The Energy–Duration Relationship in Astrophysical Self-organized Criticality Systems

Scaling laws in astrophysical systems that involve energy, geometry, and spatiotemporal evolution provide the theoretical framework for physical models of energy dissipation processes. A leading model is the standard fractal-diffusive self-organized criticality (FD-SOC) model, which is built on four fundamental assumptions: (i) the dimensionality d = 3, (ii) the fractal dimension D V = d − 1/2 = 2.5, (iii) classical diffusion L ∝ T (1/2) , and (iv) the proportionality of the dissipated energy to the fractal volume, E ∝ V . Based on these assumptions, the FD-SOC model predicts a scaling law of T ∝ E k ∝ E (4/5) = E 0.8 . On the observational side, we find only one case (out of the nine analyzed cases) that is consistent with the FD-SOC model, namely, the scaling law of T ∝ E 0.86±0.03 by A. Araujo & A. Valio. The other eight cases have scaling law coefficients clustered around values of k ≈ 0.3−0.4, which possibly can be explained by nonstandard SOC models, i.e., k = 2/( βD V ), in terms of linear expansion β = 2 and/or Euclidean dimension D V = 3. The relatively low values of time duration ranges may also indicate possible truncation biases.