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Astron. Astrophys. 339, 150-158 (1998)

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3. Fourier analysis

To find the frequencies of the pulsations that cause the line-profile variations we analysed the time series with the method as described by Gies & Kullavanijaya (1988). For each wavelength bin in the line profiles we did a Fourier analysis of the variable signal: we computed the Fourier components for frequencies between 0 and 50 cycles/day, with a frequency spacing of 0.01 cycles/day. Then we CLEANed the resulting Fourier spectrum of each wavelength bin, in order to remove the temporal window function (which is due to incomplete temporal sampling of the variational signal). We used CLEAN parameters [FORMULA] and a gain of 0.2 (Roberts et al. 1987).

For the SiIII [FORMULA] 4552 Å line the result is plotted in Fig. 4. Fig. 5 shows the one-dimensional periodogram that results from summing the variational amplitudes of the two-dimensional periodogram over the [FORMULA] 4552 Å line profile. Variational power is mainly found at frequency 15.0 cycles/day, a smaller peak is found at 13.6 cycles/day. Some of the power has leaked to one-day aliases, which shows that the CLEAN algorithm was not able to fully correct for the window function.

[FIGURE] Fig. 4. top: CLEANed Fourier analysis. For every wavelength bin (horizontal axis) a Fourier analysis of all the spectra is done. The power resulting from the Fourier analysis is plotted as a grey-value as a function of temporal frequency (vertical axis). Grey-scale cuts: 0-2[FORMULA]. A periodic signal with a frequency of 15.0 cycles/day, is detected throughout the line profile. Power is found at one-day aliases of the above frequency, and also near the harmonic frequency. bottom: Mean of all 30 spectra

[FIGURE] Fig. 5. Summed Fourier spectrum. The amplitudes (i.e. two times the square root of the power values) of the periodogram of the [FORMULA] 4552 line (Fig. 4) have been summed over the line profile. The results for the other two lines have been offset by 0.3 for clarity

For the two other lines in our spectra we also find most of the power at 15.0 cycles/day (see Fig. 5 and Table 1). A broad peak is found at 13.8 cycles/day; the broadness probably reflects a combination of a periodic signal with frequency [FORMULA]13.6 cycles/day and the one-day alias of the main frequency. Our dataset is not extensive enough to separate these frequencies: the HWHM of the main power peak in the window function is 0.09 cycles/day.


[TABLE]

Table 1. Strongest power peaks in the summed periodograms (see Fig. 5). Frequencies are given in cycles/day


The shortest timespan between subsequent exposures of this dataset corresponds to a Nyquist frequency (as defined for equidistant data sets) of [FORMULA] 25 cycles/day. Nevertheless, we find power at frequencies higher than that. Some power appears around the harmonic of the main frequency, which is expected for high amplitude line-profile variations (see Schrijvers et al. 1997, Telting & Schrijvers 1997a, 1997b). Note that this does not directly imply that the pulsation itself is non-sinusoidal (or non-linear), since the harmonic frequencies are expected to show up in the line-profile variations of linear (i.e. sinusoidal) pulsations as well. The power found at frequency 27.9 cycles/day (see Table 1) could be the first harmonic frequency of the main frequency, if the true apparent main frequency is 14.0 cycles/day (a one day alias of our previously determined main frequency). Also, if the 13.6 cycles/day frequency is real, the 27.9 cycles/day could be its harmonic. A more extensive data set is required to test these hypotheses.

We have also Fourier analysed the variations in equivalent width (EW) and centroid velocity, and find that both periodograms are noisy and do not have a maximum at frequency 15.0 cycles/day. We calculated these Fourier transforms (not CLEANed) for the [FORMULA] 4552 Å and 4567 Å lines, on a frequency domain of 0-50 cycles/day and with a frequency step of 0.01 cycles/day. The highest peaks in the frequency interval [13-17] cycles/day of each of these periodograms correspond to a maximum amplitude of the EW and centroid variations of 0.8% and of 350 m/s respectively, which are the observed constraints of the contribution to the EW and centroid variations of the main pulsation mode.

We conclude that the observed line-profile variations in [FORMULA] Sco are mainly due to one dominant pulsation mode with an apparent pulsation frequency of 15.0 cycles/day. Our dataset is not extensive enough to find other modes, nor to exclude the existence of other modes.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998
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