Reconciliation of Waiting Time Statistics of Solar Flares Observed in Hard X-Rays

We study the waiting time distributions of solar flares observed in hard X-rays with ISEE-3/ICE, HXRBS/SMM, WATCH/GRANAT, BATSE/CGRO, and RHESSI. Although discordant results and interpretations have been published earlier, based on relatively small ranges ($< 2$ decades) of waiting times, we find that all observed distributions, spanning over 6 decades of waiting times ($\Delta t \approx 10^{-3}- 10^3$ hrs), can be reconciled with a single distribution function, $N(\Delta t) \propto \lambda_0 (1 + \lambda_0 \Delta t)^{-2}$, which has a powerlaw slope of $p \approx 2.0$ at large waiting times ($\Delta t \approx 1-1000$ hrs) and flattens out at short waiting times $\Delta t \lapprox \Delta t_0 = 1/\lambda_0$. We find a consistent breakpoint at $\Delta t_0 = 1/\lambda_0 = 0.80\pm0.14$ hours from the WATCH, HXRBS, BATSE, and RHESSI data. The distribution of waiting times is invariant for sampling with different flux thresholds, while the mean waiting time scales reciprocically with the number of detected events, $\Delta t_0 \propto 1/n_{det}$. This waiting time distribution can be modeled with a nonstationary Poisson process with a flare rate $\lambda=1/\Delta t$ that varies as $f(\lambda) \propto \lambda^{-1} \exp{-(\lambda/\lambda_0)}$. This flare rate distribution represents a highly intermittent flaring productivity in short clusters with high flare rates, separated by quiescent intervals with very low flare rates.