In the lower panel of Fig.1, we show the thedominantmodes,predictinstabilityinamuchnarrower corresponding spectrum for 12 Lac. All data, including the frequency range than observed. As a possible solution of ℓ-degrees, are from Handler et al. (2006). The two spectra thisdiscrepancy,PHDproposedaccumulationofironinthe are strikingly similar. However, owing to the appearance of bump zone caused by radiation pressure, following solution twocompleteℓ=1triplets,theoneforν Eriismorereveal- of the driving problem for sdB pulsators (Charpinet at al. ing. 1996). PHD have not conducted calculations of the abun- The mean parameters of the two objects are also sim- dance evolution in the outer layers. Instead, they adopted ilar. In Fig.2 we show their positions in theoretical H-R anadhocfactor4ironenhancementshowingthatitleadsto diagram with the errors based on the parallax and pho- a considerable increase of the instability range but also al- tometry data (see caption for details). In the same figure, lowstofitthefrequencyofthetroublesomep2 dipolemode. we show also the evolutionary tracks for selected seismic However,recentcalculationsofthechemicalevolutionmade models of the stars which we will discuss in the next sec- by Bourge et al. (2007) showed that the iron accumulation tion.Both objectshavenearlystandardPopulation Iatmo- intheβ CepproceedsinadifferentmannerthaninthesdB spheric element abundances. In particular, Niemczura and stars.Unlikeinlatterobjects,inphotospheresofβCepstars Daszyn´ska-Daszkiewicz (2005) quotethefollowing valuesof theironabundanceissignificantlyenhancedforaslongasit themetalabundanceparameter[M/H]:0.05±0.09forν Eri isenhancedinthebump.Infact,thephotosphericenhance- and−0.20±0.10for12Lac.Thesenumbersagreewithinthe ment is always greater (P.-O. Bourge private communica- errorswithearlierdeterminationbyGies&Lambert(1992). tion). However, neither ν Eri nor 12 Lac shows abundance These authors also provide the Teff values which are left of anomaly in atmosphere except perhaps for thenitrogen ex- theerrorboxshowninFig.2.Theadoptedbyuspositionin cess, [N/O]=0.25±0.3, in the former object (Morel et al. the H-R diagram indicate that both stars are in advanced 2006). So there is no clear evidence for element stratifica- core hydrogen burningphase. tion and we are facing two problems: what is the cause of The projected equatorial velocity in both stars is low. element mixing in outer layers of this star, and how to ex- Abt et al.(2002) give 20 and 30 km s−1, for ν Eri and plain excitation of high frequency modes (ν > 7 cd−1) and 12 Lac, respectively, with the 9 km s−1 formal error. The theisolated very low frequency mode at ν =0.43 cd−1. corresponding numbers quoted by Gies & Lambert (1992) Unlike most of β Cep stars, where frequencies of de- are (31±3) km s−1 and (39±14) km s−1. For ν Eri, the tected modes are confined to narrow ranges around funda- two values barely agree (within the errors) but are in con- mental or first overtone of radial pulsation, modes in ν Eri flictwiththeseismicdeterminationofPHD,whoderivethe and 12 Lac are found in wide frequency ranges. Both stars value of about 6 km s−1 for the equatorial velocity which arehybridobjects withsimultaneously excitedlow-orderp- we believe tobe a more reliable value. In Section 4, we will and g-modes (typical for β Cep stars) as well high-order g- present our refined seismic estimate of the rotation rate for modes (typical for SPB stars). The greatest challenge is to thisstar and thefirst such determination for 12 Lac. explain how the latter modes are driven. Our work focuses on these two stars, which are best studied but not the only hybrid pulsators. Other such objects are γ Pegasi, ι Her, HD 13745, HD 19374 (De Cat et al. 2007). Very recently Pigulski & Pojman´ski (2008) reported a likely discovery of 3 SEISMIC MODELS AND INTERIOR fiveadditional objects. ROTATION RATE Inthenextsection,wecompareoscillation spectraand provideotherbasicobservationaldataforν Eriand12Lac. For specified input physics, element mixing recipe, and SeismicmodelsofthesetwoobjectsarepresentedinSection heavyelement mixture,weneedthefourparameters: mass, 3, after a brief description our new treatment of the over- the age, and the initial abundance parameters, (X,Z), to shooting. In the same section, we give for both objects the determinestellar model, ifdynamical effects of rotation are estimate of rotation rate gradient in the abundance vary- negligible. The latter is very well justified for the two stars ing zone outside core. Section 5 is devoted to the problem we are considering in this paper. This might suggest that ofmodeexcitation. Finally,in Section6, aftersummarizing fortestingthemixingrecipe,frequenciesformorethanfour theresults,wediscussthematterofelementmixinginouter pulsation modes are needed. This is not true because there layersandwhat isneededforprogress inBstarseismology. areadditional(non-seismic)observationalconstraintsonthe models. On theotherhand,thereare otheruncertaintiesof 2 OSCILLATION SPECTRA AND OTHER modelling. The most important concerns opacity. OBSERVATIONAL DATA Intheapproachadoptedinthispaper,likeinPHD,we fixed X at 0.7, as the changes within reasonable ranges do Themultisitephotometric(Handleretal.2004,Jerzykiewicz not haveany significant effect on frequencies. Furthermore, etal.2005)andspectroscopic(Aertsetal.2004)campaigns we considered models in the expansion phase, so that in- resultedindetectionofninenewmodesintheν Erioscilla- stead of the age we could use Teff as the model parameter, The two hybrid B-type pulsators ν Eri and 12 Lac 3 Figure 1.Oscillationspectraofν Eriand12LacbasedonJerzykiewiczetal.(2005) andHandler etal.(2006) data,respectively. The numbersabovethebarsarethemostlikelyℓ-values,asinferredfromdataonamplitudesinfourpassbands ofStro¨mgrenphotometry. whichismoreconvenient.Therearetwomodificationsinour to distance dov =αovHp(qc), the fractional hydrogen abun- modelling. Firstly, in order to asses uncertainty in opacity, dancewas assumed in theform we constructed models employing both OP (Seaton 2005) and OPAL (Iglesias & Rogers 1996) data and two heavy X =Xc+(q−qc)w[a+b(q−qc)]. (1) element mixtures, the traditional one of Grevesse & Noels Theinputparametersarewandαov.Theaandbcoef- (1993, hereafter GN93) and the new one of Asplund et al. ficientsintheexpressionabovewerecalculatedfromthetwo (2004, 2005, hereafter A04). Secondly, we allow for partial inputparametersbyimposingcontinuityofXanditsderiva- mixing in the overshoot layer. In our reference model, we tive at the top of the layer. The limit w → ∞ corresponds adopted the OP opacities calculated for the A04 mixture to the standard treatment. We wanted to see, whether this and assumed no overshooting. additional degree of freedom may help us to fit frequency To describe partial mixing, we use an additional ad- of a certain troublesome mode. In more general sense, it is justable parameter. In the partially mixed layer, extending important to know whether asteroseismology may teach us fromtheedgeoftheconvectivecoreatthefractionalmassqc about overshooting more than only its distance. 4 W. A. Dziembowski and A. A. Pamyatnykh Figure 2. ν Eri and 12 Lac in the theoretical H-R diagram. Thethe1σ errorboxforthefirststarissimilartothatadopted in PHD, except that it is moved downwards by 0.1 because we donot include now the Lutz-Kelker correction tomeasured parallax. For the second star, the error box is adopted after Han- dler et al.(2006). The evolutionary tracks correspond to seismic Figure 3.Differences between observedandcalculated frequen- models (dots) calculated assuming no overshooting and the ini- ciesofνEri.AllmodelshavebeencalculatedwiththeOPopacity tial hydrogen abundance X = 0.7. The ν Eri model has mass dataandtheA04heavyelementmixture. M =9.625M⊙ andtheheavyelementmassfractionZ=0.0185, andthatof12LachasM =11.55M⊙ andZ=0.015. parameterfreedomindescriptionoftheovershootingforfit- Inourseismic estimateoftheinteriorrotation rate,we tingsixmeasuredfrequencies.Allmodelsareconstructedto makeasimilarassumptionasPHD,thatisinchemicallyho- fitthethreelowest frequencies. Increasein αov and/or inw mogeneousenvelopeweassumeauniformrotationrate,Ωe, meansmoreefficientovershootingandreductionoftheO-C andallowitssharpinwardriseintheµ-gradientzone.Here, values for the higher frequency mode. The price to pay for we additionally assume uniform rotation rate, Ωc, within the improved fit is shown in Fig.4. All models calculated the convective core and a linear dependence of the rate Ω with αov > 0 are outside the 1σ error box in the H-R dia- on the fractional radius, x, in the adjacent µ-gradient zone gram. Still, we regard efficient overshooting and cooler star between x=xc and x=xc,0, which is thefractional radius asapossible way of fittingthethreehigh frequencymodes. corresponding to convective core mass at ZAMS. Thus, we There is, however, another problem regarding this part of usethefollowing simpleexpressionforinteriorrotationrate thespectrum.Thetwopeaksatthehighestfrequenciescan- not be interpret in terms of remaining components of the Ωc if x6xc ℓ=1,p2 tripletbecausethefrequencyseparation shouldbe Ω(x)= Ωc−(Ωc−Ωe)xcx,−0−xcxc if xc6x6xc,0 . (2) similar to that in the p1 case and it is much smaller. We  Ωe if x>xc,0 checkedthatthereisnootherlowdegreemodeinthisvicin- ity, which could be associated with any of the two peaks. Our aimis to determine Ωc and Ωe from measured or in- Any one of them may be a component of the triplet but ferred rotational splitting. certainly not both. ItisimportanttonoticeinFig.3thedifferencebetween 3.1 ν Eridani modes in the O-C changes. For the two highest frequency modes, theshift in w from 2 to8 has asimilar effect to the Seismic models of this star constructed by PHD made use shift in αov from 0.1 to 0.2. For the mode at 6.732 cd−1 ofthethreefrequencies:5.637,5.763,and6.224cd−1,which the effect of the latter shift is much larger and results in wereassociatedwith(ℓ=1,g1),(ℓ=0,p1),and(ℓ=1,p1) sign change of O-C. We identified this mode as ℓ = 4,g1. modes,respectively.Onlythemean frequenciesoftheℓ=1 Thismodewasdetectedonlywiththedatafromthesecond tripletsarematterforseismicmodels.Individualfrequencies photometric campaign (Jerzykiewicz et al. 2005) and no ℓ- probeinternalrotation.Thefrequencyof7.898cd−1 associ- value from the amplitude data has been assigned to it. In ated with ℓ=1 degree could not be reproduced with stan- our seismic models there is no lower degree mode near this dard models. PHD noted that the localized iron enhance- frequency. We may conclude that there is a prospect for ment may remove the discrepancy but, as we pointed out gettingadditionalconstraintsonovershootingoncewehave in the introduction, this is unlikely. Ausseloos et al. (2004) enough frequency data. succeededinfittingthefourfrequenciesonlyforratherunre- As already noted by PHD, the peak at 7.2 cd−1 may alistic chemical composition parameters and effective over- only correspond to ℓ = 2,p1 mode. Its O-C values are ob- shooting. tained assuming m = 0 and they would be much closer to In Fig.3, we show results of exploration of the two- zeroifm=−1(retrograde)identificationisadopted.Atthis The two hybrid B-type pulsators ν Eri and 12 Lac 5 Figure 4. ν EriinthetheoreticalH-Rdiagram.Thethe1σ er- Figure 5. Influence of the choices of the opacity data and the rorboxisinferredfromtheobservationaldata.Theevolutionary heavyelementmixtureonseismicmodelsofνEri.Theevolution- trackscorrespondtoseismicmodels(dots)calculatedfordifferent arytracks werecalculated assumingαov =0. valuesoftheovershootingparameters. point,however,wearereluctanttousethismodeinseismic Table1.TheO-C(incd−1)fortheℓ=1,p2modefrequenciesand the differences between observed effective temperature and that modelconstruction.Higherfrequencymodesaremoresensi- of the seismic model, ∆logTeff = logTeff,obs−logTeff,cal. The tivetouncertainties inouterlayers andweknowthat some observationaluncertainty ineffectivetemperatureis∆logTeff = modificationofthestructureofthislayerisrequiredbecause 0.011. Models with αov >0 were calculated with w = 8. Vrot is we haveto solve theexcitation problem. giveninkms−1. In Fig.5, we show the effect of choice of opacity data and theheavy element mixtureon seismic models. Wemay κ mixture αover O-C ∆logTeff Ωc/Ωe Vrot see that the latter is far less important. Models calculated OP A04 0.0 -0.127 -0.0103 5.55 5.93 withtheOPALdataarehotterandthereforemayaccommo- date overshooting and stay within the error box. However, OP GN93 0.0 -0.151 -0.0075 5.36 5.95 relayingontheOPALdatawillnothelptheℓ=1,p2 mode OPAL A04 0.0 -0.188 -0.0044 5.36 5.99 frequency fit, as the entries in Table 1 show. The smaller difference between measured and model effective tempera- OP A04 0.1 -0.085 -0.0159 5.82 5.91 ture, ∆logTeff, is always associated with greater O-C for OP A04 0.2 -0.034 -0.0244 5.78 5.93 this mode. Therefore, it seems that if future measurements confirm the current value of the effective temperature, the only way of reconciling theseismic and non-seismic observ- calculatedfortheselectedmodels.Thevaluesdifferonlylit- able is an increase of opacity in outer layers. In PHD this tle. Certainly, the main uncertainty of our inference is the was accomplished by an artificial enhancement of the iron x-dependenceof its rate, which was assumed in avery sim- abundance which had virtually no effect on frequencies of plistic form (Eq.(2)). the three modes used in the seismic model but caused the desired -0.1 cd−1 frequency shift of theℓ=1,p2 mode. For a specified stellar model, the two parameters, Ωc and Ωe, describing internal rotation may be directly deter- 3.2 12 Lacertae minedfrommeasuredfrequencysplitting,S =0.5(ν+−ν−), of the two ℓ=1 modes. To this aim we usethe relation In Table 2 we list frequencies and angular degrees of 1 the fivedominant modes in the oscillation spectrum of this S = dxK(x)Ω/2π, (3) star. The frequencies and the ℓ values are from Handler Z 0 et al.(2006) analysis of their photometry data. The m val- where K is the familiar rotational splitting kernel and Ω is ueswere kindlyprovidedto usbyMaarten Desmet and are givenbyEq.(2).Thekernelsforthethesemodes,whichwere based on his analysis of line profile variations. He also pro- plotted by PHD, differ significantly and this is essential for vided us the value of Ve = 52±5 km s−1 for equatorial an accurate determination of Ωc and Ωe. rotation velocity at thesurface. There aretwodifferencesrelativetothat work.Firstly, The first attempt to construct seismic model of wenowhavemeasuredfrequencyofthem=−1component 12 Lac (Dziembowski & Jerzykiewicz 1999) concentrated ofthehigherfrequencytriplet(p1)andΩc denotesnowthe on interpretation of the very nearly equidistant triplet rate in the convective core and not the mean value over (f1,f3,f5). The distances to the side peaks are -0.1558 and the [0,xc,0] range. In Table 1, we give the the values of the +0.1553 cd−1. Determination of the ℓ values by Handler et Ωc/Ωe ratio on the surface equatorial velocity, Vrot =ΩeR, al. (2006) madeall proposed interpretationsinvalid.This is 6 W. A. Dziembowski and A. A. Pamyatnykh Table 2.Thefrequenciesandmodeidentifications for12Lac. Ident. Frequency ℓ m (cd−1) f1 5.179034 1 1 f2 5.066346 1 0 f3 5.490167 2 2or1 f4 5.334357 0 0 f5 4.24062 2 ? f6 7.40705 1or2 ? f7 5.30912 2or1 ? f8 5.2162 4or2 ? f9 6.7023 1 ? f10 5.8341 1or2 ? Figure 6. The splitting kernels for three noradial modes: ℓ = not a rotationally split triplet. Equidistancy enforced by a nonlinear (cubic) resonant mode coupling is also excluded 1,g1 (solid line), ℓ = 2,g1 (dotted line) and ℓ = 2,g2 (dashed line).Trianglesmarkboundariesofthe µ-gradientzone. by the ℓ values. Thus, just a mere coincidence seems the only explanation. Construction ofseismicmodelofthisstarismorecom- Table3.Modelsfittingthefourdominantmodefrequencies(f1− plicated than that of ν Eri because it cannot be done sep- f4)atΩc=ΩeandΩc=4.7Ωe.Thequantity∆5isthefrequency arately of estimate of the internal rotation rate. Therefore, mismatchbetween f5 andν−2,2,2 incd−1. in this paper we limit ourselves to providing evidence for a significant inward rise of the rotation rate. In this pre- Ωc/Ωe Vrot M/M⊙ logTeff logL/L⊙ ∆5 mf3 liminary analysis, we consider only models calculated with theA04mixturewith Z =0.015 and without overshooting. 1.0 95.5 11.771 4.3878 4.194 0.113 1 Upon ignoring all effects of rotation, we determined stellar 4.65 47.1 11.777 4.3891 4.198 -0.134 2 mass and effectivetemperaturebyfittingfrequencies of the ℓ = 0, p1(fundamental) and ℓ = 1, g1 modes to f4 and f2, filechanges.Anadditionalproblemwiththissolutionisthe respectively.WecheckedthatwiththeconstraintsonLand Teff there is no alternative identification of radial orders of frequency distance of nearly 0.3 cd−1 between f5 and the nearest quadrupole mode, (n,ℓ,m)=(−2,2,2). We believe thetwomodesandthatmodelsinthecontractionphaseare that these two discrepancies exclude the hypothesis of uni- excluded. form rotation. The rotation rate of 12 Lac, as determined from spec- Thesplittingkernelsforthethreenonradialmodesplot- troscopy and implied by the ℓ = 1 mode splitting, is fast ted in Fig.6 are very different. This means that there is a enough so that the second order effects must be considered even at this preliminary phase. Within the Ω2 accuracy we potential for deriving significant constraints on differential rotation. However, we havedirect measurement of only one may write νm =ν(0)+D0(Ω2)+Sm+D2(Ω2)m2, (4) rotationalsplitting,theothersmustbeinferredthroughthe frequency fitting. At the equatorial velocity of rotation of whereν(0)isthefrequencycalculated ignoringalleffectsof about 100 km s−1, the mean effect of centrifugal force in- rotation,S isgiveninEq.(3),theD0 andD2 termsinclude ducesfrequencyshifts exceeding0.01 cd−1.Thus,ifwe aim secondordereffectoftheCoriolisforceandthelowestorder atsuchaprecisionofthefit,wehavetotakeintoaccountdif- effectofthecentrifugalforce.Inourtreatmentwerelyonthe ferential rotation in evolutionary models. This we leave for formalism described by Dziembowski & Goode (1992) but our future work. In the present work, the effect of the dif- simplifiedtothecaseoftheshellular(Ω=Ω(r))rotation.In ferentialrotationdescribedbyEq.(2)wasrigorouslytreated our treatment the surface-averaged effect of the centrifugal onlyforcalculationoftherotationalsplittingwhilethemean force is incorporated in evolutionary models and effects of shifts of the multiplets induced the mean centrifugal force the Coriolis force are included in zeroth-order equation for were determined assuming uniform rotation corresponding oscillations.Sotheonlyrotationalfrequencyshiftfornonra- to Ωe. dial modes calculated by means of the perturbation theory WeconsidereddifferentvaluesoftheΩc/Ωe ratiostart- isthatcausedbycentrifugaldistortion.Forradialmodeswe ingwith1andusedthef1−f2frequencydifferencetodeter- add only the shift 4/3νr2ot/ν0 caused by feedback effects of mineΩe.Then,we adjusted themass and effectivetemper- thetoroidal displacement induced by theCoriolis force. ature to fit the frequencies of the four dominant modes. As We began assuming an uniform rotation and deter- we may see in Table 2, there are two possible m values for mined its rate from the f2−f1 frequency separation using the f3 mode. With m= 1 the fit is achieved at Ωc/Ωe = 1 Eq.(4) with coefficient calculated in themodel reproducing andwith m=2at Ωc/Ωe =0.465. Dataonthetwoseismic the the frequencies of f2 and f4 identified as corresponding models are summarized in Table 3. to (n,ℓ,m) = (−1,1,0) and (1,0,0) modes. In this model We believe that our second model is more realistic be- thefrequencyofthe(n,ℓ,m)=(−1,2,1)modeisveryclose cause its surface equatorial velocity agrees within 1σ with to f3. However, the inferred equatorial velocity of over 95 observations. The mismatch between f5 and calculated fre- km s−1 is much larger than that determined from line pro- quencyofthe(n,ℓ,m)=(−2,2,2)modeissomewhathigher The two hybrid B-type pulsators ν Eri and 12 Lac 7 frequenciesshouldbeunstable.Asameasureofmodeinsta- bility we use thenormalized growth rate W η= , R dW dr 0 dr R (cid:12) (cid:12) where W is the global work(cid:12)inte(cid:12)gral. It follows from this definition that η varies in the range (−1,+1) and it is > 0 for unstable modes. The value of η is more revealing than the usual growth rate, because it is a direct measure of ro- bustnessof ourconclusion regarding mode stability. In Fig.8, we show η for ℓ = 1 and 2 modes calculated in wide frequency ranges for our best models of the two stars. The overall pattern of the η(ν) dependence in two starsissimilar.Thereistheinstabilityrangeaccountingfor excitation of most of the observed high frequency modes. Thereisalsoamaximumofηinthelowfrequencyrangenot farfromthepositionofthelowfrequencymodesdetectedin the two objects. There is enough agreement to feel that we are on the right track toward satisfactory seismic models. However, thereis a need for improvement. Comparingourresultsforν Eriwiththosepresentedby PHD,we find a significant increase of the instability range. Now, only the highest frequency modes near ν = 7.9 cd−1 and themode at ν =0.43 cd−1 fall intostable ranges. This Figure 7. Frequencies of high frequency modes detected in 12 change is a consequence of the usage of the OP data. That Lac(veriticallines)andfrequenciesoffallℓ62modes (dots) in these data lead to wider instability ranges than the OPAL the two models (see Table 3). Identified modes are shown with data, has been already noted before (see e.g. Miglio et al. encircled dots. Note than for the model with uniform rotation 2007a).WiththeOPdatawehaveunstableℓ=2high-order (upperpanel), wehavenoidentification forthe f8 peak. g-modes (n=-14 to -20), but their frequencies are still too high by factor at least 1.4. The measured frequency nearly coincideswithηmaximumforℓ=1butthevalueislessthan than in thefirst model. However, it should bestressed that thismodeismostsensitivetodetailedtreatmentofrotation zero,ηmax =−0.14.WiththeOPALdatawefindmaximum in the inner layers. We may see in Fig.6 that the splitting value at nearly the same frequency but ηmax =−0.33. Per- haps,there is still a room for improvementsin opacity that kernel,whichissimilartothemodeenergydistribution,for wouldrendermodesatthelowestfrequencyandthehighest thismodeisverymuchconcentratedintheµ-gradientzone. frequencies unstable. Frequencies of such modes are very sensitive also to over- All modes detected in 12 Lac, except of that at ν = shooting. 0.355cd−1,occurintheunstablefrequencyrange.However, Furthermore, as Fig.7 shows, the second model repro- the difficulty in explaining excitation of the low frequency duces closer frequencies of the remaining modes detected modeisevenhigherthaninthecaseofν Eri.Inpartthisis in 12 Lac. In this model, f6 may be identified as the duetolowermodefrequencyandinparttothelowerheavy p1, ℓ = 1, m = −1 mode, while f7 and f8 may be, respec- element mass fraction adopted (Z =0.015). tively, the m=0 and m=−1 components of the g1, ℓ=2 mode. It should be noted that this identification for f7 im- pliesthatthismodemaybecoupledwithradialmodeasso- ciatedwithf4,whichisveryclose.Theslightfrequencyshift causedbythisproximity(seee.g.Daszyn´ska-Daszkiewiczet 5 DISCUSSION al. 2002) should be included in the fine frequency fitting. We believe that our seismic models of ν Eridani yield a There is aplausible identification of f10 as ap1, ℓ=1. The good approximation to its internal structure and rotation only troublesome peak is f9 but only if indeed ℓ = 1, as butthereisroom for improvementand aneed for afull ex- Handler et al.(2006) proposed. There would be no problem planation of mode driving. We did not fully succeed in the if it was ℓ=2 becausethen thepeak could correspond toa interpretation of the oscillation spectrum of ν Eridani with fundamental (n=0) mode. our standard evolutionary models and our linear nonadia- batic treatment of stellar oscillations. The frequency misfit between the high frequency peak to the ℓ = 1,p2 mode is muchreduced in models built with theOPopacity data al- 4 MODE INSTABILITY lowing large overshooting but such models are much cooler (2σ) than the mean colour of thestar implies. Suchmodels Satisfactoryseismicmodelsshouldaccountnotonlyformea- also nearly avoid the driving problem. Similar conclusions suredmodefrequenciesbutalsofortheirexcitation.Inprac- regarding models allowing large overshooting distance were tice,itmeansthatmodeswewanttoassociatewithobserved reached byAusseloos et al. (2004). Asfor theconsequences 8 W. A. Dziembowski and A. A. Pamyatnykh τmcweusethewell-knownexpression(seee.g.Tassoul2000) for thespeed of themeridional flow ǫR Vmc= f(x)P2(cosθ), (5) τKH where ǫ=Ω2R3/GM, τKH = GRML2 is the Kelvin-Helmholtz time, and 4 ∂lnρ x5 Ω2 f(x)= 1− . 3(cid:16)∂lnT(cid:17)p 1−∇/∇ad (cid:18) 2πGρ(cid:19) Thelastterm,knownastheGratton-O¨pikterm,iskept thoughitisofhigherorderbecauseitisimportantnearthe surfacewheredensityρislow,evenifǫ<<1andotherwise linearizationisvalid.Inourmodelofν Eri,thetheGratton- O¨piktermis-3.4atthedepthwheremostofironissupposed to bepushed up. τKH∆x τmc= (6) ǫ |f| We evaluate τmc at thelayer where T =4×105K. and for ∆x use the distance to the bottom of the convective zone. For the ν Eri model, we have ǫ = 1.2×10−4, τKH = 5.9×104yr, x = 0.907, ∆x = 0.05, f = −15. With this Figure 8. Normalized growth rate as a function of mode fre- b quencyinmodelsofνEri(top)and12Lac(bottom)(thesameas numbers, we get from Eq.(6) τmc = 1.14×106y, which is usedinFig.2).Thefrequenciesofthedetectedmodesaremarked muchlessthantheageofthestar, 1.75×107yaccordingto with vertical lines starting at the η =0 axis. The length is pro- our seismic model. Thus, even in this slowly rotating star portionaltothemodeamplitude. theeffectofrotation cannotbeignored.Forourbestmodel of12Lac,wefindτmcbyaboutfourorderslessthaninνEri. Yet, as we may see in Fig.8, the problem with driving the of usage of the OP instead of OPAL opacity data for mode lowfrequencymodeisevengreaterthaninthecaseofν Eri. instability,ourfindingagreeswithmoregeneralobservation Meridional flow and/or turbulence developed through of Miglio et al. (2007a) that models using the former data instability induced by rotation are expected to prevent ac- predict wider frequency ranges. cumulation of iron in the driving zone and in photosphere. An assessment of theinward rise of theangular ratein This is a likely explanation of apparently normal chemical the µ-gradient zone was made for both, ν Eri and 12 Lac, atmospheric composition of β Cephei stars, including such yielding the values around five for the ratio of the core to slow rotators as ν Eri. This is also thereason to reject iron envelope rate. The surface equatorial velocities in the two accumulationinthedrivingzoneasthesolutionofthedriv- stars are very different. For the former object our seismic ingproblem,whose solution mustbesearched in adifferent estimate gives about 6 km s−1. For 12 Lac, our estimate of way. about 50 km s−1 is less certain because data on rotational Weattachgreatesthopeforsolutionofthedrivingprob- splittingaremuchpoorerbutthevalueagreeswithestimate lem with improvements in stellar opacity data. Thus, we ofDesmet(privatecommunication),basedonhisanalysisof would like to encourage further effort in this field. Reliable line profile changes. There is a large difference in rotation opacity data are essential for B star seismology. Data on rate between the two stars but unfortunately the accuracy modefrequenciesinν Eriand12Lacareabundantandac- ofourmodellingisinsufficientforaddressingthequestionof curate enough for probing rotation and structure of the µ- therelationbetweentheovershootingdistanceandrotation. gradientzoneabovetheconvectivecore.However,toanswer The surface rotation rate in ν Eri is indeed very low openquestionsregardingmacroscopictransportofelements and this was the reason why PHD suggested that chemical andangularmomentum,weneedaccuratemicroscopicinput elementstratificationinthisstarmaybesustained.Hovever, data, especially on opacity. Only if we have measurements acloserlookattheproblemrevealsthattheeffectofrotation of rotational splitting, our inference on differential rotation on element distribution should not be ignored, even at the doesnot rest on veryprecise modelof stellar structure,but equatorial velocity of few km s−1. As Bourge et al. (2007) suchdataarerare.Regardingobservationalwork,weviewas showed for their 10M⊙ model, the two-fold excess of the most important new spectroscopic observations of the two iron abundancein the driving zone is produced in the time stars aimed at improving the value of the mean effective scale comparablewith main sequencelifetime. Theprocess temperature and leading to the ℓ and m identifications for isslowerthaninlessmassiveobjects(Seaton1999)butfast a greater numberof modes. enough so that if it goes unimpeded a significant excess of ironcouldbecreated.Theeffectofrotationcouldbeignored ifthetime,τmc,formeridionalcirculationtotravelfromthe ACKNOWLEDGEMENTS depthwheremostofironismovedup(T ≈4×105K)tothe bottom of the convective zone (T ≈ 2×105K) around the This work was supported by the Polish MNiSW grant iron opacity bump is longer than the star age. To estimate No1 P03D 021 28. 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