Multi-Wavelength Observations of the Spatio-Temporal Evolution of Solar Flares with AIA/SDO: II. Hydrodynamic Scaling Laws and Thermal Energies

In this study we measure physical parameters of the same set of 155 M and X-class solar flares observed with AIA/SDO as analyzed in Paper I, by performing a {\sl differential emission measure (DEM)} analysis to determine the flare peak emission measure $EM_p$, peak temperature $T_p$, electron density $n_p$, and thermal energy $E_{th}$, in addition to the spatial scales $L$, areas $A$, and volumes $V$ measured in Paper I. The parameter ranges for M and X-class flares are: $\log(EM_p)=47.0-50.5$, $T_p=5.0-17.8$ MK, $n_p=4 \times 10^9-9 \times 10^{11}$ cm$^{-3}$, and thermal energies of $E_{th}=1.6 \times 10^{28}-1.1 \times 10^{32}$ erg. We find that these parameters obey the Rosner-Tucker-Vaiana (RTV) scaling law $T_p^2 \propto n_p L$ and $H \propto T^{7/2} L^{-2}$ during the peak time $t_p$ of the flare density $n_p$, when energy balance between the heating rate $H$ and the conductive and radiative loss rates is achieved for a short instant, and thus enables the applicability of the RTV scaling law. The application of the RTV scaling law predicts powerlaw distributions for all physical parameters, which we demonstrate with numerical Monte-Carlo simulations as well as with analytical calculations. A consequence of the RTV law is also that we can retrieve the size distribution of heating rates, for which we find $N(H) \propto H^{-1.8}$, which is consistent with the magnetic flux distribution $N(Φ) \propto Φ^{-1.85}$ observed by Parnell et al.(2009) and the heating flux scaling law $F_H \propto H L \propto B/L$ of Schrijver et al.(2004). The fractal-diffusive self-organized criticality model in conjunction with the RTV scaling law reproduces the observed powerlaw distributions and their slopes for all geometrical and physical parameters and can be used to predict the size distributions for other flare datasets, instruments, and detection algorithms.

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