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

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5. Discussion and conclusion

5.1. Energy contained in the WATCH flares

The analysis of the WATCH observations shows that in agreement with previous works, the X-ray emitting component around 10 keV does not only result from the plasma detected around a few keV by e.g. GOES. Either a hotter component or a non-thermal electron population must produce this emission. The similar time profiles observed around 10 keV and at higher energies suggest that non-thermal emission is produced in some events down to [FORMULA] 10 keV. This has previously been suggested from the analysis of other flares in both soft and hard X-ray domains (e.g. Gabriel et al. 1984; Hernandez et al. 1986; Gabriel et al. 1991; Kane et al. 1992). In some events, there is an indication of a "Neupert effect" between the WATCH time profiles at 10 keV and the derivatives of GOES time profiles around a few keV. This suggests that X-ray emission around 10 keV contains indeed a non-thermal component, and thus is a good indicator of the primary energy release in a flare.

Assuming non-thermal emission for the most energetic deka-keV bursts observed by WATCH, it is found that the peak energy flux extrapolated to the HXRBS/SMM range is that of the small hard X-ray flares (less than 1027 ergs/s above 25 keV). This suggests that the complete WATCH solar database discussed in this paper deals with smaller energy releases than those observed by HXRBS/SMM.

The observations of solar bursts by WATCH also revealed that several bursts at 10 keV may occur during a single GOES soft X-ray flare. This suggests that the injection of energy contained in suprathermal electrons occurs throughout a flare and not only at the rise phase of this flare. A lot of small soft X-ray enhancements detected by GOES are associated with small bursts observed by WATCH around 10 keV. The production of a hot (T [FORMULA] 107 K) plasma in the corona or of a suprathermal population of electrons is thus a relatively common process, which is not only limited to GOES soft X-ray flares of class C or greater.

5.2. Frequency distributions and their interpretations

The earliest attempt to account for the flare size distribution was proposed by Rosner & Vaiana (1978) in the context of the 'stochastic relaxation model'. It is based on the following three assumptions: 1) flaring is a stochastic relaxation process, 2) the energy build-up is exponential between flares, 3) all the free energy built-up between flares is released by the following flare and the system returns to its unperturbed or ground state via flaring. In the case where the built-up energy exceeds the ground-state energy, these assumptions lead to a power-law frequency distribution of released energies. In the simplest form of this model, some relationship is expected between the energy released in a flare and the time elapsed since the previous flare produced in the same flaring volume. This has been tested with WATCH observations with the result that no such relationship exists.

The more recent type of model to account for frequency distributions is based on the concept of self-organized criticality (SOC) also known as 'avalanche concept' introduced by Bak et al. (1988, 1989) and Bak & Chen (1991). This concept characterizes the behaviour of dissipative systems containing a large number of elements interacting over a short range, that evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. Frequency distributions of the output (size) parameters from the chain reaction taken over a period of time can be represented by power-laws. Lu & Hamilton (1991) proposed a model based on SOC, in which a solar flare is considered as avalanches of many small reconnection events. The frequency distributions of peak flux and total energy released are found to be well-fitted by power-laws with respective slopes -1.8 and -1.4 (Lu & Hamilton 1991). More recently Lu et al. (1993) further developed their model and found that a power-law with an exponential roll-over is a better representation for the frequency distributions. They have compared the predictions of the 'avalanche' models with the ISEE-3/ICE observations and have found a general good agreement between the predicted distributions of flare parameters and the observed ones. Their simulations also predict that there is no correlation between the size of an avalanche and the time interval since the previous one (Lu et al.,1993). This is confirmed in this paper with WATCH observations. All these observations are thus consistent with the 'avalanche' models of flares.

Further developments of the 'avalanche model' were done by Galsgaard (1995), Vlahos et al. (1995), and Georgoulis & Vlahos (1996) who either investigated the conditions of the system under which the energy release distributions are expected to be power-laws or the effects of the driving mechanism and of the instability and relaxation criteria on the slopes of the flare parameters occurence distributions.

Although, the energy range probed by WATCH may contain several emitting components, the statistical study performed on the WATCH database leads to results generally similar to those obtained on other databases at higher photon energies dealing with non-thermal emission. As was observed with HXRBS/SMM, there is a loose correlation between the total duration and the peak count rate of an event. The slope of the correlation line between the total duration and the peak count rate is found to be 0.52+/-0.07, in relatively good agreement with what was found with the HXRBS/SMM data (0.45+/-0.03) (Crosby et al. 1993).

The frequency distribution of the peak count rate above background can be represented by a power-law distribution for almost three orders of magnitude with a slope -1.58 +/- 0.02. The value of the slope differs slightly from the ones deduced from other experiments. This is probably due to the fact that the peak count rates are detector and energy dependent. Investigating the frequency distribution of the peak count rate for subgroups of events with different durations, we find that they are still well-represented by power-law distributions over several decades, while the slope systematically decreases with increasing duration. Such a behaviour is being currently modelled in the context of the "avalanche models" (Georgoulis et al., in preparation).

The total duration frequency distribution cannot be well represented by a single power-law. Although such an effect was already noticed in the HXRBS/SMM distributions (Crosby et al. 1993) and the ISEE-3 distributions Bromund et al. (1995), it is clearly seen in the WATCH observations, probably because of the larger detection of events with longer duration. The frequency distribution can be represented by two power-laws or by a single power-law ([FORMULA] = -1.08 +/- 0.03) with an exponential roll-over ([FORMULA] =2100 +/- 100 s). The value of [FORMULA] is found to be close to what Lu et al. (1993) found comparing numerical simulations of avalanches and ISEE-3 observations of X-ray emission above 20 keV. This indicates a maximum duration of flares above which the single power-law behavior of the occurence distribution breaks down. This may indicate a limit in duration of a flare above which the dynamic evolution of the system is no longer governed by a self-organized behaviour. As suggested by Lu et al. (1993) the exponential roll-over value may also be related to the size of the flaring volume. It may suggest that small flares need smaller flaring volumes while the largest and longest flares originate from larger flaring regions (up to the size of an active region). If the events are divided into sub-groups as function of peak count rate, the power-law is flatter for the sub-groups with the largest peak count rates. This behaviour is currently being investigated by Georgoulis et al. (in preparation).

Even though the peak count rate around 10 keV is not an absolute measurement of the amount of energy released in a flare, it does offer some idea of the magnitude of the flare. Therefore the fact that there is no correlation between the strength of a X-ray flare and the elapsed time since the previous X-ray flare in the same active region questions the energy storage model of Rosner & Vaiana (1978) which assumes that energy is stored exponentially and is entirely released during the following flare. The observation that there is no strong link between the strengths of successive X-ray flares (associated with the same active region) is more consistent with the "statistical flare" models.

5.3. Conclusion and perspectives

The deka-keV energy range is one of the least studied energy ranges for solar purposes. The work presented here however indicates that non-thermal electrons are observed as low as 10 keV. Therefore it is of particular interest to study this energy range ([FORMULA] 10 keV) in future, especially with higher spectral resolution. It should indeed provide clues to better estimate the energy released in the corona by the weak and numerous bursts which seems to be frequently observed in this photon range.

Furthermore the statistical results presented here show that the 'avalanche models' may provide a good context to understand the frequency distributions of solar flare parameters. However some new observational results in this paper such as the deviations from single power-law distributions and the variation of slopes in subgroups of events must now also be explained in this context. No correlation is found between the elapsed time interval between successive flares arising from the same active region and the peak intensity of the flare. This raises questions on the energy storage model of Rosner & Vaiana (1978) and provides some support to the statistical flare models.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998