Understanding solar flare waiting-time distributions

The observed distribution of waiting times At between X-ray solar flares of greater than Cl class listed in the Geostationary Operational Environmental Satellite (GOES) catalog exhibits a power-law tail similar to (Deltat)(gamma) for large waiting times (Deltat > 10 hours). It is shown that the power-law index gamma varies with the solar cycle. For the minimum phase of the cycle the index is gamma = - 1.4+/-0.1, and for the maximum phase of the cycle the index is -3.2+/-0.2. For all years 1975-2001, the index is -2.2 +/- 0.1. We present a simple theory to account for the observed waiting-time distributions in terms of a Poisson process with a time-varying rate lambda(t). A common approximation of slow variation of the rate with respect to a waiting time is examined, and found to be valid for the GOES catalog events. Subject to this approximation the observed waiting-time distribution is determined by f (lambda), the time distribution of the rate lambda. If f (lambda) has a power-law form similar tolambda(alpha) for low rates, the waiting time-distribution is predicted to have a power-law tail similar to (Deltat)(-(3+alpha)) (alpha > -3). Distributions f (lambda) are constructed from the GOES data. For the entire catalog a power-law index alpha = -0.9 +/- 0.1 is found in the time distribution of rates for low rates (lambda < 0.1 hours(-1)). For the maximum and minimum phases power-law indices α = -0.1+/-0.5 and α = -1.7 +/- 0.2, respectively, are observed. Hence, the Poisson theory together with the observed time distributions of the rate predict power-law tails in the waiting-time distributions with indices -2.2 +/- 0.1 (1975-2001), -2.9 +/- 0.5 (maximum phase) and - 1.3 +/- 0.2 (minimum phase), consistent with the observations. These results suggest that the flaring rate varies in an intrinsically different way at solar maximum by comparison with solar minimum. The implications of these results for a recent model for flare statistics (Craig, 2001) and more generally for our understanding of the flare process are discussed.