Solar Flare Geometries. II. The Volume Fractal Dimension

• Published in: The Astrophysical journal 2008-02, Vol.674 (1), p.544-553
• Main Authors: Aschwanden, Markus J, Aschwanden, Pascal D
• Format: Article
• Language: English
• Subjects: Astronomy; Earth, ocean, space; Exact sciences and technology
• ISSN: 0004-637X; 1538-4357
• DOI: 10.1086/524370

Based on the area fractal dimension D sub(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 super(D3) and the area fractal scaling 4(L) proportional to L super(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( [unk]) and volumetric filling factor qA( [unk]) as a function of the number of loops [unk], and allows one to predict the volume filling factor qv(qA) and volume fractal dimension D sub(3)(D sub(2)) from the observationally measured parameters qA and D sub(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 qvt- approximately 0.03 -0.08 (assuming solid filling for loop widths of [unk]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.