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

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4. Analysis of the observations

For the reduction procedure, the first step is to clean the raw data that sometimes come with bad points due to clouds, pointing and guiding problems, etc. After that, we subtract the contribution of the sky to both the target and the constant stars. According to the differential photometry technique, we generate two light curves in magnitudes, BQ-Comp and BW-Comp, after subtracting the comparison star to the target stars light curves. (see our Fig. 1). The observed curves are fitted by a parabola using a least-squares fitting each night to filter out long-period changes in the signal (i.e. differential color effects between target and comparison stars, transparency fluctuations or instrumental drifts). This reduces the high amplitude signal at very low frequency due to these effects, which also produces a decrease in the noise level at higher frequencies, up to a factor of 2 in amplitude in our case. Of course, potentially, intrinsic signal to the star would also be filtered after this fitting, but its effect will be only important for frequencies below [FORMULA]Hz. Finally, after removing the mean of the residuals, we obtain the final light curves in magnitudes for each one of the observed stars.

With the light curve of the stars, we then proceed to get the frequency spectra by Fourier techniques as the ISWF (Ponman 1981) employed in Belmonte et al. (1994a). As it has been described in our earlier papers, the results obtained with the ISWF and the FFT routines, are similar. Our final result is the spectra of amplitudes A and phases [FORMULA] of the stars. The resulting amplitude spectra for every star, BQ and BW Cnc, were calculated using this procedure with a sampling in frequency of 0.1 µHz.

The window function for the campaign is plotted in Fig. 2 and it shows the low amplitude of the side lobes (38% of the main lobe) and the good spectral resolution achieved in this campaign, [FORMULA]Hz, measured from the FWHM of the main lobe in the spectral window. The high spectral resolution that we have in our campaigns shows its importance in situations where very close frequencies are found as the ones we will describe later in Sect. 5.2.

[FIGURE] Fig. 2. Spectral window in amplitude for this campaign. Side lobes are at 38% of the main lobe

Once we have the amplitude spectra of the differential light curves, the main problem is to detect which one of the obtained amplitude maxima is likely to be due to a real oscillation of the star itself and not to the noise added to the signal. There are several statistical tests that can be used to determine the significance of the signal. Following Belmonte et al. (1994a) and Michel et al. (1992), we have applied two of them: the Scargle's test (Scargle 1982) and Fisher's test (Fisher 1929, Nowroozi 1967, Koen 1990). Both are based on assuming the presence of a white gaussian noise added to the signal we want to extract and, as explained below, their application has been modified as to include "coloured noise" effects. These tests offer an estimation of the significance of a given point in the amplitude spectrum, i.e. the confidence of a peak to be a real signal.

Since the observational noise is frequency-dependent, the tests are applied in 100-µHz moving boxes along the spectrum in order to have, at the same time, the largest possible number of independent frequencies and a fragment of spectrum close to white noise. The number of independent frequencies in the spectrum is only well determined in the case of evenly spaced data without gaps, with [FORMULA] independent frequencies between 0 and the Nyquist frequency for a time series of N points. However, applied to the Scargle's test, Horne & Baliunas (1986) determined a value closer to N independent frequencies for the same frequency range. Comparing both cases for evenly spaced data ([FORMULA]Hz) with only white noise (Fig. 3a), it can be noticed the small dependence of both tests on the number of independent frequencies and the very similar results between tests. We point out that an artificial level of 3.7 times the mean amplitude can reproduce the 99% confidence level obtained by these tests in each 100-µHz box.

[FIGURE] Fig. 3a and b. Statistical tests applied on spectra of a  white noise and b  BQ Cnc without the detected modes, in boxes of 100-µHz. Scargle's test (dotted) and Fisher's test (dashed) are used. A 99% confidence level is plotted. Thick lines represent the 99% level for both tests using twice the number of independent frequencies. This figure shows how the 3.7-mean-amplitude level in each box (solid line) can reproduce fairly well the 99% confidence level, no matter if the number of independent frequencies is well known or if the data have gaps b or not a (see text).

For unevenly spaced data as in a real case, there is no precise way to determine the number of independent frequencies. An estimation can be obtained from the separation of the minima in the window function (Scargle 1982), which in the evenly spaced case gives the known [FORMULA] frequency interval. This separation can be roughly estimated through the FWHM of the main lobe of the window function. For our campaign, the FWHM value is 0.60 µHz and the mean value of separations between minima found in a range of 55 µHz around the main lobe, was 0.685 µHz. Since the dependence of the confidence level on the number of independent frequencies is small, we can use the FWHM of the main lobe of the window function, as a frequency interval to determine that number. By using this, we calculated the 99% confidence level from both tests and also, the 3.7-mean-amplitude level applied on the spectrum of BQ Cnc after removing all the frequency peaks (see Fig. 3b). As it can be noticed from this figure, our artificial level can represent well the 99% confidence level in a real case with data gaps, when using the 100-µHz moving boxes and spectral windows like ours (FWHM [FORMULA] 0.60 µHz). As a consequence, all frequency peaks above this level will be considered as signals detected in the data set.

These criteria were then used to find frequency peaks in the spectra which are, with a 99% confidence level, due to the star pulsation. We use the prewhitening method to subtract a sinusoidal curve of amplitude, frequency and phase corresponding to the selected frequency from the original time series. This process is applied iteratively, getting a new frequency spectrum and a new 99% confidence level every time, until all points of the spectrum are below that amplitude level.

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© European Southern Observatory (ESO) 1998

Online publication: November 3, 1998