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Astron. Astrophys. 342, 745-755 (1999)

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3. Temporal spectroscopy

3.1. Spectral window and power spectrum

We computed a single sine function with unit amplitude at the same sample times of the entire data set (window function). The discrete Fourier transform of the window function gives the spectral window, which is shown in Fig. 2. For more completeness, both the amplitude and the power (amplitude squared) spectra of the window function are presented. The only structures which may give troubles for the unambiguous identification of the modes are the 1 cycle/day aliases, with a relative amplitude of about 0.3 (relative power of 0.09).

[FIGURE] Fig. 2. Power spectrum (top ) and amplitude spectrum (bottom ) of the entire data set. The DFT of the window function (spectral window) is also reported both in power and in amplitude; we can note that the 1 cycle/day sidelobes have maxima at [FORMULA] 10.6 µHz, corresponding to a period of 1.09 days.

Using the same discrete Fourier transform (DFT), based on the Deeming (1975) method and Kurtz (1985) algorithm, we computed the transform of the entire reduced data set. In Fig. 2 both amplitude and power spectrum are shown in units of millimodulation amplitude (mma) and micromodulation power (µmp), following the suggestion of Winget et al. (1994). We clearly see that the signals are concentrated in three main regions near 390, 470 and 950 µHz. These regions are highlighted in Fig. 3. The tested frequency resolution of our data set is 1.4 µHz, according to Loumos & Deeming (1978); such a value corresponds to obtain a half amplitude separation of two close peaks with equal amplitude.

[FIGURE] Fig. 3. Detailed power spectrum of the entire data set. Note that different vertical scales have been used.

Looking at Fig. 3, some structures appear to be not completely resolved, suggesting that the light curve could be too short that we can resolve all the present frequencies. To test this crucial point, we divided the data set into two equal parts and computed the Fourier transform of each part. The two power spectra, presented in Fig. 4, show strong differences not only in amplitude, but also in frequency. The first interpretation of such differences is simply that the data set is not long enough to "stabilize" the Fourier transform. In other words, the light curve is not completely resolved. The reliability of this hypothesis is increased by the fact that our best multisinusoidal fit of the entire data set (see next section and Table 2) gives good results also when applied only to the first or the second half of data (Fig. 4). If the DFT apparent instability is actually due to the insufficient coverage, most of the analyses reported in Sects. 3.2, 4.1 and 4.2, and based on the assumption that the DFT of HS 2324 is not time dependent on time scales shorter than our run, will need further confirmation from a new longer observational campaign.

[FIGURE] Fig. 4. Power spectrum of the first half of data (top ) versus second half (bottom ). Note the strong differences both in power and in frequency. In the small panels the regions of interest are highlighted; the dotted lines represent the spectra of the first and second half of a synthetic light curve obtained summing the 19 sinusoids listed in Table 2 and having same time sampling as the HS 2324 data. No noise was added to the synthetic light curves.


Table 2. Results of the sinusoidal fit
[FORMULA] Time of the first maximum inside the data set.
[FORMULA] These frequencies are the most reliable (see the text).

On the other hand, if the DFT time instability was real, we would need a different explanation for such peculiar behaviour. An alternative hypothesis of a fast damped oscillator has been considered and is reported in Sect. 5.

3.2. Frequency identification

Looking at Figs. 2 and 3, it is immediately evident that determining the active frequencies from the power spectrum of HS 2324 will be more difficult than in most other GW Vir stars for several reasons: the power is concentrated in only 3 crowded regions; the amplitudes are very low; the low frequencies imply that the frequency and the period spacing expected may have about same values. The high frequency region does not help much because it seems to be constituted only by linear combinations of the low-frequency peaks. Moreover we know that the frequencies are not completely resolved and therefore we certainly have errors both in frequency and in amplitude.

To distinguish the real frequencies present in the HS 2324 data from the artifacts introduced by the spectral window, we proceeded as follows. First we selected the highest peak in each of the three "active regions" near 390, 470 and 950 µHz. The separation of the three active regions guarantees that the aliases of each frequency have almost zero influence in the other two regions. Second we applied a least-squares multisinusoidal fit to the data to determine accurate amplitudes and phases of the three selected sine waves. Third we created an artificial signal adding together the three sinusoids and using the same sampling times as the data. This artificial signal was then subtracted from the data (prewhitening), and the residuals were analyzed again. Three (or less) new frequencies were selected and the whole procedure was repeated n times until the power of the prewhitened data was near the level of the noise. At each iteration we selected first those frequencies which were not coincident with the one day aliases of the strongest signals. At the end of the whole process, the frequencies, amplitudes and phases were optimized with a final least-squares fit with all the frequencies found. The resulting best fit parameters are listed in Table 2. It is important to emphasize, however, that the solution in Table 2 is not the only one. After having performed the prewhitening of 7 frequencies (marked with an asterisk in Table 2), different solutions become possible. The frequencies selected in Table 2 represent the result of several attempts. The solution that we have chosen has the advantage that it produces small residuals with a relative small number of frequencies (Fig. 5). Looking at Table 2, we can note that most (if not all) of the high frequency signals correspond to linear combinations of the high-amplitude frequencies.

[FIGURE] Fig. 5. Upper panels: amplitude spectrum of the entire data set (top ) compared with the spectrum of the 19-frequency fit (bottom ) having the same time sampling as the data. No noise was added to the synthetic data. Lower panel: spectrum of the residuals. Note that the vertical scale is the same in all panels.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999