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The Energy-Duration Relationship in Astrophysical Self-Organized Criticality Systems

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.