A correlation in the waiting-time distributions of solar flares Free

ABSTRACT

In a limited sample of isolated solar active regions, we find that the waiting times between flares may correlate well with flare magnitudes as determined by the Geostationary Operational Environmental Satellites (GOES) soft X-ray fluxes. A ‘build-up and release’ (BUR) scenario for magnetic energy storage in the solar corona suggests the existence of such a relationship, relating the slowly varying subphotospheric energy sources to the sudden coronal energy releases of flares and coronal mass ejections (CMEs). Substantial amounts of research effort had not previously found any obvious observational evidence for such a BUR process. This has posed a puzzle since coronal magnetic energy storage represents the consensus view of the basic flare mechanism. We have revisited the GOES soft X-ray flare statistics for any evidence of correlations, using two isolated active regions, and have found significant evidence for a ‘saturation’ correlation. Rather than a ‘reset’ form of this relaxation, in which the time before a flare correlates with its magnitude, the ‘saturation’ relationship results in the time after the flare showing the correlation. The observed correlation competes with the ‘obscuration’ effect of reduced GOES sensitivity following a strong event, by which weaker events can be under-reported systematically. This complicates the observed correlation, and we discuss several approaches to remedy this.

1 INTRODUCTION

Solar flares appear to result from the build up of ‘free’ magnetic energy in the corona, which can grow slowly as the result of non-radiative energy fluxes (such as wave energy or Poynting flux) injected into the solar atmosphere. These excess energies appear as non-potential field structures, describable in terms of electrical currents that connect the corona to the solar interior. A flare (or a coronal mass ejection, CME) extracts energy suddenly from this stressed field, which then must relax to a lower energy state. This well-understood scenario has broad acceptance, if only because alternative mechanisms for energy storage seem so implausible observationally (e.g. Hudson 2007). This basic picture spawned a major effort in the 1980s, the ‘Flare Build-up Study’ (Gaizauskas & Švestka 1987); its first objective was to identify signatures of the sequence of slow build-up and rapid flare energy release. This did not happen: ‘...no consistent relationship was found.. ,’ providing a motivation for the present work. Nowadays we have substantially improved observational material for characterization and for follow-up. Some of the non-potential ‘free energy’ appears to remain in the corona even after a major flare (Wang et al. 1994). We note that more numerous weak flares or even steady heating can also derive from the reservoir of coronal free energy.

A system exhibiting slow build-up and rapid release (BUR) constitutes a ‘relaxation oscillator’. The driven ‘oscillation’ need not be periodic, as was noticed at the very outset (Van der Pol & Van der Mark 1927), and can exhibit chaotic behaviour (‘irregular noise’); see Ginoux & Letellier (2012) for the full history. Many natural systems exhibiting this kind of periodic behaviour exist, as do engineering applications; in astrophysics the ‘Rapid Burster’ X-ray source gave an early example (Lewin et al. 1976). In gardens with water features one can find often a mechanical ‘dipper’ with a trickle of flowing water; this operates on the same BUR principle. The dipper form of a BUR process corresponds to a ‘reset’ of the free energy as the bucket empties and the slower refilling starts again; this predicts a correlation between the time to build up the event energy and the magnitude to reset it to zero. In the case of a steady input, such as the trickle of water in the dipper, the regular resets create a regular oscillation, but other variants of this toy model do exist.

Rosner & Vaiana (1978) gave an early theoretical description of such a BUR process for solar flares. This inspired considerable literature on flare occurrence distributions, none of which appears to have succeeded in an experimental confirmation of this kind of relationship. Flare occurrence generally follows a power-law distribution of magnitudes, much as earthquakes and many other natural systems do; this suggests a self-organized critical phenomenon (see e.g. Aschwanden et al. 2016 for a recent review). The waiting times between successive flares appear to follow a ‘piecewise Poisson’ random pattern (Wheatland 2000b).

To establish the BUR process we essentially need to compare the free energy content of the corona, and we need an estimation of the energy release in each event, in terms of observable quantities. Both tasks are difficult, but we can finesse the measurement of stored energy by assuming that it changes only slowly, on time-scales greater than those of the flare release. Many proxies for flare energy exist, and we are helped by the tendency of many of the extensive parameters to scale together (one manifestation of the ‘big flare syndrome’; Kahler 1982). In previous efforts to establish a BUR correlation, Biesecker, Ryan & Fishman (1994) used hard X-ray observations from the Burst And Transient Source Experiment (BATSE; Fishman et al. 1992), Hudson et al. (1998) used the Geostationary Operational Environmental Satellites (GOES) soft X-ray observations, and both Crosby et al. (1998) and Wheatland (2000a) used data from the the Wide Angle Telescope for Cosmic Hard X-rays (WATCH) monitor of hard X-rays, as described in an early version by Lund (1981). The relationship of any of these proxies to the total flare energy certainly contains some variance, and none represent a large fraction of the total. Proxies for the total luminous energy of a flare also miss the energy of any associated CME, a major factor for a minority of flares.

None of the earlier studies have reported a significant interval–size relationship, which would observationally support a BUR process if detected. This paper again uses the standard GOES soft X-ray data base (Section 2) but specifically seeks an alternative form of the BUR correlation, as described in Section 3. The observations clearly show an effect consistent with the BUR idea, but a systematic feature of the GOES time series complicates the picture (Section 5). We suggest further specific searches to clarify this situation.

2 DATA BASE

We again use the GOES soft X-ray proxy for flare energy, plus the metadata regarding the identification of the flaring active region. This is the standard material available from SolarSoft (Freeland & Handy 1998). The GOES event classification (e.g. ‘X1.2’ standing for 1.2 × 10⁻⁴ W m⁻²) represents irradiance (flux) rather than fluence, whereas a more appropriate proxy for total flare energy might have units of energy rather than power. Shimizu (1995) suggested a soft X-ray energy fraction of order 1 per cent, for example, but this fraction has an unknown variance and also must have, based upon models, some systematic bias across the scale of flare magnitudes. However, the peak GOES irradiance correlates with the flare soft X-ray fluence, and so it also can serve as a proxy. Even the GOES fluence itself has a systematic bias resulting from its temperature weighting, since the actual data consist of broad-band samples centred at photon energies well above flare kT values determined spectroscopically.

This study covers two active regions, NOAA 07978 (1996 July) and NOAA 10930 (2006 December). Each of them exclusively made all of the GOES flares for the intervals studied, and as can be seen from the file magnetograms in Fig. 1, each was truly isolated (for noting this about NOAA 10930, we thank M. Georgoulis, personal communication). They both produced remarkably powerful flares near the very end of their sunspot cycles, and in each case the flares had singular properties that their isolation may have helped to make detectable. SOL1996-07-09 (X2.2) spawned the first observed ‘sunquake’ (Kosovichev & Zharkova 1998); SOL2006-12-13 (X3.4) followed the event producing the first detectable MeV-energy flux of neutral atoms (Mewaldt et al. 2009).

Figs 2 and 3 show the flares used for this study, comparing the actual GOES time history at full resolution with the data base classifications. The two regions, though both isolated from the point of view of GOES flare production, had different soft X-ray background levels. The higher background flux for the 2006 region doubtless resulted from the coincidence that its major flux emergence occurred at or before its east limb arrival, whereas the 1996 region was born near disc centre. The two regions thus sample somewhat different time histories on active region time-scales. These two figures reveal the approximations involved in the use of a ‘data product’ with somewhat unknown properties. White, Thomas & Schwartz (2005) give a full scientific account of the primary GOES/X-Ray Sensor (XRS) data. In this study, we rely solely upon the tabulated National Oceanic and Atmospheric Administration (NOAA) event list, a secondary data base generated in near real time, accepting its known and unknown flaws. The classifications come from an automated detection system with some human intervention, and one can readily identify many omissions and small discrepancies in the two time series shown in the figures. The data base certainly capture every major impulsive event; the GOES spacecraft in principle have 100 per cent duty cycles.

3 AN INTERVAL–SIZE RELATIONSHIP

A BUR process should manifest itself as a correlation between the waiting interval and the flare magnitude. We can distinguish two cases (Wheatland 2000a): if each flare uses up to the entire available stock of free energy, the interval before the flare should correlate with the flare energy; the original Rosner & Vaiana (1978) model predicts this relationship. Here we term this the ‘reset’ limit. Alternatively, a limited energy release from an existing reservoir could have a correlation between the time interval after the event, on the hypothesis that the same global energy threshold enables the triggering of successive events. Hudson et al. (1998) had suggested this possibility for solar flares, and Middleditch et al. (2006) actually discovered such a relationship for the seismic ‘glitches’ observed in the period variations of the pulsar PSR J0537–6910, noting though that this behaviour is not typical for a pulsar. The glitches arise from sudden adjustments of the neutron star crust as it evolves on a longer time-scale. For this object the very strong correlation allows observers actually to predict the time of the next glitch.

We construct a toy model with these alternative features, not referring to the basic mathematical description of a relaxation oscillator (the Van der Pol equation); we note that a BUR process might not follow either the ‘saturation’ or ‘reset’ prescriptions; for example Lu (1995) states that a self-organized criticality (SOC or ‘avalanche’) model would not have the ‘reset’ property (see further discussion in Section 5). In both cases we consider a steady and constant energy input. Fig. 4 illustrates the two alternatives schematically. Note that although we think of magnetic free energy as the parameter of interest in the model, and use the GOES soft X-ray flux as a proxy for it, this simplification adds variance to any result obtained. Furthermore, other physical or geometrical parameters of the flaring system may play decisive roles. We return to discuss this further in Section 5 and in the meanwhile, we just describe the toy model in Fig. 4 as a generic parameter.

Note that this ‘model’ has no obvious physical underpinning for the saturation limit, although a garden dipper is a (typically bamboo) realization for the reset limit. Fig. 4 comes from the idl program randompowl.pro with a fixed range of event magnitudes distributed randomly according to an assumed power-law occurrence slope of −1.75, as originally suggested for flare occurrence by Akabane (1956).

We have searched for ‘reset’ and ‘saturation’ correlations in the two active regions, starting with the one at the end of Cycle 23, NOAA active region (AR) 10930. The flare timing reveals a significant ‘saturation’ correlation (Fig. 5) during the first day of its flaring life, 2006 December 6, during which an X-class flare occurred. Removal of this single point from the correlation did not change the result. The ‘reset’ alternative has no significant correlation.

To confirm this possible correlation, we also revisited AR 07978, studied inconclusively by Hudson et al. (1998). Surprisingly, as shown in Fig. 6, one 12-h interval showed an extremely strong ‘saturation’ correlation in this isolated region as well. Again, the ‘reset’ alternative did not correlate, and again the removal of the single X-class flare from the correlation gave the same results.

The correlation results for these two intervals clearly support a relaxation–oscillator (BUR) behaviour in the ‘saturation’ relationship, and conflict with the negative conclusions about waiting-time correlations by previous authors (Biesecker 1994; Crosby et al. 1998; Hudson et al. 1998; Wheatland 2000b). The difference may results from the systematic uncertainties in the data bases in use or in methodology; as discussed below there are many ways to hide the presence of even a (hypothetically) persistent correlation in the two examples of Figs 5 and 6.

We also use the AR 10930 time series to test the robustness of the ‘saturation’ correlation by looking at day-by-day event listings (Fig. 7). Table 1 gives the numerical results in the form of Pearson correlation coefficients and their uncertainties (R_s for the ‘saturation’ case, and R_r for the ‘reset’ case), along with the numbers of flares on each day and the GOES maximum class. For this single region, the day-by-day analysis shows significant ‘saturation’ correlations on several individual days, as corroborated by the power-law fits. The full time series (123 GOES events from SOL2006-12-06T05:36 through SOL2006-12-17T14:47) showed no correlation, with the Pearson correlation coefficient r = 0.06 ± 0.48. Note that Hudson et al. (1998) also had found no systematic correlation for the 1996 region, AR 9798. This suggests an intermittent behaviour or noise dominance for the correlation on longer time-scales, since significant (R_s ≫ δR_s) correlations show up for each region on individual days during the disc passage. In the context of the toy model, a slowly variable driving source would also reduce the correlation in a natural way. Interestingly, for one day (2006 December 11) we see strong correlations in both senses.

The lower panel of Fig. 7 shows the slope results of linear fits to the day-by-day data from AR 10930, the parameter α in W ∝ (Δt)^α, with W the flare peak flux and Δt the waiting time after the flare. Interestingly the fit uncertainties do not look so convincing as the results for individual days in terms of the Pearson regression coefficients. Note though that none of the 11 intervals have a significant negative slope. The slope parameters hint at α = d(ln S)/d(ln Δt) ≈ 1, but a conclusion about this would require the systematic study of a much larger sample. Note that even the existence of such a relationship, much less the constancy of its slope parameter, will remain entirely hypothetical until then.

Figs 8 and 9 show time series and correlations for the individual 1-d intervals with the best correlations for AR 10930 (indicated as boldface in Table 1). Note that these mostly correspond to the 1-d intervals with the most energetic events, although one (December 9) peaked at only B1.9, and the previous day had a comparably strong negative ‘saturation’ correlation at and a maximum flare magnitude of B4.5. The fact that December 10 (maximum C1.4) showed a solid positive ‘saturation’ correlation is noteworthy, because a C-class flare should have less obscuration, and we have noted previously that individual correlations remained for intervals with X-class flares even upon removal of their individual entries in the correlation sets. These results provide intermittent strong support for the ‘saturation’ correlation. The bottom line for these views, however, must be that the analysis is close to the limit permitted by random and systematic errors in the use of catalogue GOES data; in Section 6, we suggest several possible further steps to explore this result.

4 WHY WAS THIS CORRELATION NOT FOUND EARLIER?

Many sources of uncertainty might mask an interval–size relationship for flare waiting times. The time reference used here (the GOES soft X-ray peak) has only a crude relationship to the impulsive phase, which might mark the time of the significant energy release more exactly. The GOES peak fluxes also do not scale exactly even with the soft X-ray energy, because of variations in flare durations and spectra; another known and probably quite significant scatter also has to come from the routine treatment of the GOES observations here (no background corrections, and no correction made for the ‘obscuration’ effect noted for weak events by many; e.g. Wheatland 2001; Hudson, Fletcher & McTiernan 2014). As noted previously, a long interval may also have reduced correlation in the context of the toy model, if the driving input varies. In flares with CMEs, a large fraction of the energy (e.g. Emslie et al. 2012) may simply disappear from the view of any proxy relating to the flare electromagnetic radiation, and this would produce a substantial bias. We could also ask how the heating of the quiescent active region (and Parker’s hypothetical nanoflares; see e.g. Cargill 1994) might relate to the coronal reservoir tapped for its flare energy release.

In spite of these systematic errors and unknowns, this study has found instances of a significant correlation by using only the simplest possible flare data, namely the SolarSoft summaries of GOES soft X-ray event time, magnitude, and location. At least two previous careful searches for interval–size relationships found none, with Crosby et al. (1998) stating ‘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’. That study selected sequences of events from the same active region, but perhaps did not explicitly select isolated regions and there may have been no search for the ‘saturation’ correlation that we report here, but Wheatland (2000a) used the same data base and searched unsuccessfully for both ‘reset’ and ‘saturation’ matches. The latter paper offers several possible explanations for the non-detection of the correlation, including the idea that perhaps flare energy is not stored visibly in the corona at all! It also comments interestingly that the existence of a waiting-time correlation would tend to rule out some classes of ‘avalanche’ models (cf. Lu 1995).

5 INTERPRETATION OF THE ‘SATURATION’ CORRELATION

The top right-hand panel of Fig. 4 suggests the simplest explanation of the ‘saturation’ correlation: the available free energy, or some other parameter, builds gradually up to a limit imposed by the active-region structure, at which point a random trigger dislodges a part of the system into a lower energy state. The broad distribution of flare magnitudes (generally, a power law or a lognormal observational fit) requires that this end state lie in a continuum; homologous flares sometimes occur but only rarely. If we identify the generic parameter in the toy model with magnetic free energy in the corona, it presumably consists of inductive storage (Melrose 1995, 2017; Zaitsev et al. 1998; Khodachenko et al. 2009) in a system of non-neutralized currents (see e.g. Georgoulis, Titov & Mikić 2012). Neutralized currents may not store energy so efficiently because of their smaller inductance, but this depends upon the geometry of the system. The most powerful flare release cannot diminish these currents on short time-scales, although it can reroute them and thus alter the inductive energy storage in that way (Melrose 1995; Sun et al. 2012). Instead flares appear to leave the active-region structure in a stressed state (Wang et al. 1994), consistent with the ‘saturation’ correlation found here. Thus the total magnetically stored energy sets a firm upper limit on the magnitude of a flare, but the practical (and lower) limit would come from the properties of a ‘minimum current corona’ of some kind (Longcope 1996). Either way the power-law distribution of flare energies must roll over around this maximum point, as suggested by observation (Kucera et al. 1997; Tranquille, Hurley & Hudson 2009; Wheatland 2010) and statistical analysis (e.g. Kubo 2008).

6 CONCLUSIONS

The results presented in this paper match expectations for a BUR scenario for solar flares occurring in isolated active regions, which if confirmed would establish a pattern widely believed in, but not previously established observationally. The result found suggests a ‘saturation’ ordering, with correlated time intervals after the event, rather than the ‘reset’ relaxation often discussed, as illustrated in Fig. 4. The correlation may persist for a time-scale on the order of a day, and it exhibits intermittency, some of which must result from random and systematic error over epochs with limited ranges of flare magnitude. The triggering of the events remains as a ‘piecewise Poisson’ process (Wheatland 2000b). The unique parameter dictating the ‘saturation’ level appears could be the available free magnetic energy, since our search has been in terms of a known correlate, the GOES soft X-ray peak flux.

The appearance of the ‘saturation’ correlation in a crude analysis (we have only used imprecise catalogue information) suggests several further lines of research to check and possibly extend the relationship.

• A more thorough search of the GOES statistics, based on a quantitative analysis considering uncertainties; this would require using the primary data, rather than the existing catalogue.

• Searches with hard X-rays, which have much smaller obscuration in the time domain; here Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) data could be invaluable because of its image capability (note that its low Earth orbit restricts the uninterrupted time range).

• Time interval analysis of microflare activity within a given active region, as observed with imaging instruments (e.g. Shimizu 1995).

• Comparisons with image-based analysis of Poynting flux transport in specific active regions, specifically in well-documented homologous flare sequences (e.g. Romano, Elmhamdi & Kordi 2019).

• Theoretical work on feasible model descriptions, attempting to understand how the apparent one-parameter instability limit can lead to a continuum of final states. Avalanche models (Lu & Hamilton 1991) remain interesting (e.g. Reid et al. 2018; Farhang, Wheatland & Safari 2019) even if they have not thus far anticipated the result suggested here.

More thorough studies of flare waiting times may resolve several interesting questions involving CME occurrence, interactions between regions, time-scales associated with trans-photospheric Poynting flux, etc. The results may also offer the possibility of improving short-term prediction of flare magnitudes based on the interpretation of prior flare occurrence to reflect the magnitude of the Poynting flux responsible for the energy build-up in a specific region, noting the empirical success of the ‘after’ correlation in anticipating pulsar glitches in PSR J0537–6910 (Middleditch et al. 2006; Melatos, Howitt & Fulgenzi 2018).

ACKNOWLEDGEMENTS

A remark by Manolis Georgoulis, noting the unambiguous flare time series from the isolated active region NOAA 10930, provided the impetus for this study, which follows on from preliminary work the author had done in 1997 (Hudson et al. 1998); please see http://www.ssl.berkeley.edu/hhudson/publications/1998ASSL..229..237H.pdf for that paper, which may be difficult to obtain otherwise. Author would also like to thank Doug Biesecker, Kris Cooper, Lyndsay Fletcher, Manolis Georgoulis, Brian Welsch, Mike Wheatland, and Graham Woan for helpful discussions, and also the referee for perceptive criticism.

REFERENCES