Astron. Astrophys. 334, 299-313 (1998)

4. Statistical analysis of solar X-ray flare parameters

4.1. Frequency distributions of X-ray flare parameters

Using the WATCH solar burst catalogue, frequency distributions are derived for the following parameters: peak count rate, total duration, rise time and decay time. Events where start, end/or peak time are not observed are not included (14 events) in the analysis, thus giving a total of 1537 events for the study.

4.1.1. Flare peak count rate frequency distribution

It is first investigated whether the shape of the frequency distribution of the flare peak count rate above background is sensitive to the shift in the energy bands of WATCH discussed in Sect. 2.3. It is found that the frequency distributions obtained for the different observing periods are all well-represented by power-laws above a turn-over with a slope which does not change significantly during the three observing periods.

The analysis can thus be performed on the total database and Fig. 10 (top) illustrates the frequency distribution of the peak count rate for the total database. It can be represented, above the turn-over at 50 c/s and for almost three orders of magnitude, by a power-law with a slope , where = -1.58 +/- 0.02.

Fig. 10. The frequency distribution of the WATCH flare peak count rate for the total observing period (top) well-represented by a power-law above the turn-over at 50 c/s (1251 events) with a slope = -1.58 +/- 0.02. The frequency distribution of the WATCH peak count rate for different sub-groups of events (bottom). The steeper slope ( = -2.17+/-0.07) refers to events with duration less that 200 seconds (full line) and the flatter slope ( = -1.15+/-0.05) to events that have durations greater than 1000 seconds (dashed curve)

We furthermore divided the events into five subgroups as function of their total duration (D): 6.5 s D 200 s, 200 s D 400 s, 400 s D 700 s, 700 s D 1000 s, D 1000 s. The frequency distributions are performed on the five subgroups and the two extremes are illustrated in Fig. 10 (bottom). All five frequency distributions can be represented by power-laws above a turn-over, but it is found that the slope of the power-law systematically varies with the range of durations of the events (see Table 5). The slope is steepest for the sub-group with the shortest duration and as the duration increases for each sub-group the slope of the power-law decreases. It must be noted that this effect is systematically observed independently of the values of the durations used for the limits of the sub-groups.

Table 5. Characteristics of the frequency distributions in peak count rates for sub-groups of events.
: number of events in the fit
: total number of events in the distribution.

4.1.2. Total duration, rise and decay time frequency distributions

Fig. 11 represents the frequency distribution of the burst total duration for the whole database. A single power-law above a turn-over does not fit the distribution very well. This effect was already suggested in other databases (Crosby et al. 1993; Bromund et al. 1995), but is much more pronounced in the present study probably because of the longer day-time orbit of the spacecraft which allows to observe longer duration events. Double power-law representation or a power-law with an exponential roll-over as suggested by Lu et al. (1993) have been used to fit the distributions and the results are summarized in Tables 6 and 7. The events are then divided into subgroups defined by the range of peak count rates (P 100 c/s, P 100 c/s) to investigate if the parameters of the frequency distributions vary from one sub-group to the other (see Tables 6 and 7). It is found that the distribution is always steepest in the long time range. When the frequency distributions are fitted by a single power-law with an exponential roll-over, the power-law is flatter for the sub-group with the largest peak count rates, while the value of (s) defining the exponential roll-over systematically increases (see Table 7). A similar behaviour is obtained for frequency distributions of rise and decay times (see Crosby, 1996 for more details).

Fig. 11. The frequency distribution of the bursts total duration for the whole observing period, and the fitting by either a single power-law with a slope ( = -1.08+/-0.03) with an exponential roll-over at = 2100 +/- 100 s (dotted line) or by two power-laws with slopes: 1 = -1.09+/-0.05 and 2 = -2.28 +/- 0.08 (full lines).

Table 6. Characteristics of the frequency distributions in total durations for subgroups of events (see text for details) (2 power-law fits).

Table 7. Characteristics of the frequency distributions in total durations for subgroups of events (see text for details) (power-law slope with an exponential roll-over ).

4.2. Correlation between the different X-ray flare parameters

Fig. 12 illustrates the correlation scatter plots between the different characteristic times (total duration, rise and decay time) as function of peak count rates. The slopes of the different correlation plots are similar ( 0.5). The correlation coefficients range between 0.5 and 0.6, the coefficient between the peak count rate and the rise time being the lowest one (0.5). This is consistent with what was previously observed at higher X-ray energies with HXRBS/SMM (Crosby et al., 1993) where a loose correlation was also observed between the flare duration and the peak count rate.

Fig. 12. Scatter plots between the (1) total duration, (2) rise time and (3) decay time with the peak count rate with the following slopes (s) and correlation coefficients (cc): (1) s= 0.52+/-0.07 and cc= 0.61, (2) s= 0.46+/-0.06 and cc= 0.50, (3) s= 0.57 +/- 0.06 and cc= 0.62.

4.3. Is there a relation between successive flares in the same active region?

A large percentage of the solar bursts recorded by WATCH could be associated with an active region. We define as T the elapsed time between two events in the same active region as the difference between the peaktimes of an event and of the preceding one. Bursts that are selected to define T must originate from the same telemetry dump or from adjacent dumps forming an uninterrupted sequence and be associated with the same active region. The study performed below on several active regions is based only on the dumps in which a lot of bursts arise from the same active region. (see Crosby (1996) for details). The frequency distribution of the elapsed time T is found to be well-represented by a power-law distribution with a slope ( = -0.78 +/- 0.13) and an exponential roll-over (T= 19000 s +/- 5600). The slope is similar to what Pearce et al. (1993) found using the HXRBS/SMM database. The reason why the exponential roll-over was not observed in the HXRBS/SMM database may still be related to the fact that the SMM spacecraft had 60 minutes observation windows due to the low orbit. It is investigated whether the magnitude of bursts associated with the same active region is dependent or not on the time elapsed between successive bursts. The results are plotted in Fig. 13. No correlation is found between the elapsed time T and the size of the event. It is also checked whether T is dependent on the size of the preceding event. The triangles on Fig. 13 correspond to bursts for which the preceding event was larger than 1000 c/s. As can be noticed, there is no evidence of the need of a sufficient time interval for a burst (even a big one) to occur after a large burst in the same active region.

Fig. 13. Scatter plot between the elapsed time T since previous burst and the burst peak count rate for bursts associated with the same active region. This plot is obtained by the accumulation of 19 plots performed on several active regions. The triangles indicate that the preceding event had a large peak count rate (1000 c/s).

© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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