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-beta 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) proportional to l(-a), and the fractal Haussdor dimension D between length scales l and are areas, A(l) proportional to 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) proportional to E-alpha with a power-law coefficient of alpha = 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 approximate to 1-4 MK. We detected nano are events in extreme-ultraviolet (EUV) with the 171 and 195 Angstrom 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 powerlaw slopes of alpha approximate to 1.86 +/- 0.07 at 171 Angstrom (T-e approximate to 0.7-1.1 MK), alpha approximate to 1.8 +/- 0.10 at 195 Angstrom (T-e approximate to 1.0-1.5 MK), and alpha approximate to 1.57 +/- 0.15 in the AlMg filter (T-e approximate to 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 alpha approximate to 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 nano flares detected in EUV bands produced a too steep power-law distribution of energies with slopes of alpha approximate to 2.0-2.3 mainly because of this temperature bias. The temperature-synthesized distributions of thermal nano are energies are also found to be more consistent with distributions of nonthermal are energies determined in hard X-rays (alpha approximate to 1.4-1.6) and with theoretical avalanche models (alpha approximate to 1.4-1.5).