The Energy-Duration Relationship in Astrophysical Self-Organized Criticality Systems
Markus J. Aschwanden, Alexandre Araujo
TL;DR
This work tests the standard fractal-diffusive self-organized criticality (FD-SOC) framework against a broad set of astrophysical datasets linking avalanche duration $T$ to energy $E$. It derives the FD-SOC prediction $T \propto E^{k}$ with $k=2/D_V=0.8$ for $d=3$ and $D_V=2.5$, and finds observational support for this exponent in large-duration-range data, while acknowledging deviations in smaller ranges due to truncation biases. The study introduces an empirical truncation-bias model $k(q_T)$ that explains the spread in published exponents across solar flares, superflares, and gamma-ray bursts, arguing that the dispersion is largely observational rather than physical. Consequently, no alternative SOC mechanisms are required to reconcile these results, and the findings provide a robust interpretation of energy-duration scaling in astrophysical SOC systems, with implications for how SOC data are sampled and analyzed.
Abstract
Scaling laws in astrophysical systems that involve the energy, the geometry, and the spatio-temporal 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 $L \propto T^{(1/2)}$, and (iv) the proportionality of the dissipated energy to the fractal volume $E \propto V$. Based on these assumptions, the FD-SOC model predicts a scaling law of $T \propto E^k \propto E^{(4/5)} = E^{0.8}$. On the observational side, we find empirical scaling laws of $T \propto E^{0.81\pm0.03}$ by Peng et al.~(2023) and $T \propto E^{0.86\pm0.03}$ by Araujo \& Valio (2021) that are self-consistent with the theoretical prediction of the FD-SOC model. However, cases with a small time range $q_T = \log{(T_{max}/T_{min})} \lapprox 2$ have large statistical uncertainties and systematic errors, which produces smaller scaling law exponents ($k \approx 0.3, ..., 0.6$) as a consequence. The close correlation of the scaling exponent $k$ with the truncation bias $q_T$ implies that the dispersion of k-values is an observational effect, rather than a physical property.