Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 31 October 2018 (MN LATEX style file v2.2) The two hybrid B-type pulsators: ν Eridani and 12 Lacertae W. A. Dziembowski1,2⋆ and A. A. Pamyatnykh1,3 1 Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland 2 Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland 3 Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya Str. 48, 109017 Moscow, Russia Accepted 000. Received 2007 June 000; in original form 2007 June 000 ABSTRACT The rich oscillation spectra determined for the two stars, ν Eridani and 12 Lacertae, present an interesting challenge to stellar modelling. The stars are hybrid objects showing a number of modes at frequencies typical for β Cep stars but also one mode at frequency typical for SPB stars. We construct seismic models of these stars considering uncertainties in opacity and element distribution. We also present estimate of the interior rotation rate and address the matter of mode excitation. We use both the OP and OPAL opacity data and find significant difference in the results. Uncertainty in these data remains a major obstacle in precise modelling of the objects and, in particular, in estimating the overshooting distance. We find evidence for significant rotation rate increase between envelope and core in the two stars. Instability of low-frequency g-modes was found in seismic models of ν Eri built with the OP data, but at frequencies higher than those measured in the star. No such instability was found in models of 12 Lac. We do not have yet a satisfactory explanation for low frequency modes. Some enhancement of opacity in the driving zone is required but we argue that it cannot be achieved by the iron accumulation, as it has been proposed. Key words: stars: variables: other – stars: early-type – stars: oscillations – stars: in- dividual: ν Eridani – stars: individual: 12 Lacertae – stars: convection – stars: rotation – stars: opacity arXiv:0801.2451v1 [astro-ph] 16 Jan 2008 1 INTRODUCTION induced mixing. A comprehensive survey of properties of these variables was published by Stankov & Handler (2005). Transport of chemical elements is still not well under- stood aspect of stellar interior physics. In the case of the Pulsation encountered in β Cep stars frequently consists 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, where low order nonradial modes have mixed 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- angular momentum transport because there is a role of rota- tional splitting is a source of information about the deep tion in element mixing. The upper main sequence pulsators, interior rotation rate. Pulsation is found both in very slow β Cephei stars, are potential source of constraints on mod- (< 10 km s−1) and very rapid (> 200 km s−1) rotating stars. elling the transport processes massive stars. Recently, Miglio Thus, there is a prospect for disentangling the effect of rota- et al. (2007b) and Montalb´an et al. (2007) studied sensitivity tion in element mixing. First limits on convective overshoot- 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), and Ausselooss et al. 2004). Unfortunately, both objects are ⋆ E-mail: [email protected] very slow rotators. To disentangle the role of rotation in el- c 0000 RAS 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 the dominant modes, predict instability in a much narrower 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 this discrepancy, PHD proposed accumulation of iron in the are strikingly similar. However, owing to the appearance of bump zone caused by radiation pressure, following solution two complete ℓ = 1 triplets, the one for ν Eri is more reveal- 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 an ad hoc factor 4 iron enhancement showing that it leads to 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, lows to fit the frequency of the troublesome p2 dipole mode. we show also the evolutionary tracks for selected seismic However, recent calculations of the chemical evolution made models of the stars which we will discuss in the next sec- by Bourge et al. (2007) showed that the iron accumulation tion. Both objects have nearly standard Population I atmo- in the β Cep proceeds in a different manner than in the sdB spheric element abundances. In particular, Niemczura and stars. Unlike in latter objects, in photospheres of β Cep stars Daszy´nska-Daszkiewicz (2005) quote the following values of the iron abundance is significantly enhanced for as long as it the metal abundance parameter [M/H]: 0.05 ± 0.09 for ν Eri is enhanced in the bump. In fact, the photospheric enhance- and −0.20±0.10 for 12 Lac. These numbers agree within the ment is always greater (P.-O. Bourge private communica- errors with earlier determination by Gies & 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 the nitrogen ex- the error box shown in Fig. 2. The adopted by us position in 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 burning phase. tion and we are facing two problems: what is the cause of The projected equatorial velocity in both stars is low. −1 element mixing in outer layers of this star, and how to ex- Abt et al. (2002) give 20 and 30 km s , for ν Eri and −1 plain excitation of high frequency modes (ν > 7 cd−1) and 12 Lac, respectively, with the 9 km s formal error. The the isolated very low frequency mode at ν = 0.43 cd−1. corresponding numbers quoted by Gies & Lambert (1992) −1 −1 Unlike most of β Cep stars, where frequencies of de- are (31 ± 3) km s and (39 ± 14) km s . 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 flict with the seismic determination of PHD, who derive the −1 and 12 Lac are found in wide frequency ranges. Both stars value of about 6 km s for the equatorial velocity which are hybrid objects with simultaneously excited low-order p- we believe to be 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 this star and the first 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 & Pojma´nski (2008) reported a likely discovery of 3 SEISMIC MODELS AND INTERIOR five additional objects. ROTATION RATE In the next section, we compare oscillation spectra and provide other basic observational data for ν Eri and 12 Lac. For specified input physics, element mixing recipe, and Seismic models of these two objects are presented in Section heavy element mixture, we need the four parameters: 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 determine stellar model, if dynamical effects of rotation are estimate of rotation rate gradient in the abundance vary- negligible.