2015ApJ...814...19A

Power-law-like size distributions are ubiquitous in astrophysical instabilities. There are at least four natural effects that cause deviations from ideal power law size distributions, which we model here in a generalized way: (1) a physical threshold of an instability; (2) incomplete sampling of the smallest events below a threshold x₀; (3) contamination by an event-unrelated background x_b; and (4) truncation effects at the largest events due to a finite system size. These effects can be modeled in the simplest terms with a ''thresholded power law'' distribution function (also called generalized Pareto [type II] or Lomax distribution), where x₀> 0 is positive for a threshold effect, while x₀< 0 is negative for background contamination. We analytically derive the functional shape of this thresholded power law distribution function from an exponential growth evolution model, which produces avalanches only when a disturbance exceeds a critical threshold x₀. We apply the thresholded power law distribution function to terrestrial, solar (HXRBS, BATSE, RHESSI), and stellar flare (Kepler) data sets. We find that the thresholded power law model provides an adequate fit to most of the observed data. Major advantages of this model are the automated choice of the power law fitting range, diagnostics of background contamination, physical instability thresholds, instrumental detection thresholds, and finite system size limits. When testing self-organized criticality models that predict ideal power laws, we suggest including these natural truncation effects.