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

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4. "Classical" seismological interpretation

4.1. Frequency splitting

In the power spectrum of HS 2324 there is not any clear direct sign of frequency splitting. This may be due to different reasons. A very small rotation rate, with secondary (m[FORMULA]0) modes below the frequency resolution, seems quite unlikely because it would require a rotation period longer than about 9 days. A more realistic possibility is that the star has a low inclination, so that the amplitudes of the m[FORMULA]0 modes are near the level of the noise. A third possibility is that the concentration of the peaks is so high that we are simply not able to recognize the modes splitted by the rotation.

If a direct identification of the rotational splitting is not possible, the high number of peaks allows one to make use of statistical methods. First we constructed an histogram of the frequency separations between the signals listed in Table 2, excluding the linear combinations. The result was not significant: no preferred frequency spacings appeared. Another attempt was done using all the peaks of the power spectrum higher than a fixed level; we selected 49 frequencies and made the histogram. Here also we did not get any significant result apart that all the peaks were never higher than the one day alias near 10-11 µHz.

At this point we invoked a third method: we computed the DFT of the amplitude spectrum, using two different subsets spanning 350-1000 µHz (all signals) and 440-500 µHz (only high power signals). The resulting power spectra are shown in Fig. 6 (left panels). Considering the lower panel, the highest peaks are at 10.1 and 4.2 µHz; a third peak at 2.5 µHz is clearly visible and more evident in the upper panel. Let us focus our attention to the latter two (for the first one we will give an interpretation below): their ratio, equal to 0.594 (or 0.589), is very close (1% level) to the canonical value of 0.6 predicted by asymptotic theory for the ratio between l=1 and l=2 rotational frequency splitting. Therefore the 4.2 and 2.5 µHz peaks might correspond to the frequency separation between m and m[FORMULA]1 (l=2 and l=1) modes. The corresponding rotation period of the star would be [FORMULA] days. This result may not be considered definitive because the method used is very sensitive to noise. Moreover the signal that we are looking for is not actually coherent: the constant frequency separation between the modes of the same overtone splitted by the rotation does not correspond, in general, to the separation between successive overtones, which is not constant in frequency (but almost constant in our particular case, due to the narrowness of the high-power region). From this point of view a more appropriate - but not much less noisy - method to measure the frequency spacing is given by the autocorrelation of the DFT (Press et al. 1992). The results of the DFT autocorrelation, reported in Fig. 6 (right panel), are less significant than, but do not contradict, those obtained from the DFT of the amplitude spectrum.

[FIGURE] Fig. 6a and b. Search for the frequency spacing. Left panels: power spectrum of the data amplitude spectrum using two different subsets of the DFT: 350 [FORMULA] f [FORMULA] 1000 µHz (all the signals, top panel ) or 440 [FORMULA] f [FORMULA] 500 µHz (only high-power signals, bottom panel ). The power is normalized to the peak at 10.52 (10.06) µHz, which is partially due to the one day aliases (see the text). Right panel: autocorrelation function of the data DFT using the same subset as in the left bottom panel. The peaks between about 8 and 13 µHz are partially due to the spectral window, as it is highlighted by the autocorrelation of the spectral window (dotted line).

Looking now at the peak near 10 µHz of Fig. 6, its frequency separation is very close to that of the one day alias; therefore we could conclude that it is actually produced by all the aliases of the signals. This conclusion would give more confidence in the rotational origin of the two peaks at 4.2 and 2.5 µHz. Moreover comparing the power of these two peaks with that of the one day alias, we could suppose that the weakness of the m[FORMULA]0 modes is actually due to the low inclination of the star. But with a deeper analysis (testing the variations of the three peaks of Fig. 6 (left panels) when we subtract different signals from the HS 2324 data (prewhitening)), we can easily demonstrate that the peak at about 10 µHz has at least two components: one at 10.6 µHz actually related to the one day alias and another one related to the separation between the two signals at about 474 and 485 µHz in the data DFT. Therefore it is more difficult to derive any consideration about the weakness of the m[FORMULA]0 modes and the low inclination hypothesis does not have any support. On the other hand, we can also demonstrate that the origin of the 2.5 and 4.2 µHz peaks is strongly related to the separation between a few large amplitude signals. Conclusion: if the frequency spacings of 2.5 and 4.2 µHz are actually due to the stellar rotation, the low inclination hypothesis can not be longer followed. The new even more simple picture would be the following: there are five l=1 triplet component candidates (474.1  1 and 476.6 µHz plus 389.1, 391.7 and 393.8 µHz) and there are three l=2 quintuplet component candidates (480.9, 484.5 and 488.8 µHz). If we derive the frequency spacing from these values we obtain a rotation period of the star [FORMULA] days, slightly different from the previous one. Other possible multiplets might be present at 483.2 and 485.4 µHz (l=1), and 463.7 and 467.7 µHz (l=2).

In this context we can also try to estimate the inclination of the star using the l=1 modes. Following Pesnell (1985) 2 we obtain an indication for [FORMULA].

4.2. Period spacing and mode trapping

The two peaks in the amplitude spectrum DFT, described in the previous section, could also be due to the period spacing between modes with successive overtones. In Fig. 7 (left panels) we show the DFT of the period spectrum (amplitude spectrum in the period domain). For the upper panel we used a subset of the period spectrum with periods between 1000 and 2857 s, while for the lower panel we used a narrower part with periods spanning 2000 - 2273 s. Excluding the peak at about 46 s, which is related to the one day alias as discussed in the previous section, the most significant period spacings are 18.8 s and 10.4 s (at least in the lower graph; in the upper graph the 10.4 s peak appears more uncertain). Their ratio is close (accuracy better than 5%) to the asymptotic value of [FORMULA], suggesting that 18.8 and 10.4 s might correspond to the l=1 and l=2 period spacings. In this hypothesis, the differences between the two left panels of Fig. 7 suggest that the l=2 modes might be present only (or mainly) in the high-amplitude region between 2000 and 2273 s.

[FIGURE] Fig. 7a and b. Search for the period spacing. Left panels: Fourier transform of the period spectrum (amplitude spectrum in the period domain) in the period range 1000 [FORMULA] P [FORMULA] 2857 s (top ) and 2000 [FORMULA] P [FORMULA] 2273 s (bottom ). The power is normalized to the peak at 46.3 (46.1) s, which is mainly due to the one day aliases in the main power region between 2000 and 2273 s. The peaks at 18.8 and 10.4 s might be due to the l=1 and l=2 period spacing. The peak at 9.4 s is the first harmonic of the 18.8 s signal. Right panels: Kolmogorov-Smirnov test (top ) and Inverse Variance test (bottom ) applied to the first 12 frequencies listed in Table 2 (excluding the linear combination region). Both tests do not show any significant value for the period spacing.

An attempt to confirm the hypothesis that the modes of HS 2324 are equally spaced in period (and not in frequency) has been done applying the Kolmogorov-Smirnov (K-S) test (Kawaler 1988) and the Inverse Variance technique (O'Donoghue 1994) to the first 12 periods listed in Table 2 (excluding the linear combinations). The results, reported in Fig. 7 (right panels), do not confirm that the modes are equally spaced in period. Moreover, the lack of any significant period spacing further indicates that the period list is not complete 3.

Nothing may be said about the trapped modes phenomenon apart the following. The ratio between the frequencies of the highest peaks in the 380 and 475 µHz regions gives [FORMULA]/2 with an accuracy better than 1[FORMULA]. This number was found in other GW Vir stars, as RXJ 2117+3412 and the central star of NGC 1501 (Bond et al. 1996), and is compatible with calculated trapping coefficients (Kawaler & Bradley 1994).

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999