## 3. Methods and resultsTo 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 with the filter pulse response characteristic coefficients : The spectra of the filtered and initial series and are related by the expression The frequency and pulse response characteristics and are related by the reciprocal Fourier transformation where the frequency is given in units of the Nyquist limiting frequency and is the sampling time. The ideal low-frequency filter with a cutoff frequency with the frequency and pulse characteristics cannot be practically realized, as the number of the terms The simplest low-frequency filter is the moving-average one. The pulse coefficients of this filter are constant: , where is the filter length. The frequency characteristic of the filter is The filter passband is 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 where is the modified Bessel
function of the zero order, the
parameter that enters the filter model, 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.
The numerical values for the Kaiser filter were: the cutoff Hz, the width of transition band 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 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.
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 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.
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 may be computed as the Fourier transform of the apodizated autocovariance function is the number of photons detected during the sample time , , . We use the Tukey window where 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 is proportional to the power where is the variance of the count rate . In this case the noise peaks in the spectrum are described by the statistic For the Tukey window we have the degree of freedom and a spectral resolution at half-maximum of the spectral peak. From Eq. (12) we establish a threshold for detecting a signal at the confidence level 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).
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |