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

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3. Period analyses

3.1. Description of the methods

Two of the most widely studied methods of spectroscopic pulsation mode identifications are based on period searches on either the variations of the velocity moments of the absorption lines (the moment method, Balona 1986; Aerts et al. 1992; Aerts 1996), or on the intensity variations as a function of position in the line profile (Gies & Kullavanijaya 1988; Telting & Schrijvers 1996). For this study we apply both period finding techniques.

The first technique looks for periodicity in the change of derived quantities, such as equivalent width (EW), apparent radial-velocity ([FORMULA]), squared line width ([FORMULA]) and skewness ([FORMULA]). Hereafter we refer to this period-finding technique as the Moment Period Search (MPS). Pulsation mode identifications are done considering the characteristic changes as a function of pulsation phase for each of these quantities.

The velocity moments [FORMULA] are derived by a weighted summation of the normalized intensity across the line profile


where [FORMULA] is a reference velocity. In this paper, we calculate the first moment with the rest wavelength of the line as reference, and the second and third moment are calculated using the first moment as reference velocity [FORMULA]. The integration is carried out over an as narrow as possible wavelength region, to minimize the influence of the continuum noise in the determination of the moments. We normalized the velocity moments by dividing by the equivalent width [FORMULA] ; from here on we speak of moments when we mean normalized moments.

The second method searches for periodicity in the normalized intensity of each wavelength bin across the absorption line, resulting in diagrams that give the amplitude and phase of the profile variations as a function of position in the line profile. Then pulsation mode identification is attempted using the observed change in phase of the periodic variations as a function of wavelength. Hereafter we refer to this period-finding technique as the Intensity Period Search (IPS).

The period-finding algorithm we applied consists of an ordinary Fourier transformation for non-equidistant temporal sampling, followed by a CLEAN stage in which the window function, which is due to incomplete temporal sampling of the stellar signal, is iteratively removed from the Fourier spectrum (Roberts et al. 1987). The window function was removed in 400 iterations with a gain of 0.2. The frequency range that can be examined with the data set is approximately 0.03-400 c/d, but to cut down computing time we limited the frequency range of the computations to 0.001-30 c/d with a frequency step of 0.001 c/d. After CLEANing, the amplitude of the variations can be estimated by the relation    [FORMULA]  , where [FORMULA] is the amplitude as a function of frequency, in units of the original signal that was given as input to the Fourier routine, and where [FORMULA] is the CLEANed power spectrum.

For the four absorption lines we applied this period search algorithm to the time series of 620 spectra: we analysed the time series of the moments (MPS), and the time series of the normalized intensity of each position in the line profiles (IPS). We CLEANed the Fourier transform of the time series of the moments over the whole frequency range (0-30 c/d), to obtain the periodograms (see Fig. 4).

[FIGURE] Fig. 4. The variations in EW and in the velocity moments of the Si III [FORMULA] 4574 line (first 6 nights of observations), and the corresponding CLEANed periodograms (from all 620 spectra). Each panel consists of two parts of which the upper part's abscissa is at the top of the figure and the lower part's abscissa is at the bottom. The most prominent periodicity is due to the radial pulsation, although the variations in the second moment (i.e. line width) are not dominated by the radial pulsation. One can see a 6 day period in the equivalent width (see also Fig. 3), and a 6 day period amplitude modulation of the first moment (i.e. line centroid velocity). Even after using CLEAN the periodograms severely suffer from aliasing. We find the frequencies [FORMULA]... [FORMULA] in the periodograms of the moments, albeit at one-day aliases for some cases. We note that the frequencies associated with the rotation of the star (multiples of 0.083 c/d and their one-day aliases) are prominent especially in the equivalent width and in the third moment (i.e. skewness) of the line profiles

For the intensity variations, however, we found after the first trials that the periodograms resulting from the IPS method only contained evidence for periodicity within the intervals 0.0-3.0 c/d, 4.6-5.8, 9.8-11.0, 15.2-16.2, 20.4-21.8 and 23.0-30.0 c/d. Outside of these frequency intervals only occasional one-day aliases were present. To avoid the CLEANing of the one-day aliases we restricted the CLEAN algorithm to the above mentioned frequency intervals, thus ensuring that all power of the variations is CLEANed at the right frequencies. Of course one can only do this if one has foreknowledge of the detected frequencies; in our case the observed periodogram (see Fig. 6) and especially the repetitiveness of the frequency pattern (see next subsection) proved the validity of partially CLEANing the "dirty" Fourier spectrum. Similar partial CLEANing techniques are commonly used in the reduction of radio synthesis observations.

[FIGURE] Fig. 5. Partially CLEANed Fourier periodograms of the intensity variations in the Si III [FORMULA] 4574 line. For each wavelength bin the power as a function of frequency is plotted as a grey-value. Grey-scale cuts: 0 - 10-6. The bottom panel shows the mean, the first and the last spectrum of the data set. We find a pattern of frequencies at which variations are seen. This pattern occurs around the main pulsation frequency [FORMULA], and also around the harmonics of the main pulsation frequency. Here we show the pattern as detected around [FORMULA] =5.25 c/d and [FORMULA] =10.50 c/d
[FIGURE] Fig. 6. The variational power in the (partially) CLEANed Fourier periodograms (IPS, see Fig. 5) is summed over the wavelength range of the Si III [FORMULA] 4574 line (4573.7-4575.5Å), forming a one-dimensional periodogram. The top panel shows the window function that is removed by CLEAN. Then, from top to bottom the frequency intervals around zero, [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are shown. The horizontal scales of each of the panels are the same. Evident are the frequencies [FORMULA] (radial mode), [FORMULA] (non-radial mode), [FORMULA], [FORMULA], and [FORMULA]. The spacings [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] are 0.17 c/d each (a period of 6 days). This is equal to the frequency of subsequent high (or low) EW values of the UV resonance lines (see Fig. 3)

3.2. Results of the period analyses

3.2.1. Variations in the moments

In Fig. 4 we show the moments and their periodograms, as derived from the Si III [FORMULA] 4574 line. Consistent with the results of Aerts et al. (1994), we find that the variations of the equivalent width, the centroid velocity and the skewness of the profiles are dominated by the radial pulsation in [FORMULA] Cep.

For all 4 investigated lines except Si III [FORMULA] 4574, the variations of the second moment (squared line width) are dominated by the radial pulsation. For Si III [FORMULA] 4574 the power peak of the radial mode is of equal strength to that of other detected frequencies (which will be specified below). The relative amplitude of the variations of the equivalent width (d EW/EW) and of the second moment ([FORMULA] / [FORMULA]) are largest for the O II [FORMULA] 4591 line, whereas its average equivalent width is the smallest of the four investigated lines.

From the curves of the EW in Fig. 3 and of the moments in Fig. 4 it is evident that a 6 day period (EW) and a 6 day amplitude modulation (first and third moment) is present. Consequently, we find a peak in the periodograms of the moments at 0.17 c/d, corresponding to a period of 6 days. Since the 6 day period was already identified as half the rotation period of the star (from UV wind lines), we argue that the regular pulsational cycle-to-cycle amplitude variations as observed in the optical (e.g. Struve et al. 1953, Aerts et al. 1994) are mainly due to rotational modulation.

In addition to the three frequencies found by Aerts et al. ([FORMULA] =5.25, [FORMULA] =5.38, [FORMULA] =4.92 c/d), we find evidence for two other frequencies in the moment variations, but severe one-day aliasing makes it difficult to identify these frequencies with certainty. We note that these newly detected frequencies [FORMULA] and [FORMULA], that will be specified below, were not found in the moment variations by Aerts et al. (1994), who used the Phase Dispersion Minimization method of Stellingwerf (1978) without CLEANing by means of a window function. It is therefore clear that period analyses using the CLEANing technique can lead, in some cases, to better results compared to methods that do not make use of this technique.

Aerts et al. (1994) noted that the FWHM variation of the [FORMULA] 4552 line in [FORMULA] Cep is not dominated by the radial pulsation, but instead varies mainly with frequencies [FORMULA] and [FORMULA]. In their period analysis they only investigated the frequency interval [3.0,8.0] c/d. We analysed the FWHM variations of the [FORMULA] 4574 line using the CLEAN technique, and find in order of decreasing power: the 6 day period, [FORMULA], [FORMULA], 2 [FORMULA] and [FORMULA] + [FORMULA].

3.2.2. Variations in the normalized intensity as a function of position in the line profiles

In Fig. 5 we plot parts of the periodogram resulting from the IPS analysis of the data of the Si III [FORMULA] 4574 line. We summed the detected power of the periodograms of the variations across the lines to a one dimensional periodogram (see Fig. 6), and compare this summed periodogram with the periodograms of the velocity moments (MPS, Fig. 4). The wavelength range over which the separate periodograms are summed, is kept as small as possible to prevent the noise level from building up.

We find clear differences between the periodograms of the two different techniques, MPS and IPS. In the periodogram of the latter method we find patterns recurring around every detectable harmonic of the radial pulsation, with the same frequency spacing as around the main frequency. The advantage of this is that we can easily identify certain power peaks to be significant (see Fig. 6), which was not possible in the periodograms of the moments. This allows us to discard the one-day aliases by partially CLEANing the periodogram (Sect. 3.1).

The recurring pattern of power peaks is not due to stellar variability with the corresponding frequencies, but instead due to the unobvious mapping of the intrinsic stellar 3-dimensional variability onto a one-dimensional representation, i.e. the spectral line profile. Only the frequencies detected around the main pulsation frequency, represent true intrinsic periodic variability.

We find, apart from the already known frequencies, two other frequencies ([FORMULA] =5.08 c/d, [FORMULA] =5.42 c/d) around the main frequency, which have variations exceeding the noise level. The spacing between the frequencies are: [FORMULA] - [FORMULA] [FORMULA] 0.33 c/d, [FORMULA] - [FORMULA] [FORMULA] 0.17 c/d, and [FORMULA] - [FORMULA] [FORMULA] 0.17 c/d). Furthermore, we find in the periodograms a peak at 0.17 c/d, corresponding to a period of 6 days, and evidence for harmonics of this frequency. From UV wind line measurements, the period of 6 days has been interpreted as half the rotation period of the star (Henrichs et al. 1993, Veen et al. 1996), and hence we can identify the spacings between the main frequency [FORMULA] and the frequencies [FORMULA], [FORMULA], and [FORMULA] as multiples of the rotation frequency of the star.

We note that a small peak appears near the frequency 5.58 c/d, which is approximately [FORMULA] +0.17 c/d. Together with [FORMULA] and [FORMULA], this frequency could be the fifth component of a frequency quintuplet. However, we regard its variational power as too low to be significantly present in our data, especially since no corresponding peaks appear in the periodograms aside the harmonics of the radial mode (as is the case for [FORMULA], [FORMULA], [FORMULA] and [FORMULA], see the lower three panels of Fig. 6). A splitting of the main frequency into a quintuplet can be caused by a surface phenomenon with an angular dependence given by a spherical harmonic with degree [FORMULA] =2. Since the main mode is definitely radial (see Campos & Smith 1980, Aerts et al. 1994 and our Figs. 7 and 9), such an angular dependence must be caused by another phenomenon than the main pulsation. Some possible models are proposed in Sect. 5.

[FIGURE] Fig. 7a and b. The IPS amplitude (left) and phase (right) distributions of the line-profile variations (see Figs. 5 and 6). The top panels display the mean of the observed profiles. The middle and lower panels give the (partially) CLEANed amplitude and phase distributions for the indicated frequencies. The phase diagrams are offset by multiples of 2 [FORMULA] for clarity. The amplitude and phase distributions at the main pulsation frequency and its first harmonic (middle panel) are characteristic for a radial pulsation

The IPS period analysis gives consistent results for each of the four investigated absorption lines. The error in the detected frequencies can be estimated by the HWHM of the main peak of the window function, which is 0.011 c/d (see Fig. 6). This estimate reflects a possible systematic error due to the time sampling of the data. The multitude of absorption lines, however, allows us to make a separate estimate by computing the mean and the error in the mean of the detected frequency values (Table 1). We find that for all detected frequencies the error in the mean is in the order of the frequency spacing (0.001 c/d) that we used for the Fourier analysis.


Table 1. Observed frequencies (in cycles/day) of spectral line variations in [FORMULA] Cep. For the Si III triplet and the O II line we give the observed line centre of the mean of 620 spectra (read off from the mean spectrum sampled on 0.05Å bins). The wavelength scale is corrected for earth motion. The frequencies are derived from summing the variational power of the IPS over the wavelength region around the line centre that contains significant variability

In Fig. 5 one can see that the distribution of the power as a function of position in the line profile is rather complex. In Fig. 7 we display the IPS amplitude and phase distributions for some of the detected frequencies. The amplitude distribution at the main frequency and its first harmonic, [FORMULA] and 2 [FORMULA], can be modelled perfectly by that of a radial pulsation (see Fig. 9). Part of the complexity in the other amplitude distributions is due to incomplete sampling as a result of bad weather.

[FIGURE] Fig. 8. Model calculations of [FORMULA] Cep as a multi-mode pulsator with a radial ([FORMULA]) mode and a (non-)radial ([FORMULA]) mode. The solid lines give the IPS phase diagram of the (non-)radial mode, for different combinations of the parameters [FORMULA] and m. The phases have been shifted arbitrarily in such a way that all models could be plotted in the same figure. The observed phase diagram (of [FORMULA]) is overplotted (dots) for the models that give the best representation of the observed phase diagram: low-degree retrograde sectoral or tesseral modes
[FIGURE] Fig. 9. Observations (dots) and multi-periodic models of the line-profile variability of [FORMULA] Cep. The top panels give the mean spectrum, the middle panels give the amplitude and phase diagrams of the radial pulsation ([FORMULA]), the bottom panels give those of the non-radial pulsation ([FORMULA]) for different combinations of the parameters [FORMULA] and m. The thin line depicts the model with W =14 km/s, [FORMULA] =27 km/s, [FORMULA] =23 km/s for the radial mode ([FORMULA]). The double line depicts the model with W =18.5 km/s, [FORMULA] =25 km/s, [FORMULA] =22 km/s for the radial mode. For both models, the pulsation velocity amplitude of the non-radial mode ([FORMULA]) is adjusted to fit the observations, and varies between [FORMULA] =1.1-2.1 km/s

From Fig. 7 we conclude that the amplitude distribution at [FORMULA] and [FORMULA] + [FORMULA] is asymmetric with respect to line centre. We argue that a model for the optical line-profile variability in [FORMULA] Cep should not only account for the observed frequencies but also for the complexity of the amplitude distributions.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998