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Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

We derive universal scaling laws for the physical parameters of flarelike processes in a low-β plasma, quantified in terms of spatial length scales l, area A, volume V, electron density ne, electron temperature Te, total emission measure M, and thermal energy E. The relations are specified as functions of two independent input parameters, the power index a of the length distribution, N(l) ∝ l-a, and the fractal Haussdorff dimension D between length scales l and flare areas, A(l) ∝ lD. For values that are consistent with the data, i.e., a = 2.5 ± 0.2 and D = 1.5 ± 0.2, and assuming the RTV scaling law, we predict an energy distribution N(E) ∝ E-α with a power-law coefficient of α = 1.54 ± 0.11. As an observational test, we perform statistics of nanoflares in a quiet-Sun region covering a comprehensive temperature range of Te ≈ 1-4 MK. We detected nanoflare events in extreme-ultraviolet (EUV) with the 171 and 195 Å filters from the Transition Region and Coronal Explorer (TRACE), as well as in soft X-rays with the AlMg filter from the Yohkoh soft X-ray telescope (SXT), in a cospatial field of view and cotemporal time interval. The obtained frequency distributions of thermal energies of nanoflares detected in each wave band separately were found to have power-law slopes of α ≈ 1.86 ± 0.07 at 171 Å (Te ≈ 0.7-1.1 MK), α ≈ 1.81 ± 0.10 at 195 Å (Te ≈ 1.0-1.5 MK), and α ≈ 1.57 ± 0.15 in the AlMg filter (Te ≈ 1.8-4.0 MK), consistent with earlier studies in each wavelength. We synthesize the temperature-biased frequency distributions from each wavelength and find a corrected power-law slope of α ≈ 1.54 ± 0.03, consistent with our theoretical prediction derived from first principles. This analysis, supported by numerical simulations, clearly demonstrates that previously determined distributions of nanoflares detected in EUV bands produced a too steep power-law distribution of energies with slopes of α ≈ 2.0-2.3 mainly because of this temperature bias. The temperature-synthesized distributions of thermal nanoflare energies are also found to be more consistent with distributions of nonthermal flare energies determined in hard X-rays (α ≈ 1.4-1.6) and with theoretical avalanche models (α ≈ 1.4-1.5).