Distribution functions for magnetic fields on the quiet Sun

The statistical properties of the highly structured magnetic field of the quiet Sun are best described in terms of distribution functions, in particular the probability density functions (PDF) for the flux densities and the angular distribution for the orientations of the field vector. They are needed to test the validity of various MHD simulations, but past determinations have led to contradictory results. A main reason for these difficulties lies in the circumstance that the magnetic structuring continues on scales that are much smaller than the telescope resolution, and that this structuring strongly affects the quantities averaged over each pixel due to the non-linear relation between polarization and magnetic field. Here we use a Hinode SOT/SP data set for the disk center of the quiet Sun to explore the complex behavior of the polarized 6301–6302 Å line system and identify the observables that allow the most robust determinations of inclination angles and flux densities. These observables are then used to derive the empirical distribution functions. Our Stokes V line ratio analysis leads us to an unexpected discovery: a magnetic dichotomy with two distinct populations, representing strong (kG) and weak fields. This can be understood in terms of the convective collapse mechanism, which makes the Sun's magnetic flux end up in two states: collapsed and uncollapsed. With the linear-to-circular polarization ratio as a robust observable for the inclination angles, we find that the angular distribution is extremely peaked around the vertical direction for the largest flux densities, but gradually broadens as we go to smaller flux densities, to become asymptotically isotropic at zero flux density. The PDF for the vertical flux density, after accounting for the smearing effect of measurement noise, is found to have an extremely narrow core peak centered at zero flux density, which can be analytically represented by a stretched exponential. The PDF wings are extended and decline quadratically. The PDFs for the horizontal and total flux densities have a similar behavior. In particular we demonstrate that earlier claims that the PDF for the total flux density increases from small values at zero flux density to have a maximum significantly shifted from zero is an artefact of measurement noise