PERCOLATION THEORY AND THE GEOMETRY OF PHOTOSPHERIC MAGNETIC-FLUX CONCENTRATIONS

The magnetic field in solar active regions forms a highly structured pattern without an apparent length scale. We study this pattern in detail for a plage and its surroundings observed with the Swedish Solar Observatory on La Palma. The magnetogram has a resolution of about 1/3'' after image optimisation. We analysed the geometric properties of isolated patches of magnetic flux. Patches with a linear size up to 3'' appear to be statistically self-similar, with a fractal dimension of D(f) = 1.54 +/- 0.05 for the relation between area and linear size. This value agrees very well with the dimension D(f) = 1.56 which is found in percolation theory for clusters of tracers placed randomly on a lattice with a tracer density below a critical threshold. The distribution of observed cluster areas also agrees with that of clusters on such a random lattice. The correspondence between properties of observations and of clusters on randomly filled lattices suggests that - well after emergence - the magnetic flux on the Sun is randomly distributed at least up to sizes of about 3'' and possibly larger.