## 4. The non-radial mode in CepSmith (1977) suggested that the observed amplitude variations in 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 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 Cep, assuming a triple-periodic pulsation with frequencies , , , to interpret the variations with apparent frequency =5.38 c/d. They concluded that these variations can be modelled as a non-radial pulsation with =2 and with a low amplitude with respect to that of the radial pulsation in Cep. Here we reanalyse the same data and find two different pulsation modes that give a better description of the variability at . ## 4.1. The moment methodWith 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 Cep, using a model that includes to fit the observed moments. This fit is compared with the modelled moments for each combination of and of the non-radial mode; the comparison is carried out by the minimization of a discriminant, . Table 2 lists the best models ranked according to the minimized parameter : the mode with =2 and =1 gives the best description of the moment variations. We see that the mode with =2 corresponds to 13 km/s, which is incompatible with the outcome of the moment method ( =25 km/s) when applied to the radial mode. For appropriate values of (i.e. 25-30km/s) the value of the =2 mode is much larger than the values in Table 2, and hence we can exclude the possibility of a sectoral mode with =2 to explain the variations at . From the value of the discriminant we find that the variation at is best described by a pulsation mode with =1 and =2 or =1.
## 4.2. IPS diagnostics; the phase diagram ofFrom the moment method alone, one cannot derive whether the
non-radial mode is prograde (negative 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: (radial), (non-radial), 2 , + , 3 , 2 + , 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
( We note that the phase diagrams of the best-fitting models
(retrograde modes with =1 or
=2) are virtually identical for different values
of the inclination; for some of the other combinations of
and ## 4.2.1. The IPS amplitude and phase diagram of andThe IPS amplitude and phase diagrams form, together with the mean profile, a complete description of the line-profile variability. Besides and , we detect also other frequencies in the line-profile variability of 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 and 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 , , , and the six day period is very small, the amplitude diagram of the non-radial mode () will only be affected by beating with the radial mode (). Therefore, the modelling of the mean profile together with the amplitude and phase diagrams of and will, in the case of 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 =25 km/s, Gaussian width 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 =2,
The amplitude diagrams of 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 scales almost linearly with the pulsation amplitude of the non-radial mode. This means that for different values of the inclination we have to adjust to fit the amplitude diagrams. We note that the derived values of 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 is
greatly influenced by the value of the pulsation amplitude of the
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 , leading to a small rise of the amplitude distribution of at line-centre. From Fig. 9 we conclude that the line-profile variations at
and can successfully be
modelled with a multi-periodic star with a radial and a non-radial
mode. Although the line-profile variability at
and 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 ## 4.3. Moment method versus IPS methodWith the moment method and with the IPS method, we find consistent
identifications of the non-radial mode in Cep.
We note that with the different approaches of these methods, one can
obtain different values for the intrinsic profile width 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 and
© European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |