The Origin of the Solar Flare Waiting-Time Distribution

It was recently pointed out that the distribution of times between solar flares (the flare waiting-time distribution) follows a power law, for long waiting times. Based on 25 years of soft X-ray flares observed by Geostationary Operational Environmental Satellite (GOES) instruments it is shown that 1. the waiting-time distribution of flares is consistent with a time-dependent Poisson process, and 2. the fraction of time the Sun spends with different flaring rates approximately follows an exponential distribution. The second result is a new phenomenological law for flares. It is shown analytically how the observed power-law behavior of the waiting times originates in the exponential distribution of flaring rates. These results are argued to be consistent with a non-stationary avalanche model for flares.Comment: 7 pages, 3 figures, accepted by ApJ Letter

Time-dependent Stochastic Modeling of Solar Active Region Energy

A time-dependent model for the energy of a flaring solar active region is presented based on a stochastic jump-transition model (Wheatland and Glukhov 1998; Wheatland 2008; Wheatland 2009). The magnetic free energy of the model active region varies in time due to a prescribed (deterministic) rate of energy input and prescribed (random) flare jumps downwards in energy. The model has been shown to reproduce observed flare statistics, for specific time-independent choices for the energy input and flare transition rates. However, many solar active regions exhibit time variation in flare productivity, as exemplified by NOAA active region AR 11029 (Wheatland 2010). In this case a time-dependent model is needed. Time variation is incorporated for two cases: 1. a step change in the rates of flare jumps; and 2. a step change in the rate of energy supply to the system. Analytic arguments are presented describing the qualitative behavior of the system in the two cases. In each case the system adjusts by shifting to a new stationary state over a relaxation time which is estimated analytically. The new model retains flare-like event statistics. In each case the frequency-energy distribution is a power law for flare energies less than a time-dependent rollover set by the largest energy the system is likely to attain at a given time. For Case 1, the model exhibits a double exponential waiting-time distribution, corresponding to flaring at a constant mean rate during two intervals (before and after the step change), if the average energy of the system is large. For Case 2 the waiting-time distribution is a simple exponential, again provided the average energy of the system is large. Monte Carlo simulations of Case~1 are presented which confirm the analytic estimates. The simulation results provide a qualitative model for observed flare statistics in active region AR 11029.Comment: 25 pages, 9 figure

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.Comment: Preprint also available at http://www.lmsal.com/~aschwand/eprints/2010_wait.pd

Toward a reconnection model for solar flare statistics

A model to account for observed solar flare statistics in terms of a superposition of independent random flaring elements (assumed to be sites of magnetic reconnection in the coronal magnetic field and hence termed “separators”) is described. A separator of length is assumed to flare as a Poisson process in time, with a rate v(l) inversely proportional to the Alfvén transit time for the structure. It is shown that a relationship ξ∞lk between the mean energy of events ξ at a separator and the separator length implies a relationship E∞Tk between individual waiting times τ and energies E of events at the separator. The most plausible K=2 model is found to be compatible with simple pictures for magnetohydrodynamic energy storage prior to magnetic reconnection in a current sheet with anomalous (turbulent) resistivity. Formal inversion of the observed flare frequency-energy distribution is shown to imply a distribution P(l) ∞l-1 of the separator lengths in active regions. A simulation confirms the basic results of the model. It is also demonstrated that a model comprising time-dependent separator numbers N=N(t) can reproduce an observed power-law tail in the flare waiting-time distribution, for large waiting times

Modeling sunspot and starspot decay by turbulent erosion

Disintegration of sunspots (and starspots) by fluxtube erosion, originally proposed by Simon and Leighton, is considered. A moving boundary problem is formulated for a nonlinear diffusion equation that describes the sunspot magnetic field profile. Explicit expressions for the sunspot decay rate and lifetime by turbulent erosion are derived analytically and verified numerically. A parabolic decay law for the sunspot area is obtained. For moderate sunspot magnetic field strengths, the predicted decay rate agrees with the results obtained by Petrovay and Moreno-Insertis. The new analytical and numerical solutions significantly improve the quantitative description of sunspot and starspot decay by turbulent erosion

Time-energy correlations in solar flare occurrence

The existence of time-energy correlations in flare occurrence is still an open and much debated problem. This study addresses the question whether statistically significant correlations are present between energies of successive flares as well as energies and waiting times. We analyze the GOES catalog with a statistical approach based on the comparison of the real catalog with a reshuffled one where energies are decorrelated. This analysis reduces the effect of background activity and is able to reveal the role of obscuration. We show the existence of non-trivial correlations between waiting times and energies, as well as between energies of subsequent flares. More precisely, we find that flares close in time tend to have the second event with large energy. Moreover, after large flares the flaring rate significantly increases, together with the probability of other large flares. Results suggest that correlations between energies and waiting times are a physical property and not an effect of obscuration. These findings could give important information on the mechanisms for energy storage and release in the solar corona