ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O B-CelestialObject O O O B-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O
tokens: midplane values n e0 and n CR0 for the electron densities. For (corresponding to ), , and , we have . This is large compared to the typical Galactic magnetic field strength of a few , so Faraday rotation effects are weak. We perform the line-of-sight integration by computing on a ( θ , φ ) mesh with 36 × 72 meshpoints for distances s from the observer going up to in steps of . 4.2.3. Results from the Mean-field Model In Figure 3 we show the results for Model B of Brandenburg Furuya ( 2020 ). In spite of the parameters being unrealistic for the Galaxy, there are characteristic features that are similar to what is seen in Figure 1 for the Galaxy: positive Q at φ = 0 and 180°, along with negative E at φ = 90° and 270°. The results for U and B are not immediately obvious because there are two nearly equally big patches of opposite sign in each hemisphere. Only after φ -averaging do we recognize a latitudinal dependence that is similar to that of the Galaxy; see Figure 4 . The observed sign of emerges only when we place the observer at a distance sufficiently far away from the Galactic center ( μ = A), while for smaller distances ( μ = B), the sign changes and the profile becomes similar to that shown in the top right panel of Figure 1 of Brandenburg ( 2019 ). Faraday rotation is weak, but it contributes a profile that is symmetric about the equator. This is because the global magnetic field has quadrupolar symmetry; see Brandenburg ( 2019 ) for a corresponding result for a dipolar field. Changing the sign of α 0 results in a qualitatively different (oscillatory) dynamo, but, to our surprise, it does not change the sign of . The sign only changes when the differential rotation changes and thereby the global Galactic spiral. Changing the signs of α 0 and Ω 0 simultaneously has the advantage of leaving the dynamo properties unchanged. Figure 3. Polarization results for the mean-field model with the Faraday rotation term in Equation ( 6 ) included. Top: and . Bottom: and . Figure 4. Azimuthally averaged polarization results for the mean-field model with Faraday rotation using (solid lines) and without Faraday rotation (dashed lines). Top: and . Bottom: and . The statistical error of the mean and the standard deviation are indicated by dark and light shades, respectively, except for panels (b) and (d), where only the former is shown. An important difference between model and observation is the fact that in our model, the amplitude of is several hundred times smaller than that of , while for the Galaxy, it is only about 10 times smaller; see Table 1 . We recall that the quadratic EE correlation for dust polarization was found to be about twice that of the BB correlation (Planck Collaboration 2016 ). This came as a major surprise (Kandel et al. 2017 ) and triggered numerous investigations trying to explain this (Kritsuk et al. 2018 ). In the present Letter, however, we have not studied quadratic correlations, but the signed functions E and B themselves.