Astron. Astrophys. 327, 365-376 (1997)

4. Results

In Figs. 3 - 5we show surface plots of the average count rate per pixel, the resulting peak power at the most significant frequency and the most significant frequency above 1.5 mHz in the light curve of a pixel, for all datasets, as derived from time-series analyses without frequency binning (i.e., W=1). Our definition of significant power is the 99.9% confidence level. As can be seen not all pixels within a particular dataset show evidence of an oscillation. However, in some instances, although not in all datasets, individual pixels which do not show a clear wave-like behaviour, if added together, can sometimes show a periodicity in the 3-4 mHz range at the 99.9% confidence level. In most cases, those regions showing a clear oscillation are confined in size. Sometimes, it can be a single pixel, thus less than . In other instances, the spatial size can approach a few pixels.

Fig. 3. Surface plots of the peak frequency above the 99.9% confidence level, the peak power and the count rate for datasets 89750, 91232, 92661, 92704 and 92712 (from top to bottom). 89750:... the brighter pixels, e.g. those approaching 1000 cts per integration time, have frequencies close to 2 mHz.. by adding together pixels which do not individually show significant power we can derive a significant signal at 3.3 mHz, this region corresponds to pixels adjacent to pixels showing an excess signal, on-the-other-hand, taking a large group of pixels far removed from those showing excess signal results in a null detection and to a reduced count-rate compared to the remainder of the dataset; 91232:... as in 89750 the brighter pixels correspond to lower frequency... again by adding individual pixels which do not reach the 99.9% confidence level we can obtain significant power at 4 mHz, although there are regions where no excess signal can be detected.. the count-rate in these regions are similar to those where an excess signal is present; 92661:... nice group of pixels showing significant power at 3.0/3.3 mHz, although there are many pixels which do not show significant power; 92704:... similar to 92661; 92712:... excellent group of pixels showing 3.0/3.7 mHz signal.

Fig. 4. Surface plots of the peak frequency above the 99.9% confidence level, the peak power the count rate for datasets 92746, 92750, 92752, 92789 and 92856 (from top to bottom). 92746:... only a few pixels show a periodicity; 92750:... some very good 3.0 mHz pixels, again as in 91232 some individual pixels not reaching the 99.9% confidence level if added can become significant around 4 mHz; 92752:... only a few pixels show a periodicity... taking a large group of pixels far removed from those showing excess signal results in a null detection; 92789:... several high count-rate pixels show significant power at 3 mHz, on-the-other-hand there are several pixels with a very low count-rate that do not show evidence for periodicity even if added together; 92856:... only a few pixels show a periodicity.

Fig. 5. Surface plots of the peak frequency above the 99.9% confidence level, the peak power and the count rate for datasets 92917, 92921, 92939, 93113 and 94400 (from top to bottom). 92917:... as in 92789, several pixels with a very low count-rate even if added together do not show evidence for periodicity; 92921:... nice group of bright pixels showing significant power at 2 mHz; 92939:... only a few pixels show a periodicity; 93113:... only a few pixels show a periodicity, also as in 92789, several low count-rate pixels even if added together do not show evidence of a periodicity; 94400:... only a few pixels show a periodicity.

In Fig. 6, we show the light-curve plus the resulting power spectrum for a group of 11 pixels (showing excess signal) from the central region of dataset 92712 (note that in this figure, two adjacent frequency bins were averaged, i.e., W=2). As can be clearly seen, excess signal at 3.0 and 3.7 mHz is present corresponding to periods of 330 sec and 270 sec respectively. These periods are similar to those derived by Drake et al. (1989). In dataset 92712, there is no evidence of flare-like activity, although in some datasets the observed periodicities are flare-like. As an example, we show in Fig. 7 the light-curve and power spectrum for dataset 92921. As can be clearly seen in the light-curve, there is a series of short duration bursts which are of a flare-like appearance. The time lag between these events ranges from 6 to 10 mins. This can also be seen from the power spectrum which shows peaks corresponding to these periods, plus a range of shorter periods in the range 2-4 mins.

In many datasets, as can be seen in Fig. 6, the 5 min. period has at least 9 cycles. However, this is not the case for the longer periods perhaps because our datasets are only of 44 min duration, thus a wave with a period approaching 10 mins. can have at most 4 cycles.

Fig. 6a and b. Light-curve and power spectrum for the average of the 11 pixels corresponding to the bright region within image 92712. Note that we have applied a binning by a factor of two in frequency.

Fig. 7a and b. Light-curve and power spectrum for the average of the 8 pixels corresponding to the bright region within image 92921. Note that we have applied a binning by a factor of two in frequency.

Our data comprise the light curves of pixels. In the power spectra of 738 light curves, (29%) statistically significant power peaks were found. In most of the cases significant power was found at multiple frequencies. Above 6-10 mHz the power, in general, varies around 2, the value expected from a (Poissonian) noise distribution. For each of the light curves containing significant power, we determined the frequency above 2 mHz at which the maximum power (peak power) was found. The distribution of the number of pixels with a peak power at a given frequency is shown in Fig. 8. Because three datasets were analyzed with a slightly different frequency grid we show these separately in Figs. 8b-8d while Fig. 8a is based on the twelve other datasets. Fig. 8a shows that for the majority of the datasets the most dominant powers are found in the 2-5 mHz range with a distinct maximum around 3 mHz. Only a few pixels have their peak power in the 6-10 mHz range. For most of the datasets the number of pixels with a peak power around 2 mHz is small. However, the data set shown in Fig. 8c contains a large number of pixels with peak powers at 2 mHz (8.3 min.). These pixels are often associated with flare-like brightnings and have high count rates.

Fig. 8. Histograms showing the number of pixels as a function of frequency. For each pixel the most significant power is determined and at that frequency the pixel is included in the distribution. Most datasets were analyzed with the same frequency grid (distribution shown in a) but for three datasets a slightly different frequency grid was used (shown in b is dataset 92750, c is dataset 91232 and d is dataset 89750). Above 10 mHz no significant power peaks were found.

For all datasets, we show in Fig. 9a the distribution of the peak powers as a function of frequency and in Fig. 9b the average count rate as a function of frequency. Note that these figures show a `vertical band' structure at certain frequencies because of the different frequency grid used for three datasets. Because the power is normalized according to Eq. (5), the 99.9% confidence level corresponds to about 23. This explains the minimum power level in the figure. Also, the power at which a data point is found is a direct measure of its significance level. Because we put our detection limit at the 99.9% confidence level, the powers shown in Fig. 9a are extremely significant. The strongest powers are found in the 2-5 mHz range although the magnitude of the power strongly varies. Between 6-10 mHz the distribution of the peak powers, in terms of their significance, is relatively flat.

Fig. 9a-c. a Normalized power as a function of frequency. b Average count rate as a function of frequency. c Power, without normalization with the total number of counts, as a function of average count rate. The different symbols refer to the different distributions shown in Fig. 8: asterisks: Fig. 8a; squares: Fig. 8b; circles: Fig. 8c; triangles: Fig. 8d.

The highest count rates are found for frequencies around 2 mHz and are associated with flare-like brightnings (Fig. 9b). Most pixels with significant power in the range 2-10 mHz have average count rates of 200-600 counts/second. However, between 2 and 5 mHz, the pixels show a larger spread of the count rate than above 5 mHz. Fig. 9c shows that there is a weak correlation between the count rate and the peak power in the sense that as the count rate increases the spread in the powers increases. This could, however, be a selection effect because low count rates are necessarily related to a relatively larger Poisson noise and lower (unnormalized) powers. On the other hand, Figs. 3 - 5 demonstrate that high count rates do not always result in the detection of significant peaks in the power spectrum. From this we conclude that there is no clear correlation between the presence of a significant period and the count rate.

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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