Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

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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 n e , electron temperature T e , 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 ) ∝ l D . 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 T e ≈ 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 ( T e ≈ 0.7-1.1 MK), α ≈ 1.81 ± 0.10 at 195 Å ( T e ≈ 1.0-1.5 MK), and α ≈ 1.57 ± 0.15 in the AlMg filter ( T e ≈ 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).