Solar flare geometries. II. The volume fractal dimension

Based on the area fractal dimension D-2 of solar flares measured in Paper I, we carry out modeling of the three-dimensional (3D) flare volume here and derive an analytical relation between the volume fractal scaling V(L)proportional to L-D3 and the area fractal scaling A(L) proportional to L-D2. The 3D volume model captures a flare arcade with a variable number of flare loops; its fractal structure is not isotropic, but consists of aligned one-dimensional substructures. The geometry of the arcade model has three free parameters and makes some simplifying assumptions, such as semicircular loops, east-west orientation, location near the equator, and no magnetic shear. The analytical model predicts the scaling of the area filling factor qA(n(loop)) and volumetric filling factor q(V) (n(loop)) as a function of the number of loops nloop, and allows one to predict the volume filling factor q(V) (q(A)) and volume fractal dimension D-3(D-2) from the observationally measured parameters qA and D-2. We also corroborate the analytical model with numerical simulations. We apply this fractal model to the 20 flares analyzed in Paper I and find maximum volume filling factors with a median range of q(V) approximate to 0.03-0.08 (assuming solid filling for loop widths of less than or similar to 1 Mm). The fractal nature of the flare volume has important consequences for correcting electron densities determined from flare volume emission measures and density-dependent physical quantities, such as the thermal energy or radiative cooling time. The fractal scaling has also far-reaching consequences for frequency distributions and scaling laws of solar and stellar flares.