Magnetic field structures of galaxies derived from analysis of Faraday rotation measures, and perspectives for the SKA

Context.The forthcoming new-generation radio telescope SKA (Square Kilometre Array) and its precursors will provide a rapidly growing number of polarized radio sources. Aims.Our analysis looks at what can be learned from these sources concerning the structure and evolution of magnetic fields of external galaxies. Methods.Recognition of magnetic structures is possible from Faraday rotation measures (RM) towards background sources behind galaxies or a continuous RM map obtained from the diffuse polarized emission from the galaxy itself. We constructed models for the ionized gas and magnetic field patterns of different azimuthal symmetries (axisymmetric, bisymmetric and quadrisymmetric spirals, and superpositions) plus a halo magnetic field. RM fluctuations with a Kolmogorov spectrum due to turbulent fields and/or fluctuations in ionized gas density are superimposed. Assuming extrapolated number density counts of polarized sources, we generated a sample of RM values within the solid angle of the galaxy. Applying various templates, we derived the minimum number of background sources and the minimum quality of the observations. For a large number of sources, reconstruction of the field structure without precognition becomes possible. Results.Any large-scale regular component of the magnetic field can be clearly recognized from RM data with the help of the $\chi^2$ criterium. Under favorite conditions, a few dozen polarized sources are enough for a reliable result. A halo field with a vertical component does not affect the results of recognition. The required source number increases for small inclinations of the galaxy's disk and for larger RM turbulence. A flat number density distribution of the sources can be overcome by more sensitive observations. Application of the recognition method to the available RM data in the region around M 31 indicates that there are significant RM contributions intrinsic to the background sources or due to the foreground of the Milky Way. A reliable reconstruction of the field structure needs at least 20 RM values on a cut along the projected minor axis. Conclusions.Recognition or reconstruction of regular field structures from the RM data of polarized background sources is a powerful tool for future radio telescopes. Measuring RM at frequencies around 1 GHz with the SKA, simple field structures can be recognized in galaxies up to about 100 Mpc distance and will allow to test dynamo against primordial or other models of field origin. The low-frequency SKA array and low-frequency precursor telescopes like LOFAR may also have good RM sensitivity if background sources are still significantly polarized at low frequencies