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

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4. The non-radial mode in [FORMULA] Cep

Smith (1977) suggested that the observed amplitude variations in [FORMULA] Cep might be related to beatings, and showed that individual line profiles can be fitted with non-radial pulsations. However, Campos & Smith (1980) showed that the line-profile variations with frequency [FORMULA] can only be due to a radial pulsation mode.

Aerts et al. (1994) have analysed the moments of the line profiles of their extensive data set of [FORMULA] Cep, assuming a triple-periodic pulsation with frequencies [FORMULA], [FORMULA], [FORMULA], to interpret the variations with apparent frequency [FORMULA] =5.38 c/d. They concluded that these variations can be modelled as a non-radial pulsation with [FORMULA] [FORMULA] [FORMULA] =2 and with a low amplitude with respect to that of the radial pulsation in [FORMULA] Cep. Here we reanalyse the same data and find two different pulsation modes that give a better description of the variability at [FORMULA].

4.1. The moment method

With the moment method one can identify the pulsation mode from the variations in the moments of the line profiles. We used the new version of the moment method (Aerts 1996, who uses a definition of the moments that is different from that used in this paper) to derive the parameters of the non-radial mode in [FORMULA] Cep, using a model that includes [FORMULA] to fit the observed moments. This fit is compared with the modelled moments for each combination of [FORMULA] and [FORMULA] of the non-radial mode; the comparison is carried out by the minimization of a discriminant, [FORMULA]. Table 2 lists the best models ranked according to the minimized parameter [FORMULA]: the mode with [FORMULA] =2 and [FORMULA] =1 gives the best description of the moment variations. We see that the mode with [FORMULA] [FORMULA] [FORMULA] =2 corresponds to [FORMULA] [FORMULA] 13 km/s, which is incompatible with the outcome of the moment method ([FORMULA] =25 km/s) when applied to the radial mode. For appropriate values of [FORMULA] (i.e. 25-30km/s) the [FORMULA] value of the [FORMULA] [FORMULA] [FORMULA] =2 mode is much larger than the values in Table 2, and hence we can exclude the possibility of a sectoral mode with [FORMULA] =2 to explain the variations at [FORMULA]. From the value of the discriminant we find that the variation at [FORMULA] is best described by a pulsation mode with [FORMULA] =1 and [FORMULA] =2 or [FORMULA] =1.


Table 2. Results of the analysis of the line-profile variations with apparent frequency [FORMULA] =5.38c/d with the moment method (Aerts 1996); the mode with the lowest value of [FORMULA] fits the moment variations best. We list the pulsation amplitude as defined by Aerts (1996, [FORMULA]) and as defined by Schrijvers et al. (1996, [FORMULA]).

4.2. IPS diagnostics; the phase diagram of [FORMULA]

From the moment method alone, one cannot derive whether the non-radial mode is prograde (negative m) or retrograde (positive m). In order to overcome this limitation we fit the observed IPS phase diagram of [FORMULA] with that of a model of a multi-periodic star. Gies & Kullavanijaya (1988) and Telting & Schrijvers (1996) have shown that the phase diagrams hold essential information for mode identification. To generate spectra for a star with a radial and a non-radial mode both simultaneously present, we used the pulsation model as described by Schrijvers et al. (1996) and Telting & Schrijvers (1996), with [FORMULA] =27 km/s, inclination i = [FORMULA], Gaussian intrinsic profile width W =14 km/s, and linear limb-darkening coefficient [FORMULA] =0.36 (see Aerts et al. 1994). For the radial mode we used: observed frequency [FORMULA] =5.25 c/d, pulsation velocity amplitude [FORMULA] =23 km/s, ratio of rotation and pulsation frequency [FORMULA] =0.016 (for a rotation period of 12 days), and 5% variability (peak to peak) in the equivalent width of the lines with the EW in phase with the radial displacement, as is observed for the SiIII triplet. For the non-radial mode we used: [FORMULA] =5.38 c/d, [FORMULA] =0.015, [FORMULA] =1.9 km/s, and ratio of horizontal to vertical amplitudes k =0.03, which was determined by the expression [FORMULA], where we used M=15 [FORMULA] and R=11 [FORMULA] (see e.g. Heynderick et al. 1994). We varied the pulsation parameters [FORMULA] and m of the non-radial mode, and generated spectra with the same time sampling as that of the 620 observed spectra. For each of the generated time series we did an IPS analysis as described in Sect. 3.1, resulting in periodograms, and amplitude and phase diagrams.

The IPS periodograms of the models of a combined high-amplitude radial and low-amplitude non-radial mode show the same recurrence of the frequency pattern as in the case of the observations: [FORMULA] (radial), [FORMULA] (non-radial), 2 [FORMULA], [FORMULA] + [FORMULA], 3 [FORMULA], 2 [FORMULA] + [FORMULA], etc. (see Fig. 6). Harmonics of the non-radial mode are too small to be detectable in our data set. In Fig. 8 we compare the modelled and observed phase diagrams.

For prograde modes we expect that the variations of the non-radial mode give rise to bumps and troughs that move from blue to red through the line profile (Vogt & Penrod 1983). Equivalently, the phase diagrams of prograde modes have a negative slope from blue to red. From Fig. 8 we conclude that the phase diagrams of zonal (m =0) and prograde (m [FORMULA] 0) non-radial modes cannot fit the observed phase diagram: the non-radial mode must be retrograde. The phase diagrams of the modelled retrograde [FORMULA] =3 modes are too steep; only for [FORMULA] [FORMULA] 35 km/s these slopes are consistent with the data, but this value of [FORMULA] is too large to properly fit the line-profiles. Higher values of [FORMULA] will give even steeper phase diagrams, and hence we can exclude the possibility that the line-profile variations with frequency [FORMULA] are due to a mode with [FORMULA] [FORMULA] 2.

We note that the phase diagrams of the best-fitting models (retrograde modes with [FORMULA] =1 or [FORMULA] =2) are virtually identical for different values of the inclination; for some of the other combinations of [FORMULA] and m, the phase diagrams change somewhat as a function of i, but they never fit well. The phase diagrams hardly change for small changes in the adjustable model parameters [FORMULA], W, [FORMULA], and [FORMULA] (see next subsection).

4.2.1. The IPS amplitude and phase diagram of [FORMULA] and [FORMULA]

The IPS amplitude and phase diagrams form, together with the mean profile, a complete description of the line-profile variability. Besides [FORMULA] and [FORMULA], we detect also other frequencies in the line-profile variability of [FORMULA] Cep, which are probably due to other phenomena than just pulsations (see Sect. 5). Whereas the observed line profiles are affected by these other sources of variability, the phase diagrams at [FORMULA] and [FORMULA] are not. This is because the IPS Fourier analysis separates the different variabilities in frequency space (see e.g. Telting & Schrijvers 1996). The corresponding amplitude diagrams can be affected by the effects of beating, but since the profile variability at frequencies [FORMULA], [FORMULA], [FORMULA], and the six day period is very small, the amplitude diagram of the non-radial mode ([FORMULA]) will only be affected by beating with the radial mode ([FORMULA]). Therefore, the modelling of the mean profile together with the amplitude and phase diagrams of [FORMULA] and [FORMULA] will, in the case of [FORMULA] Cep, give better constraints on the pulsational characteristics than model fits to the line profiles themselves.

In Fig. 9 we show the observed and modelled mean profile and amplitude and phase diagrams. We compare the multi-periodic model as described in Sect. 4.2 (thin solid line) with a multi-periodic model with [FORMULA] =25 km/s, Gaussian width W =18.5 km/s, and amplitude of the radial mode [FORMULA] =22 km/s (double line). We adjusted the pulsation amplitude of the non-radial mode to fit the observed amplitude diagrams; with i = [FORMULA] we find that [FORMULA] ranges from 1.1 km/s to 2.1 km/s for the different combinations of [FORMULA] and m in Fig. 9.

The phase diagrams of both models are virtually identical. The first model gives good agreement with the observed amplitude diagram of the non-radial mode (especially for [FORMULA] =2, m =1), the latter model gives a better description of the amplitude diagram of the radial mode. This is because the large value of W leads to a decrease in the line-profile variability, which is needed to fit the observed amplitude distribution at [FORMULA] (see the left middle panel in Fig. 9). However, a smaller value of the intrinsic width W gives better fits to the amplitude and phase diagram of the non-radial mode.

The amplitude diagrams of [FORMULA] change as a function of the inclination, with as major difference a scaling of the amplitudes of the line-profile variability. Similarly, the amplitude distribution at [FORMULA] scales almost linearly with the pulsation amplitude [FORMULA] of the non-radial mode. This means that for different values of the inclination we have to adjust [FORMULA] to fit the amplitude diagrams. We note that the derived values of [FORMULA] will be slightly different for different intrinsic-profile shapes, and also for other values of the limb-darkening coefficient.

The shape of the amplitude distribution at [FORMULA] is greatly influenced by the value of the pulsation amplitude of the radial mode ([FORMULA]), whereas the corresponding phase diagram is not.

Within the range of EW variations allowed by the observations (see Fig. 3), the pulsational EW variations have little influence on the amplitude diagrams; the EW variations give rise to line-centre variability at [FORMULA], leading to a small rise of the amplitude distribution of [FORMULA] at line-centre.

From Fig. 9 we conclude that the line-profile variations at [FORMULA] and [FORMULA] can successfully be modelled with a multi-periodic star with a radial and a non-radial mode. Although the line-profile variability at [FORMULA] and [FORMULA] cannot be fitted perfectly with our model, the identification of the non-radial mode from the shape and slope of the IPS phase diagrams and the general shape of the IPS amplitude diagrams (Figs. 8 and 9) is independent of variations in the model parameters (inclination, intrinsic width, pulsation amplitude, etc.). We find from the IPS diagnostics that a mode with m =1 and [FORMULA] =2 or [FORMULA] =1 gives the best description of the line-profile variability at [FORMULA].

4.3. Moment method versus IPS method

With the moment method and with the IPS method, we find consistent identifications of the non-radial mode in [FORMULA] Cep. We note that with the different approaches of these methods, one can obtain different values for the intrinsic profile width W, for the pulsation amplitude [FORMULA], and for [FORMULA].

With the implementation of the moment method that is currently in use, it is not possible to tune the stellar and pulsational parameters by fitting both the radial and the non-radial mode simultaneously. Such an approach is possible by fitting the mean profile and each of the relevant IPS amplitude and phase diagrams simultaneously (Fig. 9).

With the constraints on the values of [FORMULA] and m from the moment method and from the fits to the IPS amplitude and phase diagrams, we conclude that the line-profile variations at [FORMULA] are due to a retrograde non-radial mode with m =1 and [FORMULA] =2 or [FORMULA] =1. With a rotation period of 12 days the pulsation frequency in the corotating frame becomes [FORMULA] =5.46 c/d. As is the case for most [FORMULA] Cephei stars that exhibit a non-radial pulsation with a small to moderate amplitude, the other components of the triplet (in the case of [FORMULA] =1) or of the quintuplet (in the case of [FORMULA] =2) belonging to [FORMULA] =5.46 c/d are not detected and thus must have amplitudes below our detection treshold.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998