Global Energetics of Solar Flares. XII. Energy Scaling Laws

In this study we test 30 variants of 5 physical scaling laws that describe different aspects of solar flares. We express scaling laws in terms of the magnetic potential field energy $E_p$, the mean potential field strength $B_p$, the free energy $E_{free}$, the dissipated magnetic flare energy $E_{diss}$, the mean loop length scale $L$, the mean helically twisted flux tube radius $R$, the sunspot radius $r$, the emission measure-weighted flare temperature $T_w$, the electron density $n_e$, and the total emission measure $EM$, measured from a data set of $\lapprox 400$ GOES M- and X-class flare events. The 5 categories of physical scaling laws include (i) a scaling law of the potential-field energy, (ii) a scaling law for helical twisting, (iii) a scaling law for Petschek-type magnetic reconnection, (iv) the Rosner-Tucker-Vaiana scaling law, and (v) the Shibata-Yokoyama scaling law. We test the self-consistency of these theoretical scaling laws with observed parameters by requiring two conditions: a cross-corrleation coefficient of CCC$>$0.5 between the observed and theoretically predicted scaling laws, and a linear regression fit with a slope of $\alpha \approx 1$. With these two criteria we find that 10 out of the 30 tested scaling law variants are consistent with the observed data, which strongly corroborates the existence and validity of the tested flare scaling laws.