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Asteroseismology and Oblique Pulsator Model of β Cephei H. Shibahashi Department of Astronomy, School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan and 0 0 0 C. Aerts1 2 Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, n Celestijnenlaan 200 B, B-3001 Leuven, Belgium a J 1 2 1 ABSTRACT v 2 We discuss the oscillation features of β Cephei, which is a magnetic star in which 7 the magnetic axis seems to be oblique to the rotation axis. We interpret the observed 3 1 equi-distant fine structure of the frequency spectrum as a manifestation of a magnetic 0 perturbation of an eigenmode, which would be a radial mode in the absence of the 0 0 magnetic field. Besides these frequency components, we interpret another peak in the / frequency spectrum as an independent quadrupole mode. By this mode identification, h p we deduce the mass, the evolutionary stage, the rotational frequency, the magnetic field - o strength, and the geometrical configuration of β Cephei. r t s a Subject headings: stars: early-type — stars: individual (β Cephei) — stars: magnetic : v — stars: pulsation i X r a 1Postdoctoral FellowoftheFundforScientificResearch,Flanders,Belgium 1 1. Introduction being the dominant variation. From these results, it is concluded that the magnetic field of β Cephei is Beta Cephei (B1III, V = 3.2, V sini = 25 km/s) e oblique to the rotation axis. is the prototype of a group of early-type pulsating In this paper, we try to understand the cause of stars. Since many of the β Cephei-type stars show the quintupletandtoextractinformationontheevo- multi-periodic pulsations, it is expected that a lot of lutionary stage and the geometrical configuration of information can be extracted from the pulsations of β Cephei by using the oscillation data and the most thesestars. Recently,Aertsetal. (1994)andTelting, recent magnetic data. A previous similar study was Aerts, & Mathias (1997) investigated the line-profile presentedin Shibahashi& Aerts (1998)but they still variability of β Cephei in detail, by means of an ex- used the older magnetic field measurements. tensive data set of high-resolution, high S/N spectra obtainedatthe Observatoirede Haute Provencedur- 2. Oscillations of a Rotating Magnetic Star ing onemonth, anddeducedthatthe variationsshow periodicitywithatleastfivefrequencies. Thefrequen- Henrichsetal. (1999)findavariablemagneticfield cies obtained by them from the intensity variation in with a mean value of only 0±5G. We assume here the Si III λ 4574 line are listed in table 1. Note that that the dominant component of the weak magnetic threefrequenciesamongthem(f3,f4,andf5)aresep- field is dipolar. Being influenced by such a magnetic arated from the main frequency f1 =5.250 day−1 by field, the eigenmode, which would be a pure radial −2/6 day−1, −1/6 day−1, and +1/6 day−1, respec- mode in the absence of a magnetic field, is deformed tively. Hence, weregardthatthe four peaks (andthe to haveanaxially symmetric quadrupolecomponent, smallpeakatf1+2/6day−1)areapartofafrequency whose symmetric axis coincides with the magnetic quintuplet with equalspacing. The centralfrequency axis. Hence, the eigenfunction at the surface is char- f1 hasbeenidentifiedastheradialpulsationmodeby acterizedbymeansofasuperpositionofthespherical Aerts et al. (1994) and by Telting et al. (1997). The harmonic with ℓ = m = 0 and that of ℓ = 2 and second highest peak in the periodogram, f2 seems to m=0 with respect to the magnetic axis (Shibahashi be an independent mode, and has been identified as 1994, Shibahashi & Takata 1995); i.e., an ℓ=2 mode by Telting et al. (1997). Y0(θ ,φ )+αY0(θ ,φ ) exp(iωt), (1) (cid:2) 0 B B 2 B B (cid:3) Table 1: Observedfrequenciesofspectral linevariations. whereαisdeterminedbythemagneticfieldofthestar andtheunperturbedeigenfunctionofthemode. Since Frequencies (day−1) Amplitude we are assuming that the magnetic axis is oblique f 4.923 6.89×10−4 to the rotation axis, the aspect angle of the pulsa- 3 f 5.082 1.10×10−3 tion axis varies with the rotation of the star. There- 4 f 5.250 1.00 fore, the contribution of the quadrupole component 1 f 5.380 1.41×10−3 of the eigenfunction to the apparent intensity vari- 2 f 5.417 1.40×10−3 ation changes with time and produces a quintuplet 5 f ? 5.583 1.62×10−4 fine structure with an equal spacing of the rotation 6 frequency in the power spectrum. Mathematically, a spherical harmonic expressed in the spherical coordi- Besidesthepulsationsintheluminosityandinthe nates with respect to the magnetic axis (θ ,φ ) is B B velocity field, a magnetic field has been reported at written in terms of (2ℓ +1) spherical harmonics of several occasions in the past. For a compilation of the same degree ℓ with respect to the spherical coor- these results, see Veen (1993). No clear periodicity dinates (θ ,φ ): L L was found in these old measurements. Very recently ℓ ℓ hnoetwicevfieerl,dHaenndrifcohusnedtaalc.le(a1r9l9y9v)arrei-ambleeafisuerldedwtihthemapage-- Yℓm(θB,φB) = X X (−1)m′d(0ℓm)′(β)dm(ℓ)′m′′(i) m′=−ℓm′′=−ℓ riodof12daysandasemi-amplitudeof90±6G.The UVwindlinesofthisstaralsorevealaperiodicvaria- Yℓm′′(θL,φL)exp(−im′Ωt), (2) tion with both a 6 day and a 12 day periodic compo- where {d(ℓ) (β)} and{d(ℓ) (i)} arethe matricesto nent (Henrichs et al. 1993), the 6 day period clearly mm′ m′m′′ transformthesphericalharmonicsexpressedinterms 2 of the spherical coordinates with respect to the mag- magnetic field strength. Let r be the ratio between neticaxis,tothoseexpressedintermsofthespherical the apparentminimum andthe maximumofthe field coordinates with respect to the line-of-sight. The ex- strength, plicit form of {d(ℓ) (β)} for ℓ = 2 can be found in Bobs,min mm′ r≡ . (4) Kurtz et al. (1989). The angles β,i ∈ [0◦,180◦] are Bobs,max the anglebetweenthe magneticaxisandthe rotation Then it is related with the geometrical configuration axis and that between the line-of-sight and the ro- by (Stibbs 1950) tation axis, respectively. The observable variation is obtained by integrating equation (1) over the visible 1−r tanβtani= . (5) disc(θL =[0,π/2]andφL =[0,2π])aftersubstituting 1+r equation(2)intoequation(1). Sincetheintegralwith respect to φ becomes zero unless m′′ = 0, the am- Substitution of the observedvalues gatheredby Hen- L plitude of the component at ω+m′Ω is proportional richs et al. (1999) together with their error, r ∈ [−2.5,−0.6], leads to tanβtani ∈ [−∞,−2.3] or to d(2) (β)d(2) (i)(Shibahashi1986). Ifanadditional 0m′ m′0 [4.1,∞]. This result is seemingly in contradiction quadrupole component of the magnetic field is taken with the one derived from the amplitude ratio, but into account, the eigenfunction is deformed to have we will show in the Sect.5 that the two estimates are ℓ= 1, ℓ =3, and ℓ= 4 components as well and then compatible with each other and point to almost the the expected power spectrum becomes a nonuplet (a same geometry. 9-foldmultiplet)ratherthanaquintuplet(Shibahashi 1994, Takata & Shibahashi 1994). The additional 4. Identification of the Evolutionary Stage outerside-componentsaredueto the ℓ=3 andℓ=4 components,andhencetheiramplitudesareexpected Thesecondhighestpeakf inthe powerspectrum 2 to be muchsmallerthan the centralfive components. hasbeenidentifiedbyTeltingetal. (1997)asamode with ℓ = 2 and m = +1. Telting et al. (1997) have 3. Deduction of tanβ and tani chosen the rotation axis as the symmetry axis of the pulsationintheirwork. Ifthemagneticeffectsonthe Ifthe staris rotatingwitha periodof6days,then pulsationdominateovertheeffectsduetotheCoriolis the power spectrum of the eigenmode, which would force,asapplieshere,thesymmetryaxisoftheeigen- be the radial mode in the absence of the magnetic functionisthemagneticaxisratherthantherotation field, is expected to reveala quintuplet with anequal axis. Following Telting et al.’s (1997) mode identi- spacing of 1/6 day−1. This can be explained by the fication, we assume that f belongs to a quadrupole observed quintuplet fine structure of f , f , f , f , 2 1 3 4 5 (ℓ = 2) mode, but we assume m = 0 with respect to and an additional very small peak in the power spec- the magnetic axis. As will be shown later, the fol- trum (which we call f ). The relative ratios of the 6 lowing conclusionabout the evolutionarystage is the side-peaks’amplitudes to the central peak amplitude same even in the case of m 6= 0. Under the influence depend onthe strengthof the magnetic field. On the ofthemagneticfield,thismodeisnolongerdescribed other hand, the relative ratios among the side-peak by a single spherical harmonic of ℓ = 2 and m = 0, amplitudes depend onthe anglesβ andi. Byanalyz- and it is deformed to have components of some other ing them, we can determine these angles. If we write ℓ. However,thefinestructureofthismodeisexpected the amplitude of the component at ω+m′Ω as Am′, to be difficult to detect, sincethe amplitude ofthef then (A2+A−2)/(A1+A−1) is given by (Shibahashi component itself is small. 2 1986) Sincewehavetwoindependentfrequencies,f and 1 AA21++AA−−21 =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dd(0(02221))((ββ))dd(2(12200))((ii))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)= 41|tanβtani|. (3) Ffst2iag,ruw.reeTch1aenslheidfotewnpstainftyhelethsfheroeewqvusoeltunhtceiyornavadarriyaialsttmiaoogndeeoofffrtaehqe1u0esMtnac⊙ry. Substitutionoftheobservedamplitudesintotheabove variation,whiletherightpanelshowsthecaseofℓ=2 equation leads to |tanβtani| ≃ 2.2, where we have modes. As the star evolves from the zero age main usedA2 =A−2 sincethepresenceofafrequencypeak sequency,itsradiusincreasesandhencethefrequency atf isnotwellestablished. Thefactortanβtanican decreases. The situation changes during the contrac- 6 be independently estimated from the variationin the tion phase when hydrogen is exhausted in the stellar 3 center, and the frequency increases again in the hydrogen shell-burning phase. The frequency variation oftheℓ=2modeismorecomplicatedbecausetheµ- gradient zone around the convective core grows with evolution as a consequence of conversionof hydrogen into helium. The frequencies of the modes trapped there become higher. When the frequencies of two modes happen to be very close, they never degener- ate but repel each other. This is known as “avoided crossing” (cf. Unno et al. 1989). Fig. 2.—TheHR-diagramof8–20M⊙stars. Thetickmarks on each of the evolutionary tracks indicate Models 1, 10, 20, ···. Thestarswhoseradialmodefrequencycoincideswiththe observed frequency f1 are connected with the solid lines, and those whose quadrupole mode frequency matches with f2 are shown by the dashed lines. The crossing points of the solid and dashed lines are the candidate locations for β Cephei on theHR-diagram. 5. Deduction of i and β The oscillation mode f is identified as the radial 1 Fig. 1.—Frequencyvariationofradialmodes(leftpanel)and fundamental mode in the case (i) or (iii) in the pre- ℓ = 2 modes (right panel) of a 10M⊙ star with stellar evolu- vious section. On the other hand, it is identified as tion. Theverticalscaleisinunitofday−1,andthehorizontal the radial first harmonic in the case (ii). First, let scaleshowsthemodelnumbers(Model1isthezero-agemain- us consider the case (iii). The radius of the star for sequencestar,andModel33isthemodelattheturningpoint on the HR-diagram.). The solid lines indicate the observed the case (iii) is estimated from the stellar evolution frequenciesf1 (leftpanel)andf2 (rightpanel). calculation as R≃6.5R⊙. Since in our consideration 2π/Ω = 6 days and V sini ≃ 25 km/s, this means e By sweeping out evolutionary models of various i ≃ 30◦ or 150◦. Combining this with the first esti- masses,we searchfor stellarmodelswhose radialand mate of |tanβtani| ≃ 2.2 deduced from the power quadrupole mode eigenfrequencies coincide with the spectrum, we obtain β ≃ 75◦ or 105◦. The estimate observedfrequency f1 and f2. Figure 2 shows the se- tanβtani≃−6.14deducedfromtherecentmagnetic ries of these models on the HR diagram, calculated fieldmeasurementsresultsinβ ≃95◦. Weobtainthat by using Pacyn´ski’s (1970) program (X0 = 0.73 and the solutions from both estimates of tanβtani point Z0 =0.02). The two dashedlines showthe models of towardsanalmostequalgeometry. Theorientationof which the quadrupole mode has the same frequency the magnetic axis can only be derived from the mea- as f2, and the solid lines show the models whose ra- surements of the magnetic field. We conclude that dialmodefrequencycoincideswithf1. βCepheimust the anglebetweenthe rotationaxisandthe magnetic be on one of the crossingpoints of the solidlines and axis amounts to some 100◦ in β Cephei in the case of the dashed lines. The candidate is (i) a ∼18M⊙ star scenario (iii). at the middle of the hydrogencore-burning phase, or If we take the case (ii), the radius of the star is (ii) a ∼ 12M⊙ star near the turning point, or (iii) a larger than the case (iii) (R ≃ 8.5R⊙), and hence i ∼ 9M⊙ star at the late stage of the hydrogen core- becomessmaller(i≃20◦ ori≃160◦)andβ becomes burning phase. closeto90◦ (β ≃80◦ or100◦ if|tanβtani|≃2.2and 4 β ≃ 93◦ if tanβtani ≃ −6.1). Though the case (ii) cannot be ruled out, we think the case (iii) is more likely because the radial fundamental mode is more easily excited. The case (i) seems unlikely because of a high mass required. Inthecaseofm6=0,thefrequencyofthequadrupole mode is shifted by the Coriolis force by mCΩcosβ (Shibahashi & Takata 1993). Here C is determined by the equilibrium structure and the eigenfunction, and it is of the order of ∼ 0.1 for the low order p- modes of ℓ = 2. Since |cosβ| ∼ 0.17, the frequency shift due to the Coriolis force is so small that we do not need to change the conclusion about the evolutionary stage discussed in the previous section. We have adopted the case (iii), and have calculated the theoretically expected power spectra and compared them with the observations. Figure 3 was Fig. 3.—Theoreticallyexpectedpowerspectrum. Themodel calculated with an assumption of B = 90 G, is a 9M⊙ star at the late stage of the hydrogen core-burning obs,max phase. The mode f1 is identified as the radial fundamental i=30◦, andβ =95◦ (the upper panel)andβ =105◦ mode. The angle between the rotation axis and the line of (the lower panel). The magnetic field was assumed sight is assumed to be 30◦ and the magnetic axis is assumed to be mainly dipolar with a 10% contribution from a to be inclined to the rotation axis by 95◦ (upper panel) and 105◦ (lowerpanel). Themagneticfieldstrengthisassumedto quadrupole component. (Note that the combination beBobs,max=90G,andthefieldisassumedtobeadipole+ of i and β is reversible.) Figure 3 resembles the ob- quadrupolefield,andthelattercontributionis10%. servedpowerspectrum,anditimpliesthatouridenti- ficationofthepulsationmodes,theevolutionarystage ing of 1/6 day−1 with the rotation period of 12 days ofthestar,andthegeometricalconfigurationarerea- from the oblique pulsator model, we have to assume sonable. that the magnetic field is almost entirely quadrupo- larratherthandipolarandchooseanappropriatege- 6. Discussion ometrical configuration to give only the five compo- TheUVlinevariationofβ Cepheiimpliesthatthe nents among the nonuplet fine structure of the vis- rotationperiodiseither6daysor12dayswiththelat- ible amplitude. The former condition is necessary, ter being more likely as the UV line equivalent-width because, otherwise, the eigenfunction would have an revealsavariationwithalternatingdeepandless-deep ℓ=1 component,whichwouldinduce apairofpeaks minima (Henrichs et al. 1993). Based on this obser- separated from the central peak in the power spec- vation, Henrichs et al. (1999) prefer an oblique rota- trumby1/12day−1. Inthecaseofapurequadrupole tor model with a dipolar magnetic field and with a magnetic field, the eigenfunction is characterized by rotation period of 12 days, the more so since this is means of a superposition of the spherical harmonic the mainperiodfoundin their14recentobservations with ℓ = m = 0 and those of ℓ = 2 and m = 0 and of the averaged value of the magnetic field over the ℓ = 4 and m = 0 (Takata & Shibahashi 1994).The stellar disk. Such a model can explain the UV data ℓ = 4 component induces a nonuplet fine structure. if an equator-on view is assumed. However, if the But, in the case of i ≃ 90◦ or β ≃ 90◦, both of the rotation period is 12 days and the magnetic field is amplitudesatω±Ωandatω±3Ωhappentobecome a pure dipole, then the power spectrum of the line- much smaller than those at ω±2Ω and ω±4Ω, and profile variations must show a quintuplet of an equal thefinestructurelookslikeaquintupletwithanequal spacing of 1/12 day−1 for i 6= 90◦ and β 6= 90◦, or a spacing of1/6day−1 (see figure 4). The combination triplet of an equal spacing of 1/6 day−1 for i = 90◦ of2π/Ω=12daysandV sini≃25km/sindeedleads e or β = 90◦; a quintuplet fine structure of an equal to i ≃ 90◦, and one might consider that this would spacing of 1/6 day−1 is in that case unrealistic. be favorable to explain the apparent quintuplet fine Inorderto getanapparentquintupletofthe spac- structure. However, in the case of a pure quadrupole 5 magnetic field, the observed magnetic field strength We wouldlike to expressour sincere thanks to Dr. should vary as John Telting for helpful discussions. This work was supportedinpartbyaGrant-in-AidforScientificRe- Bobs ∝P2(cosΘ)∝3cos2Θ−1, (6) searchof the JapanSociety for the Promotionof Sci- ence (No. 11440061). where REFERENCES cosΘ=cosβcosi−sinβsinicosΩt. (7) Aerts, C., Mathias, P., Gillet, D., & Waelkens, C. Then, in the case of i ≃ 90◦, the observed magnetic 1994,A&A, 286, 109 field strength is expected to vary with a period of 6 days ratherthan 12 days,andthis is incontradiction Henrichs, H. F., Bauer, F., Hill, G. M., Kaper, L., with the observation (Henrichs et al. 1999). NicholsBohlin,J.S., &Veen, P.M. 1993,inProc. IAU Colloq.139,New PerspectivesonStellar Pul- sation and Pulsating Variable Stars, ed. J. M. Ne- mec & J. M. Matthews (Cambridge: Cambridge Univ. Press), 295 Henrichs,H.F.,deJong,J.A.,Donati,J.-F.,Catala, C., Shorlin, S., Wade, G. A., & Veen, P. M. 1999, A&A, in preparation Kurtz, D. W., Matthews, J. M., Martinez, P., See- man, J., Cropper, M., Clemens, J. C., Kreidl, T. J.,Sterken,C.,Schneider,H., Weiss,W., Kawaler, S. D., & Kepler, S. O. 1989, MNRAS, 240, 881 Pacyn´ski, B. 1970, Acta Astron., 20, 47 Rudy, R. J., & Kemp, J. C. 1978, MNRAS, 183, 595 Shibahashi,H.1986,inHydrodynamicandMagneto- Fig. 4.—Thesameasfigure3butforthealmostquadrupole hydrodynamic Problems in the Sun and Stars, ed. magnetic field and i = 90◦ and β = 50◦. The value of i has Y. Osaki (Tokyo: Univ. of Tokyo), 195 been chosen sothat it is consistent with R≃6.5R⊙, 2π/Ω = 12daysandVesini=25km/s. Thevalueofβhasbeenchosen Shibahashi, H. 1994, in ASP Conf. 76, GONG ’94: sothatr≃−1.4withanassumptionofaquadrupolemagnetic Helio- and Asteroseismology from the Earth and field and i=90◦. The dipole field contribution is assumed to beonly0.01%. Space, ed. R. K. Ulrich, E. J. Rhodes, Jr., & W. Däppen (San Francisco: ASP), 618 Inordertosolvethecontroversyabouttherotation Shibahashi, H., & Aerts, C. 1998,in IAU Symp. 185, periodofthestar,itishighlydesirablethatnumerous New Eyes to See Inside the Sun and Stars, ed. F.- newmagneticfieldmeasurementsbeperformedovera L.Deubner,J.Christensen-Dalsgaard,&D.Kurtz much longertime base than achievedso far. We note (Dordrecht: Kluwer), 395 that the older magnetic field measurements pointed towards very different values of the mean field, rang- Shibahashi, H., & Takata, M. 1993, PASJ, 45, 617 ingfrom70Gupto800G(Rudy&Kemp1978,Veen 1993). The new data obtained by Henrichs et al. Shibahashi, H., & Takata,M. 1995,in ASP Conf. 83, (1999),however,are ofmuch better quality. It would AstrophysicalApplication of Stellar Pulsation, ed. beextremelyimportanttoconfirmtheresultsbyHen- R. S. Stobie & P. A. Whitelock (San Francisco: richsetal. (1999)andtoachieveabetterprecisionof ASP), 42 the strength and the geometry of the magnetic field. Stibbs, D. W. N. 1950, MNRAS, 110, 305 This would allow a critical evaluation of the current theory of pulsations in hot magnetic stars. Takata, M., & Shibahashi, H. 1994, PASJ, 46, 301 6 Telting, J. H., Aerts, C., & Mathias, P. 1997, A&A, 322, 493 Unno, W., Osaki, Y., Ando, H., Saio, H., & Shiba- hashi,H.1989,NonradialOscillationofStars(2nd Edition) (Tokyo: Univ. of Tokyo Press) Veen, P. M. 1993, Undergraduate Thesis, University of Amsterdam, The Netherlands This 2-column preprint was prepared with the AAS LATEX macrosv4.0. 7