Different scaling and autocorrelation characteristics and their application to astronomical images are discussed: the structure function, the autocorrelation function, Fourier spectra and wavelet spectra. We recommend as the optimal mathematical tool the wavelet spectrum with a suitable choice of the analysing wavelet. We introduce the wavelet cross-correlation function which enables to study the correlation between images as a function of scale. The wavelet cross-correlation coefficient strongly depends on the scale. The classical cross-correlation coefficient can be misleading if a bright, extended central region or an extended disk exists in the galactic images. An analysis of the scaling and cross-correlation characteristics of 9 optical and radio maps of the nearby spiral galaxy NGC 6946 is presented. The wavelet spectra allow to separate structures on different scales like spiral arms and diffuse extended emission. Only the images of thermal radio emission and Halpha emission give indications of 3-dimensional Kolmogorov-type turbulence on the smallest resolved scales (160-800 pc). The cross-correlations between the images of NGC 6946 show strong similarities between the images of total radio emission, red light and mid-infrared dust emission on all scales. The best correlation is found between total radio emission and dust emission. Thermal radio continuum and Halpha emission are best correlated on a scale of about 1' \simeq 1.6 kpc, the typical width of a spiral arm. On a similar scale, the images of polarised radio and Halpha emission are anticorrelated, which remains undetected with classical ross-correlation analysis.Comment: 15 pages with 12 figures. Accepted for publication in MNRA

Coherent structures in the distribution of the Faraday rotation measure (RM) of extragalactic radio sources are isolated using wavelet transformation techniques. A new algorithm of wavelet analysis for data points non‐uniformly distributed on a sphere is developed and implemented. Signatures of the magnetic fields in the local (Orion) arm, the Sagittarius arm (and its extension, the Carina arm), the synchrotron Loop I and, possibly, the Perseus arm have been revealed using the RM catalogues of Simard‐Normandin et al. (551 sources) and Broten et al. (663 sources). Unlike earlier analyses of the RM sky, our approach has allowed us to assess the stability of the results with respect to modifications of the data sample. Only the aforementioned features remain stable under mild sample modifications. We consider separately low‐latitude sources at |b|<10° and, using the model of electron density distribution of Cordes et al., we estimate magnetic field strength by comparing the model wavelet transform with that of the real data. Independent estimates of the mean magnetic field strength in the Orion arm using low‐ and high‐latitude sources converge to 1.4±0.3 μG. Rotation measures of low‐latitude sources provide a clear indication of a magnetic field reversal at a distance 0.6–1 kpc towards the Galactic Centre. Our analysis has revealed for the first time the extension of the reversal in the Carina arm. Low‐latitude sources from the catalogue of Broten et al. indicate a magneto‐ionic structure in the direction of the Perseus arm with the magnetic field direction reversed with respect to that in the Orion arm. The extent of the region with reversed magnetic field near the Sun is 3 kpc or more in the azimuthal direction. The average pitch angle of magnetic field in the nearby spiral arms is 15°, and the mean field strength in the Sagittarius–Carina and Perseus arms is 1.7±0.3 μG and 1.4±1.2 μG, respectively. The line‐of‐sight magnetic field in Loop I is estimated as 0.9±0.3 μG. We find firm evidence of a dominant even symmetry of the local mean magnetic field with respect to the Galactic equator. Our results are compatible with a moderate large‐scale north–south asymmetry, with the magnetic field in the southern hemisphere being stronger in a region of at least 3 kpc in size. It cannot be excluded, however, that the asymmetry is local and results from vertical bending of magnetic lines in a region of about 400 pc in size, with the Sun being located close to the top of a magnetic loop, the magnetic field of which is 0.5 μG stronger than the average field.

Interstellar magnetic fields and the propagation of cosmic ray electrons have an important impact on the radio-infrared (IR) correlation in galaxies. This becomes evident when studying different spatial scales within galaxies. We investigate the correlation between the IR and free-free/synchrotron radio continuum emission at 20 cm from the two local group galaxies M 31 and M 33 on spatial scales between 0.4 and 10 kpc. The multi-scale radio-IR correlations have been carried out using a wavelet analysis. The free-free and IR emission are correlated on all scales, but on some scales the synchrotron emission is only marginally correlated with the IR emission. The synchrotron-IR correlation is stronger in M 33 than in M 31 on small scales (<1 kpc), but it is weaker than in M 31 on larger scales. Taking the smallest scale on which the synchrotron-IR correlation exists as the propagation length of cosmic ray electrons, we show that the difference on small scales can be explained by the smaller propagation length in M 33 than in M 31. On large scales, the difference is due to the thick disk/halo in M 33, which is absent in M 31. A comparison of our data with data on NGC 6946, the LMC and M 51 suggests that the propagation length is determined by the ratio of ordered-to-turbulent magnetic field strength, which is consistent with diffusion of CR electrons in the ISM. As the diffusion length of CR electrons influences the radio-IR correlation, this dependence is a direct observational evidence of the importance of magnetic fields for the radio-IR correlation within galaxies. The star-formation rate per surface area only indirectly influences the diffusion length as it increases the strength of the turbulent magnetic field.

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 χ 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.