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Astron. Astrophys. 322, 493-506 (1997)

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5. Interpretation of the frequency splitting

The observed frequency spectrum could be the signature of five different (non-)radial pulsation modes. This would be a generalisation of the model proposed by Aerts et al. (1994), who interpreted [FORMULA] and [FORMULA] as being due to small-amplitude non-radial pulsations. However, the then unexplained equal frequency spacing between [FORMULA], [FORMULA], [FORMULA], and [FORMULA] (which could not be noted by Aerts et al. since they did not find [FORMULA] and [FORMULA]), renders this model unlikely. Below, we describe two models that can produce a frequency splitting.

5.1. Rotational modulation; temperature spots

Considering the similarity between the changes in equivalent width of the optical photospheric lines and the UV wind lines of [FORMULA] Cep (Fig. 3) and the constant frequency spacing between the frequency of the main radial pulsation mode and [FORMULA], [FORMULA], and [FORMULA], we argue that the latter frequencies might be caused by rotational modulation induced by the changing conditions of the photosphere when the magnetic poles are passing through the line of sight.

The frequency modulation can be interpreted in terms of surface temperature spots that are associated with the presence of a magnetic field. The geometric structure of the spots is then expected to be related to the geometry of this field. For a magnetic dipole we assume that the temperature distribution can be represented by an axisymmetric spherical harmonic of degree [FORMULA] =2 along the magnetic axis, in such a way that the magnetic poles are cooler than the magnetic equator. We take a temperature distribution of the form :


where [FORMULA] are spherical coordinates with respect to the magnetic axis. In Equation (2) we use the spherical harmonics [FORMULA] = [FORMULA] and the associated Legendre polynomials [FORMULA] as defined by Aerts et al. (1992), which differ slightly from the classical definition as given by e.g. Unno et al. (1989). We use a temperature difference between the magnetic poles and equator of 10% of [FORMULA]. From the phasing of the equivalent-width variations and the radial-velocity curve, in combination with the strength of the SiIII triplet as a function of temperature, one can derive that [FORMULA] Cep must be hotter than [FORMULA] [FORMULA] 23000 K (e.g. Cugier 1993). Gies & Lambert (1992) and Heynderickx et al. (1994) give [FORMULA] =26700 K and [FORMULA] =24500 K, respectively.

To calculate the influence of the temperature spots on the line-profile variations, we need to determine the corresponding temperature distribution as a function of position on the visible stellar disc. This distribution is time dependent due to the rotation of the star and can be determined by performing two consecutive transformations. A first time-dependent transformation gives the temperature distribution along the rotation axis, which is oblique to the magnetic axis with angle [FORMULA]  :


where [FORMULA] are spherical coordinates with respect to the rotation axis and where the functions [FORMULA] are defined in Aerts et al. (1992). A second time-independent transformation leads to the temperature distribution as seen by the observer, who is inclined to the rotation axis with angle i  :


where [FORMULA] are spherical coordinates with respect to the line of sight. The temperature variations give rise to local brightness variations which can be approximated by means of atmosphere models, e.g. the ones by Kurucz (1992): [FORMULA]. The temperature distribution due to the magnetic field then leads to a distribution of the specific intensity through the relation [FORMULA]. The apparent intensity distribution is then further multiplied by the usual limb-darkening factor [FORMULA] with [FORMULA] =0.36, in order to obtain the total intensity field of the visible hemisphere.

We only model the brightness variation associated with the temperature variation; although the observed changes in equivalent width clearly have a 6 day periodicity (see Figs. 3 and 4), we ignore these changes in our model, for reasons of simplicity. We also ignore the temperature and equivalent width changes induced by the radial pulsation.

We have calculated theoretical line profiles for a radial pulsation with parameters appropriate for [FORMULA] Cep (i.e. rotation period 12 days, [FORMULA] =25 km/s, W =16 km/s), taking into account the above described temperature variation. We have generated 16 sets of 250 line profiles with observation times that are randomly spread in the time interval of our [FORMULA] Cep data. The different sets are numbered according to the values for the geometric angles i and [FORMULA] as given in Table 3. We then performed an IPS frequency analysis on sets [FORMULA] to see if our simple temperature model can reproduce the observed frequency splitting around the main pulsation frequency. The periodograms resulting from this analysis are shown in Fig. 10.


Table 3. The numbering of the different sets of theoretically generated line profiles, as a function of angles i and [FORMULA].

[FIGURE] Fig. 10a and b. Parts of the CLEANed IPS periodograms for the sets 1, [FORMULA],16, which were generated with the temperature spot model described in Sect. 5.1, with the geometric angles i and [FORMULA] given in Table 1. The top periodogram is of the observed SiIII 4574 line in [FORMULA] Cep (see also Fig. 6). The dashed vertical lines indicate the 6-day period and its one-day aliases (left) and the frequencies [FORMULA], [FORMULA] and [FORMULA] (right)

From Fig. 10 we find that models 6, 7, 8, 10, 11, 12, 14, 15, and 16 give rise to a frequency splitting around [FORMULA] that is symmetric in both power and frequency. The sets 8, 12, 14, 15, and 16 give rise to the correct frequency splitting concerning [FORMULA], [FORMULA] and [FORMULA], and the corresponding splittings around the harmonics of [FORMULA] (see Fig. 6). These models also give rise to the observed peak at 0.17 c/d (6 day period), and give no or a relatively small peak at 0.083 (12 day period). These sets have [FORMULA] and non-zero i, or [FORMULA] and non-zero [FORMULA]. From the analysis of the non-radial mode ([FORMULA], see Sect. 4) with the moment method, we found that for the best fitting modes the inclination of [FORMULA] Cep should be around [FORMULA]. With this value of the inclination, we find that the angle between the rotation axis and the magnetic axis is larger than [FORMULA].

None of the models is able to reproduce a peak at frequency [FORMULA] or at the harmonic of the 6 day period, i.e. 0.33 c/d. It is clear that, if the observed frequency splitting is caused by a temperature distribution due to the magnetic field, then the latter has to be more complex than the assumed constant dipole field. An off-centre dipole (see e.g. Hatzes 1990, for the case of the Ap star HR5857) or a quadrupole field with unequal components might give better results, but such detailed modelling is beyond the scope of our paper.

In Fig. 11 we plot the observed and modelled amplitude distributions of the variations at [FORMULA] and [FORMULA]. The figure shows the results of models 11 and 16; the other models give similar amplitude distributions, but with different maximum amplitude. Although the modelled amplitude distributions of [FORMULA] show, like the observed one, a triple peak structure, the general agreement between model and observations is poor. The observed amplitude distributions have a much higher degree of complexity (see also Fig. 7). We have performed an IPS analysis on a generated time series with the same time sampling as the data of [FORMULA] Cep, and with the individual spectra scaled to match the observed variations of the equivalent width. This approach proved to give rise to more structure in the amplitude distributions, and hence we argue that more detailed modelling, including equivalent width variations, is necessary to fit the line-profile variations as found at frequencies [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 11. Observed (Si III [FORMULA] 4574, dotted line) and modelled amplitude distributions at frequencies [FORMULA] and [FORMULA]. The solid line represents temperature spot model 16, the dashed line is for model 11 (see Table 3)

5.2. The oblique pulsator model

The oblique pulsator model (e.g. Kurtz & Shibahashi 1986) describes non-radial pulsations in a star that are subject to the Coriolis force due to the rotation and to the Lorentz force due to the presence of a magnetic field. With this model, it is assumed that the pulsation and magnetic axes are aligned and are oblique to the rotation axis. The model is quite successful in explaining the observed photometric light curves of rapidly oscillating Ap stars (roAp stars, see Kurtz 1990 for a review). Refinements of the theory of an oblique pulsator in the case of a magnetic dipole for which the influence of the Lorentz force is of equal importance as the influence of the Coriolis force (Shibahashi & Takata 1993) have led to the theoretical explanation of the observed equally-split frequency septuplet in the roAp star HR 3831 (Kurtz 1992).

The analogy of the frequency splitting observed in roAp stars and in [FORMULA] Cep has led us to wonder if the Lorentz force is an important clue to the understanding of the line-profile variations of [FORMULA] Cep. The ratio of the powers of the different frequency peaks in the periodogram is much larger for roAp stars than for [FORMULA] Cep. These powers are a direct measure of the strength of the magnetic field, which can amount to 2000 Gauss in the case of the roAp stars. For [FORMULA] Cep, Rudy & Kemp (1978) reported a field strength of 810 [FORMULA] 170 G, and concluded that the field was variable. Veen et al. (1996) find that the magnetic field of [FORMULA] Cep is highly variable, with a mean value of about 200 Gauss. For such a weak magnetic field we estimate that the effects of the Lorentz force are very small, but of the order of the effects of the Coriolis force. The Coriolis force is proportional to the ratio of the rotation frequency and the pulsation frequency. For [FORMULA] Cep we have [FORMULA] and [FORMULA] Gauss, while in the case of the roAp star HR 3831 the corresponding values are [FORMULA] and [FORMULA] Gauss. It thus seems worthwhile to study the oblique pulsator model in more detail to see if a spectroscopical application of this model can explain the observed frequency splitting, and if it can lead to an estimate of the geometrical angles i and [FORMULA] and an estimate of the magnetic field strength. A more detailed study of the model in the case of [FORMULA] Cep is currently being undertaken (Shibahashi & Aerts 1996).

5.3. More detailed modelling

Similar to our simple temperature model, the oblique pulsator model as such does not predict equivalent width variations. In order to match the observed EW variations, one has to include temperature variations in the oblique pulsator model, and relate these temperature variations to changes in the equivalent width of the local intrinsic profile. After integration of the local profiles over the visible stellar disc, one can compare the modelled and observed EW variations (see e.g. Smith 1977, Cugier 1993, Gies 1996, Townsend 1996).

To give a complete description of the data, it is not sufficient to model the observed temporal frequency spectrum alone. In addition, future models should account for the variations in the EW, FWHM and the moments of the line profiles, or equivalently, the IPS amplitude and phase diagrams. Hopefully, such detailed modelling will lead to a satisfactory description of the superb data set that we analysed in this paper, and will enable us to understand the physical properties that give rise to the complex optical line-profile variations in [FORMULA] Cep.

At present, the moments and the IPS amplitude and phase diagrams of the line-profile variations at [FORMULA] (radial mode) and [FORMULA] (non-radial mode, see Sect. 4) are fairly well understood. The physical processes that give rise to the 6 or 12 day period and the modulation of the observed amplitude of the radial mode ([FORMULA], [FORMULA], [FORMULA]) remain uncertain; it seems inevitable that the stellar rotation plays an important role in the origin of the observed frequency pattern.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998