The two hybrid B-type pulsators \nu Eri and 12 Lac

The two hybrid B-type pulsators:  Eridani and 12 Lacertae

W. A. Dziembowski and A. A. Pamyatnykh E-mail: wd@astrouw.edu.pl   
Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland
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,  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 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  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.

keywords:
stars: variables: other – stars: early-type – stars: oscillations – stars: individual:  Eridani – stars: individual: 12 Lacertae – stars: convection – stars: rotation – stars: opacity

1 Introduction

Transport of chemical elements is still not well understood aspect of stellar interior physics. In the case of the upper main sequence there are uncertainties regarding the extent of mixing of the nuclear reaction products beyond the convective core, known as the overshooting problem, as well as, regarding the survival of element stratification in outer layers caused by selective radiation pressure and diffusion. Closely related is another difficult problem of the angular momentum transport because there is a role of rotation in element mixing. The upper main sequence pulsators, Cephei stars, are potential source of constraints on modelling the transport processes massive stars. Recently, Miglio et al. (2007b) and Montalbán et al. (2007) studied sensitivity of the oscillation frequencies in B stars to the rotationally induced mixing. A comprehensive survey of properties of these variables was published by Stankov & Handler (2005).

Pulsation encountered in Cep stars frequently consists in excitation of a number of modes, which differ in their probing properties. They are found most often in evolved objects, where low order nonradial modes have mixed character: acoustic-type in the envelope and gravity-type in the deep radiative interior. Frequencies of such modes are very sensitive to the extent of overshooting. Their rotational splitting is a source of information about the deep interior rotation rate. Pulsation is found both in very slow ( km s) and very rapid ( km s) rotating stars. Thus, there is a prospect for disentangling the effect of rotation in element mixing. First limits on convective overshooting and some measures of differential rotation have been already derived from data on the Cep stars HD 129929 (Aerts et al. 2003),  Eri (Pamyatnykh et al. 2004 (PHD), and Ausselooss et al. 2004). Unfortunately, both objects are very slow rotators. To disentangle the role of rotation in element mixing, we need corresponding constraints for more rapidly rotating objects.

It seems that we understand quite well the driving effect responsible for mode excitation in Cep stars. The effect arises due to the metal (mainly iron) opacity bump at temperature near 200 000 K. Yet, the seismic models of  Eri, which reproduce exactly the measured frequencies of the dominant modes, predict instability in a much narrower frequency range than observed. As a possible solution of this discrepancy, PHD proposed accumulation of iron in the bump zone caused by radiation pressure, following solution of the driving problem for sdB pulsators (Charpinet at al. 1996). PHD have not conducted calculations of the abundance evolution in the outer layers. Instead, they adopted an ad hoc factor 4 iron enhancement showing that it leads to a considerable increase of the instability range but also allows to fit the frequency of the troublesome p dipole mode. However, recent calculations of the chemical evolution made by Bourge et al. (2007) showed that the iron accumulation in the Cep proceeds in a different manner than in the sdB stars. Unlike in latter objects, in photospheres of Cep stars the iron abundance is significantly enhanced for as long as it is enhanced in the bump. In fact, the photospheric enhancement is always greater (P.-O. Bourge private communication). However, neither  Eri nor 12 Lac shows abundance anomaly in atmosphere except perhaps for the nitrogen excess, [N/O]=, in the former object (Morel et al. 2006). So there is no clear evidence for element stratification and we are facing two problems: what is the cause of element mixing in outer layers of this star, and how to explain excitation of high frequency modes ( cd) and the isolated very low frequency mode at  cd.

Unlike most of Cep stars, where frequencies of detected modes are confined to narrow ranges around fundamental or first overtone of radial pulsation, modes in  Eri and 12 Lac are found in wide frequency ranges. Both stars are hybrid objects with simultaneously excited low-order p- and g-modes (typical for  Cep stars) as well high-order g-modes (typical for SPB stars). The greatest challenge is to 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  Pegasi,  Her, HD 13745, HD 19374 (De Cat et al. 2007). Very recently Pigulski & Pojmański (2008) reported a likely discovery of five additional objects.

In the next section, we compare oscillation spectra and provide other basic observational data for  Eri and 12 Lac. Seismic models of these two objects are presented in Section 3, after a brief description our new treatment of the overshooting. In the same section, we give for both objects the estimate of rotation rate gradient in the abundance varying zone outside core. Section 5 is devoted to the problem of mode excitation. Finally, in Section 6, after summarizing the results, we discuss the matter of element mixing in outer layers and what is needed for progress in B star seismology.

2 Oscillation spectra and other observational data

The multisite photometric (Handler et al. 2004, Jerzykiewicz et al. 2005) and spectroscopic (Aerts et al. 2004) campaigns resulted in detection of nine new modes in the  Eri oscillation spectrum, in addition to four, which have been known for years. Furthermore, data from these campaigns allowed to determine (or at least constrain) the angular degrees for most of the modes detected in this star. A schematic oscillation spectrum for this star is shown in the upper panel of Fig. 1. The angular degrees, , are adopted after De Ridder et al. (2004). In the lower panel of Fig. 1, we show the corresponding spectrum for 12 Lac. All data, including the -degrees, are from Handler et al. (2006). The two spectra are strikingly similar. However, owing to the appearance of two complete triplets, the one for  Eri is more revealing.

Figure 1: Oscillation spectra of  Eri and 12 Lac based on Jerzykiewicz et al. (2005) and Handler et al. (2006) data, respectively. The numbers above the bars are the most likely -values, as inferred from data on amplitudes in four passbands of Strömgren photometry.

The mean parameters of the two objects are also similar. In Fig. 2 we show their positions in theoretical H-R diagram with the errors based on the parallax and photometry data (see caption for details). In the same figure, we show also the evolutionary tracks for selected seismic models of the stars which we will discuss in the next section. Both objects have nearly standard Population I atmospheric element abundances. In particular, Niemczura and Daszyńska-Daszkiewicz (2005) quote the following values of the metal abundance parameter [M/H]: for  Eri and for 12 Lac. These numbers agree within the errors with earlier determination by Gies & Lambert (1992). These authors also provide the values which are left of the error box shown in Fig. 2. The adopted by us position in the H-R diagram indicate that both stars are in advanced core hydrogen burning phase.

The projected equatorial velocity in both stars is low. Abt et al. (2002) give 20 and 30 km s, for  Eri and 12 Lac, respectively, with the 9 km s formal error. The corresponding numbers quoted by Gies & Lambert (1992) are  km s and  km s. For  Eri, the two values barely agree (within the errors) but are in conflict with the seismic determination of PHD, who derive the value of about 6 km s for the equatorial velocity which we believe to be a more reliable value. In Section 4, we will present our refined seismic estimate of the rotation rate for this star and the first such determination for 12 Lac.

Figure 2:  Eri and 12 Lac in the theoretical H-R diagram. The the 1 error box for the first star is similar to that adopted in PHD, except that it is moved downwards by 0.1 because we do not include now the Lutz-Kelker correction to measured parallax. For the second star, the error box is adopted after Handler et al. (2006). The evolutionary tracks correspond to seismic models (dots) calculated assuming no overshooting and the initial hydrogen abundance . The  Eri model has mass and the heavy element mass fraction , and that of 12 Lac has and .

3 Seismic models and interior rotation rate

For specified input physics, element mixing recipe, and heavy element mixture, we need the four parameters: mass, the age, and the initial abundance parameters, , to determine stellar model, if dynamical effects of rotation are negligible. The latter is very well justified for the two stars we are considering in this paper. This might suggest that for testing the mixing recipe, frequencies for more than four pulsation modes are needed. This is not true because there are additional (non-seismic) observational constraints on the models. On the other hand, there are other uncertainties of modelling. The most important concerns opacity.

In the approach adopted in this paper, like in PHD, we fixed at 0.7, as the changes within reasonable ranges do not have any significant effect on frequencies. Furthermore, we considered models in the expansion phase, so that instead of the age we could use as the model parameter, which is more convenient. There are two modifications in our modelling. Firstly, in order to asses uncertainty in opacity, we constructed models employing both OP (Seaton 2005) and OPAL (Iglesias & Rogers 1996) data and two heavy element mixtures, the traditional one of Grevesse & Noels (1993, hereafter GN93) and the new one of Asplund et al. (2004, 2005, hereafter A04). Secondly, we allow for partial mixing in the overshoot layer. In our reference model, we adopted the OP opacities calculated for the A04 mixture and assumed no overshooting.

To describe partial mixing, we use an additional adjustable parameter. In the partially mixed layer, extending from the edge of the convective core at the fractional mass to distance , the fractional hydrogen abundance was assumed in the form

(1)

The input parameters are and . The and coefficients in the expression above were calculated from the two input parameters by imposing continuity of and its derivative at the top of the layer. The limit corresponds to the standard treatment. We wanted to see, whether this additional degree of freedom may help us to fit frequency of a certain troublesome mode. In more general sense, it is important to know whether asteroseismology may teach us about overshooting more than only its distance.

In our seismic estimate of the interior rotation rate, we make a similar assumption as PHD, that is in chemically homogeneous envelope we assume a uniform rotation rate, , and allow its sharp inward rise in the -gradient zone. Here, we additionally assume uniform rotation rate, , within the convective core and a linear dependence of the rate on the fractional radius, , in the adjacent -gradient zone between and , which is the fractional radius corresponding to convective core mass at ZAMS. Thus, we use the following simple expression for interior rotation rate

(2)

Our aim is to determine and from measured or inferred rotational splitting.

3.1  Eridani

Seismic models of this star constructed by PHD made use of the three frequencies: 5.637, 5.763, and 6.224 cd, which were associated with (, g), (, p), and (, p) modes, respectively. Only the mean frequencies of the triplets are matter for seismic models. Individual frequencies probe internal rotation. The frequency of 7.898 cd associated with degree could not be reproduced with standard models. PHD noted that the localized iron enhancement may remove the discrepancy but, as we pointed out in the introduction, this is unlikely. Ausseloos et al. (2004) succeeded in fitting the four frequencies only for rather unrealistic chemical composition parameters and effective overshooting.

Figure 3: Differences between observed and calculated frequencies of  Eri. All models have been calculated with the OP opacity data and the A04 heavy element mixture.

In Fig. 3, we show results of exploration of the two-parameter freedom in description of the overshooting for fitting six measured frequencies. All models are constructed to fit the three lowest frequencies. Increase in and/or in means more efficient overshooting and reduction of the O-C values for the higher frequency mode. The price to pay for the improved fit is shown in Fig. 4. All models calculated with are outside the error box in the H-R diagram. Still, we regard efficient overshooting and cooler star as a possible way of fitting the three high frequency modes. There is, however, another problem regarding this part of the spectrum. The two peaks at the highest frequencies cannot be interpret in terms of remaining components of the triplet because the frequency separation should be similar to that in the p case and it is much smaller. We checked that there is no other low degree mode in this vicinity, which could be associated with any of the two peaks. Any one of them may be a component of the triplet but certainly not both.

Figure 4:  Eri in the theoretical H-R diagram. The the 1 error box is inferred from the observational data. The evolutionary tracks correspond to seismic models (dots) calculated for different values of the overshooting parameters.

It is important to notice in Fig. 3 the difference between modes in the O-C changes. For the two highest frequency modes, the shift in from 2 to 8 has a similar effect to the shift in from 0.1 to 0.2. For the mode at 6.732 cd the effect of the latter shift is much larger and results in sign change of O-C. We identified this mode as . This mode was detected only with the data from the second photometric campaign (Jerzykiewicz et al. 2005) and no -value from the amplitude data has been assigned to it. In our seismic models there is no lower degree mode near this frequency. We may conclude that there is a prospect for getting additional constraints on overshooting once we have enough frequency data.

As already noted by PHD, the peak at 7.2 cd may only correspond to mode. Its O-C values are obtained assuming and they would be much closer to zero if (retrograde) identification is adopted. At this point, however, we are reluctant to use this mode in seismic model construction. Higher frequency modes are more sensitive to uncertainties in outer layers and we know that some modification of the structure of this layer is required because we have to solve the excitation problem.

In Fig. 5, we show the effect of choice of opacity data and the heavy element mixture on seismic models. We may see that the latter is far less important. Models calculated with the OPAL data are hotter and therefore may accommodate overshooting and stay within the error box. However, relaying on the OPAL data will not help the mode frequency fit, as the entries in Table 1 show. The smaller difference between measured and model effective temperature, , is always associated with greater O-C for this mode. Therefore, it seems that if future measurements confirm the current value of the effective temperature, the only way of reconciling the seismic and non-seismic observable is an increase of opacity in outer layers. In PHD this was accomplished by an artificial enhancement of the iron abundance which had virtually no effect on frequencies of the three modes used in the seismic model but caused the desired -0.1 cd frequency shift of the mode.

Figure 5: Influence of the choices of the opacity data and the heavy element mixture on seismic models of  Eri. The evolutionary tracks were calculated assuming .
mixture O-C
OP A04 0.0 -0.127 -0.0103 5.55 5.93
OP GN93 0.0 -0.151 -0.0075 5.36 5.95
OPAL A04 0.0 -0.188 -0.0044 5.36 5.99
OP A04 0.1 -0.085 -0.0159 5.82 5.91
OP A04 0.2 -0.034 -0.0244 5.78 5.93
Table 1: The O-C (in cd) for the =1, mode frequencies and the differences between observed effective temperature and that of the seismic model, . The observational uncertainty in effective temperature is . Models with were calculated with . is given in km s.

For a specified stellar model, the two parameters, and , describing internal rotation may be directly determined from measured frequency splitting, , of the two modes. To this aim we use the relation

(3)

where is the familiar rotational splitting kernel and is given by Eq. (2). The kernels for the these modes, which were plotted by PHD, differ significantly and this is essential for an accurate determination of and .

There are two differences relative to that work. Firstly, we now have measured frequency of the component of the higher frequency triplet () and denotes now the rate in the convective core and not the mean value over the range. In Table 1, we give the the values of the ratio on the surface equatorial velocity, , calculated for the selected models. The values differ only little. Certainly, the main uncertainty of our inference is the -dependence of its rate, which was assumed in a very simplistic form (Eq. (2)).

3.2 12 Lacertae

In Table 2 we list frequencies and angular degrees of the five dominant modes in the oscillation spectrum of this star. The frequencies and the values are from Handler et al.(2006) analysis of their photometry data. The values were kindly provided to us by Maarten Desmet and are based on his analysis of line profile variations. He also provided us the value of  km s for equatorial rotation velocity at the surface.

Ident. Frequency
(cd)
5.179034 1 1
5.066346 1 0
5.490167 2 2 or 1
5.334357 0 0
4.24062 2 ?
7.40705 1 or 2 ?
5.30912 2 or 1 ?
5.2162 4 or 2 ?
6.7023 1 ?
5.8341 1 or 2 ?
Table 2: The frequencies and mode identifications for 12 Lac.

The first attempt to construct seismic model of 12 Lac (Dziembowski & Jerzykiewicz 1999) concentrated on interpretation of the very nearly equidistant triplet . The distances to the side peaks are -0.1558 and +0.1553 cd. Determination of the values by Handler et al. (2006) made all proposed interpretations invalid. This is not a rotationally split triplet. Equidistancy enforced by a nonlinear (cubic) resonant mode coupling is also excluded by the values. Thus, just a mere coincidence seems the only explanation.

Construction of seismic model of this star is more complicated than that of  Eri because it cannot be done separately of estimate of the internal rotation rate. Therefore, in this paper we limit ourselves to providing evidence for a significant inward rise of the rotation rate. In this preliminary analysis, we consider only models calculated with the A04 mixture with and without overshooting. Upon ignoring all effects of rotation, we determined stellar mass and effective temperature by fitting frequencies of the , p(fundamental) and , g modes to and , respectively. We checked that with the constraints on and there is no alternative identification of radial orders of the two modes and that models in the contraction phase are excluded.

The rotation rate of 12 Lac, as determined from spectroscopy and implied by the mode splitting, is fast enough so that the second order effects must be considered even at this preliminary phase. Within the accuracy we may write

(4)

where is the frequency calculated ignoring all effects of rotation, is given in Eq. (3), the and terms include second order effect of the Coriolis force and the lowest order effect of the centrifugal force. In our treatment we rely on the formalism described by Dziembowski & Goode (1992) but simplified to the case of the shellular () rotation. In our treatment the surface-averaged effect of the centrifugal force is incorporated in evolutionary models and effects of the Coriolis force are included in zeroth-order equation for oscillations. So the only rotational frequency shift for nonradial modes calculated by means of the perturbation theory is that caused by centrifugal distortion. For radial modes we add only the shift caused by feedback effects of the toroidal displacement induced by the Coriolis force.

We began assuming an uniform rotation and determined its rate from the frequency separation using Eq. (4) with coefficient calculated in the model reproducing the the frequencies of and identified as corresponding to and modes. In this model the frequency of the mode is very close to . However, the inferred equatorial velocity of over 95 km s is much larger than that determined from line profile changes. An additional problem with this solution is the frequency distance of nearly 0.3 cd between and the nearest quadrupole mode, . We believe that these two discrepancies exclude the hypothesis of uniform rotation.

Figure 6: The splitting kernels for three noradial modes: (solid line), (dotted line) and (dashed line). Triangles mark boundaries of the -gradient zone.

The splitting kernels for the three nonradial modes plotted in Fig. 6 are very different. This means that there is a potential for deriving significant constraints on differential rotation. However, we have direct measurement of only one rotational splitting, the others must be inferred through the frequency fitting. At the equatorial velocity of rotation of about 100 km s, the mean effect of centrifugal force induces frequency shifts exceeding 0.01 cd. Thus, if we aim at such a precision of the fit, we have to take into account differential rotation in evolutionary models. This we leave for our future work. In the present work, the effect of the differential rotation described by Eq. (2) was rigorously treated only for calculation of the rotational splitting while the mean shifts of the multiplets induced the mean centrifugal force were determined assuming uniform rotation corresponding to .

95.5 11.771 4.3878 4.194 0.113 1
47.1 11.777 4.3891 4.198 -0.134 2
Table 3: Models fitting the four dominant mode frequencies () at and . The quantity is the frequency mismatch between and in cd.

We considered different values of the ratio starting with 1 and used the frequency difference to determine . Then, we adjusted the mass and effective temperature to fit the frequencies of the four dominant modes. As we may see in Table 2, there are two possible values for the mode. With the fit is achieved at and with at Data on the two seismic models are summarized in Table 3.

We believe that our second model is more realistic because its surface equatorial velocity agrees within with observations. The mismatch between and calculated frequency of the mode is somewhat higher than in the first model. However, it should be stressed that this mode is most sensitive to detailed treatment of rotation in the inner layers. We may see in Fig. 6 that the splitting kernel, which is similar to the mode energy distribution, for this mode is very much concentrated in the -gradient zone. Frequencies of such modes are very sensitive also to overshooting.

Figure 7: Frequencies of high frequency modes detected in 12 Lac (veritical lines) and frequencies off all modes (dots) in the two models (see Table 3). Identified modes are shown with encircled dots. Note than for the model with uniform rotation (upper panel), we have no identification for the peak.

Furthermore, as Fig. 7 shows, the second model reproduces closer frequencies of the remaining modes detected in 12 Lac. In this model, may be identified as the p mode, while and may be, respectively, the and components of the g mode. It should be noted that this identification for implies that this mode may be coupled with radial mode associated with , which is very close. The slight frequency shift caused by this proximity (see e.g. Daszyńska-Daszkiewicz et al. 2002) should be included in the fine frequency fitting. There is a plausible identification of as a p, . The only troublesome peak is but only if indeed , as Handler et al.(2006) proposed. There would be no problem if it was because then the peak could correspond to a fundamental () mode.

4 Mode instability

Satisfactory seismic models should account not only for measured mode frequencies but also for their excitation. In practice, it means that modes we want to associate with observed frequencies should be unstable. As a measure of mode instability we use the normalized growth rate

where is the global work integral. It follows from this definition that varies in the range and it is for unstable modes. The value of is more revealing than the usual growth rate, because it is a direct measure of robustness of our conclusion regarding mode stability.

In Fig. 8, we show for and 2 modes calculated in wide frequency ranges for our best models of the two stars. The overall pattern of the dependence in two stars is similar. There is the instability range accounting for excitation of most of the observed high frequency modes. There is also a maximum of in the low frequency range not far from the position of the low frequency modes detected in the two objects. There is enough agreement to feel that we are on the right track toward satisfactory seismic models. However, there is a need for improvement.

Comparing our results for  Eri with those presented by PHD, we find a significant increase of the instability range. Now, only the highest frequency modes near and the mode at fall into stable ranges. This change is a consequence of the usage of the OP data. That these data lead to wider instability ranges than the OPAL data, has been already noted before (see e.g. Miglio et al. 2007a). With the OP data we have unstable high-order g-modes (=-14 to -20), but their frequencies are still too high by factor at least 1.4. The measured frequency nearly coincides with maximum for but the value is less than zero, . With the OPAL data we find maximum value at nearly the same frequency but . Perhaps, there is still a room for improvements in opacity that would render modes at the lowest frequency and the highest frequencies unstable.

All modes detected in 12 Lac, except of that at , occur in the unstable frequency range. However, the difficulty in explaining excitation of the low frequency mode is even higher than in the case of  Eri. In part this is due to lower mode frequency and in part to the lower heavy element mass fraction adopted ().

Figure 8: Normalized growth rate as a function of mode frequency in models of  Eri (top) and 12 Lac (bottom) (the same as used in Fig. 2). The frequencies of the detected modes are marked with vertical lines starting at the axis. The length is proportional to the mode amplitude.

5 Discussion

We believe that our seismic models of  Eridani yield a good approximation to its internal structure and rotation but there is room for improvement and a need for a full explanation of mode driving. We did not fully succeed in the interpretation of the oscillation spectrum of Eridani with our standard evolutionary models and our linear nonadiabatic treatment of stellar oscillations. The frequency misfit between the high frequency peak to the mode is much reduced in models built with the OP opacity data allowing large overshooting but such models are much cooler () than the mean colour of the star implies. Such models also nearly avoid the driving problem. Similar conclusions regarding models allowing large overshooting distance were reached by Ausseloos et al. (2004). As for the consequences of usage of the OP instead of OPAL opacity data for mode instability, our finding agrees with more general observation of Miglio et al. (2007a) that models using the former data predict wider frequency ranges.

An assessment of the inward rise of the angular rate in the -gradient zone was made for both,  Eri and 12 Lac, yielding the values around five for the ratio of the core to envelope rate. The surface equatorial velocities in the two stars are very different. For the former object our seismic estimate gives about 6 km s. For 12 Lac, our estimate of about 50 km s is less certain because data on rotational splitting are much poorer but the value agrees with estimate of Desmet (private communication), based on his analysis of line profile changes. There is a large difference in rotation rate between the two stars but unfortunately the accuracy of our modelling is insufficient for addressing the question of the relation between the overshooting distance and rotation.

The surface rotation rate in  Eri is indeed very low and this was the reason why PHD suggested that chemical element stratification in this star may be sustained. Hovever, a closer look at the problem reveals that the effect of rotation on element distribution should not be ignored, even at the equatorial velocity of few km s. As Bourge et al. (2007) showed for their model, the two-fold excess of the iron abundance in the driving zone is produced in the time scale comparable with main sequence life time. The process is slower than in less massive objects (Seaton 1999) but fast enough so that if it goes unimpeded a significant excess of iron could be created. The effect of rotation could be ignored if the time, , for meridional circulation to travel from the depth where most of iron is moved up (K) to the bottom of the convective zone (K) around the iron opacity bump is longer than the star age. To estimate we use the well-known expression (see e.g. Tassoul 2000) for the speed of the meridional flow

(5)

where , is the Kelvin-Helmholtz time, and

The last term, known as the Gratton-Öpik term, is kept though it is of higher order because it is important near the surface where density is low, even if and otherwise linearization is valid. In our model of  Eri, the the Gratton-Öpik term is -3.4 at the depth where most of iron is supposed to be pushed up.

(6)

We evaluate at the layer where K. and for use the distance to the bottom of the convective zone. For the  Eri model, we have , yr, , , . With this numbers, we get from Eq. (6) y, which is much less than the age of the star, y according to our seismic model. Thus, even in this slowly rotating star the effect of rotation cannot be ignored. For our best model of 12 Lac, we find by about four orders less than in  Eri. Yet, as we may see in Fig. 8, the problem with driving the low frequency mode is even greater than in the case of  Eri.

Meridional flow and/or turbulence developed through instability induced by rotation are expected to prevent accumulation of iron in the driving zone and in photosphere. This is a likely explanation of apparently normal chemical atmospheric composition of  Cephei stars, including such slow rotators as  Eri. This is also the reason to reject iron accumulation in the driving zone as the solution of the driving problem, whose solution must be searched in a different way.

We attach greatest hope for solution of the driving problem with improvements in stellar opacity data. Thus, we would like to encourage further effort in this field. Reliable opacity data are essential for B star seismology. Data on mode frequencies in  Eri and 12 Lac are abundant and accurate enough for probing rotation and structure of the -gradient zone above the convective core. However, to answer open questions regarding macroscopic transport of elements and angular momentum, we need accurate microscopic input data, especially on opacity. Only if we have measurements of rotational splitting, our inference on differential rotation does not rest on very precise model of stellar structure, but such data are rare. Regarding observational work, we view as most important new spectroscopic observations of the two stars aimed at improving the value of the mean effective temperature and leading to the and identifications for a greater number of modes.

Acknowledgements

This work was supported by the Polish MNiSW grant No 1 P03D 021 28.

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