The Poissonian origin of power laws in solar flare waiting time distributions
Markus, J. Aschwanden Submitted: 2021-06-11 12:51
In this study we aim for a deeper understanding of the power law slope, α, of waiting time dis-
8 tributions. Statistically independent events with linear behavior can be characterized by binomial,
9 Gaussian, exponential, or Poissonian size distribution functions. In contrast, physical processes with
10 nonlinear behavior exhibit spatio-temporal coherence (or memory) and “fat tails” in their size distribu-
11 tions that fit power law-like functions, as a consequence of the time variability of the mean event rate,
12 as demonstrated by means of Bayesian block decomposition in the work of Wheatland et al. (1998).
13 In this study we conduct numerical simulations of waiting time distributions N(τ) in a large pa-
14 rameter space for various (polynomial, sinusoidal, Gaussian) event rate functions λ(t), parameterized
15 with an exponent p that expresses the degree of the polynomial function λ(t) ∝ tp. We derive an
16 analytical exact solution of the waiting time distribution function in terms of the incomplete gamma
17 function, which is similar to a Pareto type-II function and has a power law slope of α = 2 + 1/p, in
18 the asymptotic limit of large waiting times. Numerically simulated random distributions reproduce
19 this theoretical prediction accurately. Numerical simulations in the nonlinear regime (p ≥ 2) predict
20 power law slopes in the range of 2.0 ≤ α ≤ 2.5. The self-organized criticality model yields a prediction
21 of α = 2. Observations of solar flares and coronal mass ejections (over at least a half solar cycle) are
22 found in the range of αobs ≈ 2.1 - 2.4. Deviations from strict power law functions are expected due
23 to the variability of the flare event rate λ(t), and deviations from theoretically predicted slope values
24 α occur due to the Poissonian weighting bias of power law fits.
Authors: Aschwanden,M.J., Johnson,J.R., and Nurhan,Y.
Publication Status: (2021), submitted
Last Modified: 2021-06-12 00:56