Reconciling Power Law Slopes in Solar Flare and Nanoflare Size Distributions

Abstract: We unify the power laws of size distributions of solar flare and nanoflare energies. We present three models that predict the power law slopes $\alpha_E$ of flare energies defined in terms of the 2-D and 3-D fractal dimensions ($D_A, D_V$): (i) The spatio-temporal standard SOC model, defined by the power law slope $\alpha_{E1}=1+2/(D_V+2)=(13/9)\approx 1.44$; (ii) the 2-D thermal energy model, $\alpha_{E2}=1+2/D_A=(7/3)\approx 2.33$, and (iii) the 3-D thermal energy model, $\alpha_{E3}=1+2/D_V=(9/5)\approx 1.80$. The theoretical predictions of energies are consistent with the observational values of these three groups, i.e., $\alpha_{E1}=1.47 \pm 0.07$; $\alpha_{E2}=2.38 \pm 0.09$, and $\alpha_{E3}=1.80 \pm 0.18$. These results corroborate that the energy of nanoflares does not diverge at small energies, since $(\alpha_{E1}<2$) and $(\alpha_{E3}<2)$, except for the unphyiscal 2-D model $(\alpha_{E2}>2)$. This conclusion adds an additional argument against the scenario of coronal heating by nanoflares.