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Astron. Astrophys. 340, 149-159 (1998)

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5. Results

We have applied the considerations indicated above to the data obtained in this three-site campaign, and the results are given in Table 3. The prewhitening process to obtain the frequencies detected above a 99% confidence level in BQ and BW Cnc is represented in Fig. 4.


Table 3. Modes detected in our target stars above a 99% confidence level. Frequencies are accurate in the same order than the resolution ([FORMULA]Hz). Estimated errors in amplitude and phase are of the order of [FORMULA] 0.1 mmag and [FORMULA] 0.1 rad, respectively. Origin of [FORMULA] is at HJD 2 449753.0

[FIGURE] Fig. 4. Amplitude spectrum and prewithening process in BQ and BW Cnc . In each step, one peak above the 99% confidence level (solid line) is selected and removed from the time series and a new spectrum is obtained. During this process, the amplitude of the harmonics decrease. In the case of BQ Cnc (top ), three modes above this level were detected in this way (dot-dashed lines). For BW Cnc (bottom ), nine frequencies were detected this time (dot-dashed lines). Four of them are grouped in two pairs of very close frequencies each - [FORMULA] [138.7 µHz] and [FORMULA] [139.1 µHz], and [FORMULA] [260.6 µHz] and [FORMULA] [261.5 µHz] (see Table 3 and Fig. 8). The low noise level reached after prewhitening all the frequencies.

The low noise level reached in this campaign for our target stars, has to be noticed. As it is well known, this noise increases at low frequency range while it is almost constant at higher frequencies (above [FORMULA]500 µHz, in our case). For BQ Cnc, the mean noise level was 150 µmag at 200 µHz and 50 µmag at 600 µHz. For BW Cnc, the noise levels were 190 and 60 µmag for the same positions, respectively.

Figs. 5 and 6 show the light curves for BQ and BW Cnc respectively, after we applied the procedure described in the paper. The solid line is the synthetic light curve of the detected modes for each star that reproduces the observations (see Table 3). We can see that the "peak to peak" amplitudes may be larger than 20 mmag. Data averaged every minute are represented by dots.

[FIGURE] Fig. 5. Differential light curve for BQ Cnc . Data averaged every minute are represented by dots. The solid line is the synthetic light curve reproduced by using the three frequencies as shown in Table 3.

[FIGURE] Fig. 6. Differential light curve for BW Cnc . Data averaged every minute are represented by dots. The solid line is the synthetic light curve reproduced by using the nine frequencies as shown in Table 3.

5.1. BQ Cancri

BQ Cnc shows three well marked observed pulsation frequencies, as shown in Table 3. These frequencies are concentrated below 200 µHz, as for other Praesepe [FORMULA] Scuti stars observed (EP Cnc, Breger et al. 1994; KW 284, Belmonte et al. 1994b; BT Cnc, Hernández et al. 1998).

Earlier observations of BQ Cnc, report a frequency at 156.4 µHz (Breger 1973) after only 3 hours of observations from a single observatory. Vidal et al. (1993), after 19.6 hours of observation from one observatory, found three frequencies from CCD observations. Their highest amplitude frequency (11.9 mmag with a period of 11.11 h) is most probably a harmonic frequency of a day periodicity (1/2 a day). They found two other frequencies at 119 and 174 µHz, that probably correspond to the same modes we have detected ([FORMULA] and [FORMULA]), but their results lack the needed precision, due to the poor spectral window and the low signal-to-noise ratio of their data.

As we already noticed, BQ Cnc is a binary system. Considering that the system is not eclipsing, the total luminosity should be the sum of the components luminosity. Only a limited number of combinations for the colour indices of the components can give the observed colour index. Furthermore, the luminosity and colour of each component must be located on the cluster isochrone.

With this information, we can restrict the position of both components in the HR diagram. This should allow us to investigate which star is actually pulsating. Since the primary is the brightest component, this should have similar observational features to the observed system. Colour excess estimations offer a [FORMULA] for the primary in the range of 7500-7800 K and 7000-7300 K for the secondary approximately, differences in agreement to what is expected for a SB2 system (Andersen 1991). Adding the restriction in luminosity (i.e. the system luminosity is the sum of the components' luminosity), we obtain a smaller range in [FORMULA] for the estimated positions of the primary and secondary on the isochrone (see Fig. 7). From this figure, it can be seen that the primary should be the pulsating component since the secondary is found beyond the red edge of the instability strip.

[FIGURE] Fig. 7. Estimated positions of the primary and secondary components (dashed boxes) of BQ Cnc on the cluster isochrone. These positions are derived from colour and luminosity restrictions on the components (see Sect. 5.1). Only the primary is found between the blue (BE) and red edges (RE) of the instability strip. As a reference, an isochrone belonging to Z = 0.025, that matches the HIPPARCOS distance used, has been plotted as a dashed line.

Taking into account this position on the cluster isochrone for the pulsating component and the HIPPARCOS distance modulus ([FORMULA] = 6.24 [FORMULA] 0.12 mag.), the following information for the primary can be derived: [FORMULA] = 2.20-2.55 mag., [FORMULA] = 7400-7500 K, M = 1.55-1.76 [FORMULA], and [FORMULA] = 4.16-4.22 dex. This gives a pulsation constant, Q of 0.045-0.052 day for the frequency at [FORMULA], which is far above the Q of the fundamental frequency ([FORMULA]=0.033 day.) showing that this frequency must correspond to a g-mode, even considering the decrease of the frequency due to rotation (Pérez Hernández et al. 1995). For the other two modes, the Q values are 0.034-0.039 day and 0.031-0.035 day, for [FORMULA] and [FORMULA], respectively, so both could be the fundamental mode. Models within suitable age ranges for Praesepe cluster such as 550-650 Myr. and 700-800 Myr. for Z=0.03 and Z=0.019, respectively, were taken to evaluate the pulsation constants (Michel et al. 1998).

5.2. BW Cancri

BW Cnc shows a richer frequency spectrum, covering the full range from 60 to 400 µHz. Breger (1973), after 3 hours of observations, reported a main frequency at 160.7 µHz that agrees, within its spectral resolution, with the mode of highest amplitude detected during our campaign [FORMULA] = 171.8 µHz. Other stars observed by us also show these rich frequency spectra (BN Cnc and BU Cnc, Belmonte et al. 1994a). BW Cnc covers the whole range, while frequencies of BN Cnc and BU Cnc are concentrated in a smaller range.

It should be noticed that the detection of the frequency reported at 68.3 µHz above the 99% confidence level could be affected by the parabola filtering, since the filtering leaves a spectrum "less white" at low frequencies. So, close to this zone ([FORMULA]Hz), the noise is less white than for the rest of the spectrum and, therefore, application of the tests will be more doubtful. Due to that, the amplitude of that frequency has to be taken with caution.

BW Cnc shows two pairs of very close frequencies ([FORMULA] and [FORMULA], and [FORMULA] and [FORMULA]) that can be resolved only when we take the whole campaign spectrum. In fact, in the case of [FORMULA] and [FORMULA], they have a separation under the resolution limit (0.60 µHz), but the larger width of the peak and the existence of a residual frequency above the 99% confidence level, when one of them is removed during the prewhitening process, (see Fig. 8) gives some confidence that this pair of close oscillation frequencies are indeed intrinsic. We also noticed that the sum of two of the detected frequencies, [FORMULA] and [FORMULA], offers a value (361.7 µHz) close to another detected frequency at [FORMULA], which could be casual, although the presence of non-linear effects cannot be discarded neither.

[FIGURE] Fig. 8. BW Cnc. Comparison between the amplitude spectra of the two pairs of close frequencies found at [FORMULA]-[FORMULA] and [FORMULA]-[FORMULA] (see Table 3), before and after subtracting the peak of largest amplitude. The second pair is clearly resolved, while the first is unresolved. In spite of that, both frequencies in the first pair are above the 99% confidence level (dashed line) after the subtraction.

The possibility of investigating the evolution of the spectrum along the campaign (29 days) when pairs of peaks are excited, is very interesting. In this way, we splitted the whole period in smaller periods of three days, to study individually the behaviour of these peaks along the time. We noticed the almost disappearance of the lower frequency pair in the first half of the observations, while the higher frequency pair was hidden in the noise in the second half. To appreciate better this situation, we selected two sets of data with similar number of points in order to have comparable conditions, and as much separated as possible in time. Considering this, the first 12 days of the campaign (HJD 2 449737.11-749.62, 4817 minutes) and the last 9 days (HJD 2 449755.89-765.28, 4032 minutes) were chosen. The amplitude spectrum corresponding to each one is plotted in Fig. 9 (left). The change in amplitude of the peaks at the frequency position belonging to the pairs is clear, in comparison to the other modes detected.

[FIGURE] Fig. 9. Left: Amplitude spectrum of BW Cnc in two different observing periods, at the beginning and at the end of the campaign. Arrows represent the positions of the detected modes in the whole campaign spectrum, as listed in Table 3. Arrows with 2's mark the position of the two pairs of close frequencies, not resolved in these smaller sets. A 99% confidence level has been applied in both spectra (solid line). It can be noticed the dramatic amplitude changes experimented at the position of the pairs compared to the other detected modes along the observations. Right: Amplitude spectrum of 4 artificial sinusoids corresponding to the components of the two pairs of close frequencies (taking the A, [FORMULA] and [FORMULA] from Table 3) for the same observing periods. It can be seen the effect of the mutual interference in each pair on the amplitude, which is shown to be the main source of the amplitude changes at the same positions in the observed spectra (left ).

This is mainly due to interference phenomena where the close frequencies and the similar amplitudes of the elements of each pair produce constructive and destructive interferences in different epochs of observation. To illustrate this fact, we generate four sinusoids with the frequencies, amplitudes and phases corresponding to each element of the two pairs. With the same time sampling as for the two previous data sets, we obtained the amplitude spectrum for each set, with only the four sinusoids. In Fig. 9 (right), it can be seen that such mutual interferences are indeed the reason for the amplitude changes of a factor, from 2 to 3 between the two sets.

Considering that this close frequency phenomenon can be found frequently, as it can also be seen in BN and BU Cnc (Belmonte et al. 1994a), the importance of having long observational campaigns is confirmed again by this sort of effects. Moreover, this means that we can have different modes detected depending on the observing epoch, although we observe a long enough series of data. In this way, studies of secular amplitude changes of modes where typical 1-week observing periods are usually included, can be strongly affected by this effect.

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© European Southern Observatory (ESO) 1998

Online publication: November 3, 1998