Waiting Times of Solar Hard X-Ray Flares

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Number: 123
1st Author: Markus Aschwanden
2nd Author: Jim McTiernan
Published: 15 March 2010
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Delightful traffic engineering! A special Nuggets prize offered for the first reader to identify this intersection (hint:is it in California?).

You might drive a car in a foreign city and have to stop at many red traffic lights (e.g., Figure 1). From the statistics of waiting times you probably can quickly figure out which signals operate independently and which ones operate the smart way with inductive-loop traffic detectors in a so-called demand-actuated mode. Thus, the statistics of waiting times bears crucial information how a system works, either having independent elements that act randomly, or consisting of elements with long-range connections that enable coupling and synchronization. In geophysics, aftershocks have been found to exhibit different waiting time statistics (Omori’s law; cf Omori) than the main shocks of earthquakes. In magnetospheric physics, the waiting-time statistics is used to distinguish prosaic Poisson random processes from more sophisticated kinds of variability: self-organized criticality, intermittent turbulence, finite system size effects, or clusterization. Such processes have been described for auroral emission, the auroral electron jet (AE) index, or substorms at the Earth’s magnetotail.

Waiting times for solar flares

Waiting-time statistics is studied intensely in solar physics, where most flares are found to be produced by a Poissonian random process, but there are also so-called sympathetic flares that have a causal connection or trigger each other. The waiting-time statistics of solar flares has been studied in hard X-rays, in soft X-rays, for coronal mass ejections (CMEs), for solar radio bursts, and for the solar wind. In astrophysics, waiting-time distributions have been studied for flare stars as well as for black-hole candidates such as Cygnus X-1.

Waiting-time distributions for various hard X-ray observaitons of solar flares; see reference [1] for details.

Reference [1] focused on waiting time distributions of solar flares detected in hard X-rays. The most comprehensive sampling of solar flare waiting times was gathered in soft X-rays so far, using the extensive (25-year) catalog of GOES soft X-ray flares observations, but three different interpretations have been proposed using the very same data! First, a non-stationary (time-dependent) Poisson process [2], second, a shell-model of MHD turbulence [3], or third, a L´evy flight model of self-similar processes with some memory [4]. All three interpretations can produce the observed powerlaw-like distribution of waiting times. On the other side, self-organized criticality models predict a Poissonian random process, which has an exponential distribution of waiting times for a stationary (constant) flare rate, but can produce powerlaw-like waiting time distributions with slope p less than 3 for nonstationary variations of the flare rate. Therefore, the finding of a powerlaw-like distribution of waiting times of solar flares has ambiguous interpretations. The situation in solar flare hard X-rays is very discordant, from powerlaws, near-exponential, to double-hump distribution with an overabundance of short waiting times. In [1] we have analyzed flare catalogs from the hard X-ray observations of several satellites, starting with RHESSI's, and are able to model all observed hard X-ray waiting time distributions with a unified model in terms of a nonstationary Poisson process in the limit of high intermittency. We resolve also the discrepancy between exponential and powerlaw-like waiting time distributions, in terms of selected fitting ranges. It can be shown analytically that nonstationary Poisson processes with slowly-varying flare rates produce a waiting-time distribution that has a powerlaw tail with a slope less than 3, while highly intermittently fluctuating flare rates produce a powerlaw tail with a slope of about 2. The observations from our data sets are closer to the latter case and can all be fitted with the same waiting time distribution (Fig. 2). Thus the waiting-time distribution for solar flares indicates a highly intermittent flare productivity: short clusters with high flare rates, separated by quiescent intervals with very low flare rates. A similar intermittent behavior is obseved in earthquakes and in X-ray pulses from accretion disks.


Events such as solar flares (and many other natural phenomena) may occur in great profusion. We may or may not understand the details of the physics in each case, but often there are organizing principles derivable from the statistics of the event occurrence.

Solar flares are about the best-studied phenomenon that exhibits self-organized criticality (SOC). A necessary (but not sufficient) criterion for SOC is the statistical independence of individual events, which primarily can be tested with waiting-time distributions. The new result is that the statistical distributions of waiting times observed during three solar cycles with different instruments are fully consistent with a nonstationary Poisson process; this corroborates the conclusion that solar flares are indeed a SOC phenomenon.


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

[2] The Origin of the Solar Flare Waiting-Time Distribution

[3] Power Laws in Solar Flares: Self-Organized Criticality or Turbulence?

[4] Solar Flare Waiting Time Distribution: Varying-Rate Poisson or Lévy Function?

Biographical note

Markus Aschwanden and Jim McTiernan are senior RHESSI scientists at Lockheed Martin (Palo Alto) and the Space Sciences Laboratory (UC Berkeley), respectively.

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