Thresholded Power Law Size Distributions of Instabilities in Astrophysics

TL;DR

This paper introduces a generalized thresholded power law distribution model to better fit astrophysical instability size data, accounting for natural effects like thresholds, background noise, and system size limits, and applies it to various astrophysical datasets.

Contribution

The paper develops and analytically derives a thresholded power law distribution model that accounts for natural deviations from ideal power laws in astrophysical data.

Findings

The model fits most observed astrophysical data well.

It enables diagnostics of background contamination and detection thresholds.

It suggests improvements for self-organized criticality models.

Abstract

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 $x_0$; (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 simplest terms with a "thresholded power law" distribution function (also called generalized Pareto [type II] or Lomax distribution), $N(x) dx \propto (x+x_0)^{-a} dx$, where $x_0 > 0$ is positive for a threshold effect, while $x_0 < 0$ 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 $x_0$. 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 inastability thresholds, instrumental detection thresholds, and finite system size limits. When testing self-organized criticality models, which predict ideal power laws, we suggest to include these natural truncation effects.