Electric current density calculation and error analysis of solar active regions

This investigation centers upon calculating the vertical electric current densities from the two transverse components of vector magnetograms. The electric current density calculated by using the differential form of Ampere's law, jz=1μ0(∇×B)z, on the basis of spatially sampled transverse magnetic fields varies widely over an active region. In order to investigate how errors in the vector magnetograph measurements produce errors in the vertical electric current densities, we have numerically simulated the effects of random noise on a standard photospheric magnetic configuration produced by electric currents satisfying the force-free field conditions. We have also compared the results of the densities calculated by three different methods, and analyzed the reason inducing the differences in the results. The three methods are (1) to calculate the density using the differential form of Ampere's law, jz=1μ0(∇×B)z; (2) to calculate the density using the integral form of Ampere's law along the square with edges at (x-Δx, y-Δy), (x+Δx, y-Δy), (x+Δx, y+Δy), (x-Δx, y+Δy) (hereafter small integral path), jz=(1μ0∮C1Bt·dl)/dS; (3) to calculate the density by using the integral form of Ampere's law along the square with edges at (x-2Δx, y-2Δy), (x+2Δx, y-2Δy), (x+2Δx, y+2Δy), (x-2Δx, y+2Δy) (hereafter big integral path), jz=(1μ0∮C2Bt·dl)/dS. It is found that the uncertainty of the measured electric current density increases along with the decreasing of the spatial pixel size ds. When the spatial pixel size of vector magnetograms is small enough, properly expanding the integral path can help us decrease the effect of random noise and demodulate the “true” electric current density from the measured transverse magnetograms. Close attention should be paid to the fact that some true fine-structure of the electric current density will be lost when the integral path exceeds the linear size of the smallest spatially resolved structure in the studied magnetogram.