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Astron. Astrophys. 364, 641-645 (2000)

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3. Methods and results

To detect small oscillations, large intrinsic intensity variations must first be removed. To achieve this, we apply high-pass filtering. This may be realized by subtraction of low-frequency components from the time series. After that, the residuals are devided by the mean quiescent intensity to obtain only the high-frequency fractional intensity variations.

Frequency filtering can be performed by convolving the series counts [FORMULA] with the filter pulse response characteristic coefficients [FORMULA]:

[EQUATION]

The spectra of the filtered and initial series [FORMULA] and [FORMULA] are related by the expression

[EQUATION]

The frequency and pulse response characteristics [FORMULA] and [FORMULA] are related by the reciprocal Fourier transformation

[EQUATION]

where the frequency [FORMULA] is given in units of the Nyquist limiting frequency [FORMULA] and [FORMULA] is the sampling time.

The ideal low-frequency filter with a cutoff frequency [FORMULA] with the frequency and pulse characteristics

[EQUATION]

[EQUATION]

cannot be practically realized, as the number of the terms l must be infinite in order for the condition (4) to be fulfilled.

The simplest low-frequency filter is the moving-average one. The pulse coefficients of this filter are constant: [FORMULA], where [FORMULA] is the filter length. The frequency characteristic of the filter is

[EQUATION]

The filter passband is [FORMULA] However, this filter has many disadvantages, among which is the aliasing effect, whereby some fraction of the signal power may percolate through the filter side lobes. This difficulty is easily resolved by the use of the more refined near-ideal Kaiser filter (Rabiner & Gold 1975). The filter coefficients are

[EQUATION]

where [FORMULA] is the modified Bessel function of the zero order, [FORMULA] the parameter that enters the filter model, l the number of filter coefficient pairs. Three basic input parameters, i. e. (1) the filter passband [FORMULA], (2) the width of transition band and (3) the stopband loss in the decibel scale, completely determine quantities [FORMULA], l and the filter as a whole (for more details see Kaiser & Reed 1977). Thus, using the Kaiser filter we can set limits on signals that would have been aliased through side lobes into the stopband frequency domain.

We have used both the moving-average and Kaiser convolutions to eliminate the main outburst light curve from the sets of EV Lac data. In our cases mentioned below, the cutoff frequency values of 0.067, 0.055, and 0.033 Hz were employed. This removes any variations in time scales greater than 15, 18 and 30 s, respectively. From comparison studies of the moving-average and Kaiser convolutions, it has been found that the oscillation pattern does not change, disregarding minor variations occurring in the vicinity of sharply defined changes in the light curve.

Fig. 1b shows a portion of the high-pass filtered flare light curve of EV Lac, obtained at Peak Terskol with the 2 m telescope. Readings in the U band were taken every 0.2 s. Well-defined short-period oscillations are superimposed on to the very intense main flare plotted in Fig. 1a. The gap in the plot in Fig. 1b is due to the transition process resulting from filtering in the vicinity of the sharp flare maximum in that case.

[FIGURE] Fig. 1a and b. The EV Lac outburst light curve (upper panel) and superimposed high-frequency oscillations (lower panel), as seen by the 2 m telescope at Peak Terskol on October 8, 1998, 18:03:01 UT (max), U-band.

The numerical values for the Kaiser filter were: the cutoff [FORMULA] Hz, the width of transition band [FORMULA] Hz, the stopband loss is 60 decibels and the time span of the filter is 35 s.

The flare event in Fig. 1 is a clear example of an extreme strong excitation of the HFO with a sudden onset and a decay similarly to that of HII 2411 reported by Rodonó.

The above inferences about high-frequency oscillations were tested by means of the results of many-site synchronous observations. Fig. 2a shows the simultaneous observations of the oscillations on EV Lac at the Crimean and at the Stephanion observatories. Fig. 2b shows the main flare curve in B band with sampling time of 1.2 s. Time variations in oscillations were detected by subtraction of a moving average over 15 points (18 sec) and normalized to the quiescent intensity as was mentioned above. A comparison between these two measurements obtained from synchronous observations at different sites showed obvious correlations significant at greater than 99% confidence level (Fig. 3). A period of 12.8 [FORMULA] 0.7 s (0.078 Hz) was obtained from the times of maxima plotted in Fig. 2a and a mean amplitude of 0.025 mag was calculated. From these facts we can be assured that both an atmospheric and an instrumental origin for the high-frequency oscillations can be ruled out with a high degree of confidence.

[FIGURE] Fig. 2a and b. Portions of the high-frequency B-band remains (upper panel) and of the main light curve (lower panel) from the sets of EV Lac data obtained with the Crimean (solid line) and Greek (dotted line) telescopes on September 11, 1998, 21:55:02 UT (max).

[FIGURE] Fig. 3. The cross-correlation between the sets of high-frequency oscillation data shown in Fig. 2a. The dashed lines show the 99% confidence level for the sample correlation coefficient of two mutually independent normal variables.

During our observations, multicolor monitoring of EV Lac was being carried out constantly only in Crimea. But in some cases, flare events were registered simultaneously in different bands from different sites. From the many-site observations, there is experimental evidence that oscillations occur around the flare maximum phase in B color (Fig. 2). Now we have a good chance of following its characteristics, including color variations, during the whole flare light curve.

Fig. 4 gives some insight into the way in which high-frequency oscillations arise and develop during the progress of a flare. An illustrative example of a strong excitation of oscillations is furnished by Fig. 4a. These oscillations first arise at the earliest stage of the flare development, with a frequency of 0.039 Hz (the period = 25.7 [FORMULA] 1.8 s). Some time later they transform into a wave of a twofold frequency. Their amplitude may reach 10 % of the quiescent intensity in the U band and about five times lower in the B. A further example of a highly-blue oscillation color is given in Fig. 5. This figure shows the oscillations obtained at the Belogradchik (U-band) and Stephanion (B-band) observatories synchronously. The high-frequency B-band residuals magnified by a factor of six practically coincide with the U ones. At the same time the expected dU/dB ratio between U and B fluxes, caused by the atmospheric scintillation, lies in the range from 1 to 1.2, depending on the aperture of the telescope (Stecklum 1985). The result lends additional support to the reality of the high-frequency oscillations.

[FIGURE] Fig. 4a and b. Time variations in high-frequency oscillations in the U (solid) and B (dotted) bands from the sets of EV Lac data obtained at Crimea on September 11, 1998, 21:55:02 UT (max), (upper panel). The lower panel represents the raw main U and B light curves. Both B curves are magnified by a factor of 5.

[FIGURE] Fig. 5. The magnified B-band and raw U-band light curves of the EV Lac flare on September 15, 1998, 22:30:26 UT (max) (upper panel) as seen simultaneously by the Stephanion 30 inch telescope (dotted line) and the Belogradchik 60 cm telescope (solid line). The lower panel shows time variations in high-frequency oscillations; both curves are strongly correlated at a more than 99% confidence level.

To investigate the oscillation frequency spectra, the high-frequency residuals of the outburst light curve in the U band were subjected to a power spectrum analysis with the Tukey spectral window, as described by Jenkins & Watts (1969). The power spectral density [FORMULA] may be computed as the Fourier transform of the apodizated autocovariance function [FORMULA]

[EQUATION]

[EQUATION]

[FORMULA] is the number of photons detected during the sample time [FORMULA], [FORMULA], [FORMULA]. We use the Tukey window

[EQUATION]

where [FORMULA] is a cut off portion of the total number of measurements N that allows adjustment of the spectral resolution.

Two kinds of noise are typical for time-series photometry of stars: scintillation noise from the atmosphere and stochastic Poisson noise due to the limited number of photons detected, with the uniform spectral density. If the latter prevails (which is common with faint stars, such as EV Lac), the signal-to-noise ratio [FORMULA] is proportional to the power

[EQUATION]

where [FORMULA] is the variance of the count rate [FORMULA]. In this case the noise peaks in the spectrum are described by the [FORMULA] statistic

[EQUATION]

For the Tukey window we have the degree of freedom [FORMULA] and a spectral resolution [FORMULA] at half-maximum of the spectral peak. From Eq. (12) we establish a threshold for detecting a signal at the confidence level [FORMULA]

[EQUATION]

The power spectra in Fig. 6 indicate clearly that an oscillation feature occurs during the outburst phase. This oscillation feature is absent both in the preflare state and at the late flare tail. Two harmonics were detected at 0.039 and 0.078 Hz, both during the early rise and the early decline phase of the outburst. The remarkable fact is that EV Lac also exhibits the short-period harmonic at 13 s, as in the case of HII 2411 reported by Rodonó (1974).

[FIGURE] Fig. 6. Power spectra of high-frequency oscillations calculated from three sets of data shown in Fig. 4, transformed into a signal-to-noise ratio scale. The solid line indicates an early rise and decline region of the outburst. The dashed and dotted lines correspond to a preflare and flare tail regions, respectively. The 99% confidence level for white noise data is shown as the dashed horizontal line.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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