Global Energetics of Solar Flares. XII. Energy Scaling Laws

In this study we test 18 versions of five fundamental energy scaling laws that operate in large 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 magnetic length scale L, the thermal length scale L _th, the mean helically twisted flux tube radius R, the sunspot radius r, the emission-measure-weighted flare temperature T _e, the electron density n _e, and the total emission measure EM, measured from a data set of 173 GOES M- and X-class flare events. The five 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 criteria: a cross-correlation coefficient of CCC > 0.5 between the theoretically predicted scaling laws and observed values, and a linear regression fit with a slope of α ≈ 1 within one standard deviation σ. These two criteria enable us (i) to corroborate some existing (or modified) scaling laws, (ii) to reject other scaling laws that are not consistent with the observations, (iii) to probe the dimensionality of flare geometries, and (iv) to predict various energy parameters based on tested scaling laws.