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 ($Δt \approx 10^{-3}- 10^3$ hrs), can be reconciled with a single distribution function, $N(Δt) \propto λ_0 (1 + λ_0 Δt)^{-2}$, which has a powerlaw slope of $p \approx 2.0$ at large waiting times ($Δt \approx 1-1000$ hrs) and flattens out at short waiting times $Δt \lapprox Δt_0 = 1/λ_0$. We find a consistent breakpoint at $Δt_0 = 1/λ_0 = 0.80\pm0.14$ 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, $Δt_0 \propto 1/n_{det}$. This waiting time distribution can be modeled with a nonstationary Poisson process with a flare rate $λ=1/Δt$ that varies as $f(λ) \propto λ^{-1} \exp{-(λ/λ_0)}$. 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.
In isolated solar active regions, we find that the waiting times between flares correlate with flare magnitudes as determined by the GOES soft X-ray fluxes. A "build-up and release" scenario (BUR) 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 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 repres…
We use Renewal Theory for the estimation and interpretation of the flare rate from the Geostationary Operational Environmental Satellite (GOES) soft X-ray flare catalogue. It is found, that in addition to the flare rate variability with the solar cycles, a much faster variation occurs. The fast variation on time scales of days and hours down to minute scale appears to be comparable with time intervals between two successive flares (waiting times). The detected fast non-stationarity of the flaring rate is discussed in the framework of the previously published stochastic models of the waiting ti…
The original concept of self-organized criticality (Bak et al.~1987), applied to solar flare statistics (Lu and Hamilton 1991), assumed a slow-driven and stationary flaring rate, which warrants time scale separation (between flare durations and inter-flare waiting times), it reproduces power-law distributions for flare peak fluxes and durations, but predicts an exponential waiting time distribution. In contrast to these classical assumptions we observe: (i) multiple energy dissipation episodes during most flares, (ii) violation of the principle of time scale separation, (iii) a fast-driven and…
Waiting time distributions of solar flares and {\sl coronal mass ejections (CMEs)} exhibit power law-like distribution functions with slopes in the range of $α_τ \approx 1.4-3.2$, as observed in annual data sets during 4 solar cycles (1974-2012). We find a close correlation between the waiting time power law slope $α_τ$ and the {\sl sunspot number (SN)}, i.e., $α_τ$ = 1.38 + 0.01 $\times$ SN. The waiting time distribution can be fitted with a Pareto-type function of the form $N(τ) = N_0$ $(τ_0 + τ)^{-α_τ}$, where the offset $τ_0$ depends on the instrumental sensitivity, the detection threshold…
Flare waiting time as a novel proxy of stellar magnetic activity
National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文学与空间科学学院）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学前沿天文与天体物理研究所）；New Cornerstone Science Laboratory, National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台新基石科学实验室）
In this study we aim for a deeper understanding of the power law slope, $α$, of waiting time distributions. Statistically independent events with linear behavior can be characterized by binomial, Gaussian, exponential, or Poissonian size distribution functions. In contrast, physical processes with nonlinear behavior exhibit spatio-temporal coherence (or memory) and "fat tails" in their size distributions that fit power law-like functions, as a consequence of the time variability of the mean event rate, as demonstrated by means of Bayesian block decomposition in the work of Wheatland et al.~(19…
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 t…
The distributions of solar flare energies and waiting times have not been described simultaneously by a single physical model, yet. In this research, we investigate whether recent avalanche models can describe the distributions for both the released energies and waiting times of flares in an active region. Flaring events are simulated using the modified Lu and Hamilton model (Lu and Hamilton (1991), ApJ, 380, 89) and also the optimized model (Farhang et al. (2018), ApJ, 859, 41). Applying a probability balance equation approach, we study the statistics of the simulated flaring events and inves…
A Convolutional Neural Network-Derived Catalog of Solar Flares from Soft X-Ray Observations
Sydney Institute for Astronomy, University of Sydney（悉尼大学天体物理研究所）；School of Physics, University of Melbourne（墨尔本大学物理学院）；OzGrav-Melbourne, Australian Research Council Centre of Excellence for Gravitational Wave Discovery（澳大利亚研究委员会引力波发现卓越中心）
We show examples of excitation of coronal waves by flare-related abrupt eruptions of magnetic rope structures. The waves presumably rapidly steepened into shocks and freely propagated afterwards like decelerating blast waves that showed up as Moreton waves and EUV waves. We propose a simple quantitative description for such shock waves to reconcile their observed propagation with drift rates of metric type II bursts and kinematics of leading edges of coronal mass ejections (CMEs). Taking account of different plasma density falloffs for propagation of a wave up and along the solar surface, we d…
We explore the tail of various waiting time datasets of processes that follow a nonstationary Poisson distribution with a sinusoidal driver. Analytically, we find that the distribution of large waiting times of such processes can be described using a power law slope of -2.5. We show that this result applies more broadly to any nonstationary Poisson process driven periodically. Examples of such processes include solar flares, coronal mass ejections, geomagnetic storms, and substorms. We also discuss how the power law specifically relates to the behavior of driver near its minima.
Waiting time distributions allow us to distinguish at least three different types of dynamical systems, such as (i) linear random processes (with no memory); (ii) nonlinear, avalanche-type, nonstationary Poisson processes (with memory during the exponential growth of the avalanche rise time); and (iii) chaotic systems in the state of a nonlinear limit cycle (with memory during the oscillatory phase). We describe the temporal evolution of the flare rate $λ(t) \propto t^p$ with a polynomial function, which allows us to distinguish linear ($p \approx 1$) from nonlinear ($p \gapprox 2$) events. Th…
We present a study of the waiting time distributions (WTDs) of solar energetic particle (SEP) events observed with the spacecraft $WIND$ and $GOES$. Both the WTDs of solar electron events (SEEs) and solar proton events (SPEs) display a power-law tail $\sim Δt^{-γ}$. The SEEs display a broken power-law WTD. The power-law index is $γ_{1} =$ 0.99 for the short waiting times ($<$70 hours) and $γ_{2} =$ 1.92 for large waiting times ($>$100 hours). The break of the WTD of SEEs is probably due to the modulation of the corotating interaction regions (CIRs). The power-law index $γ\sim$ 1.82 is derived …
We have characterized the typical temporal behaviour of the bright X-ray flares detected from the three Supergiant Fast X-ray Transients showing the most extreme transient behaviour (XTEJ1739-302, IGRJ17544-2619, SAXJ1818.6-1703). We focus here on the cumulative distributions of the waiting-time (time interval between two consecutive X-ray flares), and the duration of the hard X-ray activity (duration of the brightest phase of an SFXT outburst), as observed by INTEGRAL/IBIS in the energy band 17-50 keV. Adopting the cumulative distribution of waiting-times, it is possible to identify the typic…
We analyzed Kepler short-cadence M dwarf observations. Spectra from the ARC 3.5m telescope identify magnetically active (H$α$ in emission) stars. The active stars are of mid-M spectral type, have numerous flares, and well-defined rotational modulation due to starspots. The inactive stars are of early-M type, exhibit less starspot signature, and have fewer flares. A Kepler to U-band energy scaling allows comparison of the Kepler flare frequency distributions with previous ground-based data. M dwarfs span a large range of flare frequency and energy, blurring the distinction between active and in…
Solar active regions (ARs) that produce major flares typically exhibit strong plasma shear flows around photospheric magnetic polarity inversion lines (MPILs). It is therefore important to quantitatively measure such photospheric shear flows in ARs for a better understanding of their relation to flare occurrence. Photospheric flow fields were determined by applying the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) method to a large data set of 2,548 co-aligned pairs of AR vector magnetograms with 12-min separation over the period 2012-2016. From each AR flow-field ma…
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 productivi…
V2487 Oph is a recurrent nova with detected eruptions in 1900 and 1998. Startlingly, V2487 Oph shows flares, called `Superflares', with up to 1.10 mag amplitude, fast rises of under one-minute, always with an initial impulsive spike followed by a roughly-exponential tail, typically one-hour durations, and with random event times averaging once-per-day. The typical flare energy $E$ is over 10$^{38}$ ergs, while the yearly energy budget is 10$^{41}$ ergs. V2487 Oph Superflares obey three relations; the number distribution of flare energies scales as $E^{-2.34\pm0.35}$, the waiting time from one …
Active region 11029 was a small, highly flare-productive solar active region observed at a time of extremely low solar activity. The region produced only small flares: the largest of the $>70$ Geostationary Observational Environmental Satellite (GOES) events for the region has a peak 1--$8Å$ flux of $2.2\times 10^{-6} {\rm W} {\rm m}^{-2}$ (GOES C2.2). The background-subtracted GOES peak-flux distribution suggests departure from power-law behavior above $10^{-6} {\rm W} {\rm m}^{-2}$, and a Bayesian model comparison strongly favors a power-law plus rollover model for the distribution over a si…
The coronal mass ejection productivity of solar active region 13664/8
Sun Yat-sen University（中山大学）；CAS center for Excellence in Comparative Planetology（中国科学院比较行星学卓越中心）；University of Science and Technology of China（中国科学技术大学）；Institute of Deep Space Sciences, Deep Space Exploration Laboratory（深空科学研究所，深空探测实验室）
A Monte-Carlo approach to solving a stochastic jump transition model for active-region energy (Wheatland and Glukhov, Astrophys. J. 494, 1998; Wheatland, Astrophys. J. 679, 2008) is described. The new method numerically solves the stochastic differential equation describing the model, rather than the equivalent master equation. This has the advantages of allowing more efficient numerical solution, the modelling of time-dependent situations, and investigation of details of event statistics. The Monte-Carlo approach is illustrated by application to a Gaussian test case, and to the class of flare…
Solar flares stem from the reconnection of twisted magnetic field lines in the solar photosphere. The energy and waiting time distributions of these events follow complex patterns that have been carefully considered in the past and that bear some resemblance with earthquakes and stockmarkets. Here we explore in detail the tangling motion of interacting flux tubes anchored in the plasma and the energy ejections resulting when they recombine. The mechanism for energy accumulation and release in the flow is reminiscent of self-organized criticality. From this model we suggest the origin for two i…
Fast radio bursts (FRBs) are bright milliseconds radio transients with large dispersion measures. Recently, FRB 200428 was detected in temporal coincidence with a hard X-ray flare from the Galactic magnetar SGR 1935+2154, which supports that at least some FRBs are from magnetar activity. Interestingly, a portion of X-ray flares from magnetar XTE J1810-197 and the Sun are also accompanied by radio bursts. Many features of Galactic FRB 200428 and cosmological FRBs resemble solar radio bursts. However, a common physical origin among FRBs, magnetar radio pulses and solar radio bursts has not yet b…
Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pu…
