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$\begin{figure} \par\includegraphics[width=18cm,clip]{133.f1} \end{figure}$ Figure 1: Polarized intensity (contours) of M 31 with the orientation of the emission B-vector also shown (dashes, not corrected for Faraday rotation) with their lengths proportional to the degree of polarization, observed at $\lambda6~{\rm cm}$ with the Effelsberg radio telescope (Berkhuijsen et al. 2003). Note that the foreground RM of $-90~{\rm rad~m^{-2}}$ (Table 2) corresponds to Faraday rotation of about $20\hbox {$^\circ $ }$ so that the intrinsic B-vectors are roughly azimuthal. The beam width is $3\hbox {$^\prime $ }$ and the rms noise is 0.2 mJy/beam. Contour levels are $1,2,3,4,6 \times (5\times 10^{-4})$ Jy/beam. A length of B-vectors of $3\hbox {$^\prime $ }$ corresponds to a degree of polarization of 36%. The northern major axis is to the left and the ellipses show the radial range of the data analyzed in this paper, $6\le r\le14~{\rm kpc}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f2} \end{figure}$ Figure 2: Polarization angles ( $\psi$ , measured from the local radial direction in the plane of M 31) against azimuth ( $\theta$ ) for the ring 6-8 $~{\rm kpc}$ . Fit (solid) and observations (squares with error bars, horizontal lines with error bars show excluded points) are shown for $\lambda6~{\rm cm}$ , $\lambda11~{\rm cm}$ and $\lambda20.5~{\rm cm}$ from top to bottom. The error bars show the $1\sigma$ deviations.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f3} \end{figure}$ Figure 3: As in Fig. 2 but for the ring 8-10 $~{\rm kpc}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f4} \end{figure}$ Figure 4: As in Fig. 2 but for the ring 10-12 $~{\rm kpc}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f5} \end{figure}$ Figure 5: As in Fig. 2 but for the ring 12-14 $~{\rm kpc}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f6} \end{figure}$ Figure 6: Face-on view of M 31 showing sectors and regular magnetic field vectors obtained from the fits shown in Table 2. The grid radii are 6, 8, 10, 12 and 14 kpc. The length of the vectors is proportional to $\vec~B$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f7} \end{figure}$ Figure 7: For the ring $10<r<12~{\rm kpc}$ . a) Rotation measures derived from the fitted magnetic field shown in Table 2. b) The normalized amplitude of the component of the regular magnetic field lying in the sky plane, $~B_{\perp}$ , derived from the fitted magnetic field shown in Table 2. c) The observed degree of polarization (P₆= polarized intensity/total nonthermal intensity) at $\lambda6~{\rm cm}$ . d) The observed degree of polarization at $\lambda20~{\rm cm}$ (P₂₀).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f8} \end{figure}$ Figure 8: Observed (squares with error bars) and expected (solid line) depolarization between $\lambda20~{\rm cm}$ and $\lambda6~{\rm cm}$ , DP_20/6, for the ring $10<r<12~{\rm kpc}$ assuming various depolarization mechanisms. The azimuthal angle $\theta$ is measured counterclockwise from the northern major axis. Depolarization due to a) differential Faraday rotation using Eq. (A.1), b) RM gradients in the synchrotron source and differential Faraday rotation from Eq. (A.4) and c) RM gradients in a Faraday screen given by Eq. (A.5) are represented by solid lines. RM and the gradient in RM are derived from the fitted magnetic field described in Sect. 5. We conclude that foreground RM gradients, illustrated in panel c), dominate other wavelength dependent depolarization mechanisms.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f9} \end{figure}$ Figure 9: Observed (squares with error bars) and expected (solid line) depolarization between $\lambda20~{\rm cm}$ and $\lambda6~{\rm cm}$ , DP_20/6, due to RM gradients and differential Faraday rotation using Eqs. (8), (11) and (12), for the ring $10<r<12~{\rm kpc}$ and with different ratios of the synchrotron to thermal disk scale heights, $q=h_{{\rm syn}}/h_{{\rm th}}$ .
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In the text

$\begin{figure} \par\includegraphics[width=6.5cm,clip]{133.f10} \end{figure}$ Figure 10: Sketch showing the scale heights of the thermal disk, $h_{{\rm th}}$ , synchrotron disk, $h_{{\rm syn}}$ , and the depth in the thermal disk from which polarized emission is visible at $\lambda20~{\rm cm}$ , $\Delta z$ . The galactic midplane is at z=0, i.e., at the bottom of the figure. The deeper layers of the synchrotron disk are invisible in polarized emission because of internal Faraday dispersion. This sketch is only an illustration: note that $\Delta z$ can exceed $h_{{\rm th}}$ if more than half the disk thickness is visible. However, $h_{{\rm th}}-h_{{\rm syn}}\leq \Delta z\leq h_{{\rm th}}+h_{{\rm syn}}$ with the extreme values corresponding to DP_20/6=0 and 1, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f11} \end{figure}$ Figure 11: Observed (squares with error bars representing $1\sigma$ error) and expected (solid line) depolarization between $\lambda20~{\rm cm}$ and $\lambda6~{\rm cm}$ , DP_20/6, due to Faraday effects using Eq. (8), (11) and (12), for each of four radial rings. The expected depolarization is fitted by varying the ratio of the synchrotron to thermal disk scale heights, $q=h_{{\rm syn}}/h_{{\rm th}}$ and depolarization due to internal Faraday dispersion, $DP_{{\rm in}}$ . The fitted values of q are 0.6, 0.4, and 0.3 and of $DP_{{\rm in}}$ 0.1, 0.3 and 0.2 in panels b), c) and d), respectively. The solid line was obtained using RM derived from the regular magnetic field fits of Table 2.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{133.f12} \end{figure}$ Figure A.1: Sketch showing wavelength independent depolarization. The double headed arrows represent E-vectors.

In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{133.f13} \end{figure}$ Figure A.2: Sketch of synchrotron emitting layer and a foreground Faraday screen.

In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{133.f14} \end{figure}$ Figure A.3: Differential Faraday rotation occurs within the synchrotron emitting layer. Emission from different depths along the same line of sight undergoes different amounts of Faraday rotation, causing depolarization. For clarity the sketch separates emission from different depths.

In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{133.f15} \end{figure}$ Figure A.4: Sketch illustrating Faraday dispersion. When this effect occurs within the synchrotron emitting layer it is called internal Faraday dispersion; occurrence in a Faraday screen (no synchrotron emission) is external Faraday dispersion. Note that the required variation of rotation measure within the beam could also be caused by fluctuations of thermal electron density. In that case the magnetic field could be totally regular but Faraday dispersion would still occur.

In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{133.f16} \end{figure}$ Figure A.5: Depolarization due to gradients in rotation measure: if RM varies across the beam, different parts of the beam area contribute with different polarization angles. Note that variations of thermal electron density or magnetic field strength can also produce RM gradients even in a totally regular magnetic field.

In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f3} \end{figure}$	Figure 3: As in Fig. 2 but for the ring 8-10 $~{\rm kpc}$ .
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f4} \end{figure}$	Figure 4: As in Fig. 2 but for the ring 10-12 $~{\rm kpc}$ .
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f5} \end{figure}$	Figure 5: As in Fig. 2 but for the ring 12-14 $~{\rm kpc}$ .
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f6} \end{figure}$	Figure 6: Face-on view of M 31 showing sectors and regular magnetic field vectors obtained from the fits shown in Table 2. The grid radii are 6, 8, 10, 12 and 14 kpc. The length of the vectors is proportional to $\vec~B$ .
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{133.f9} \end{figure}$	Figure 9: Observed (squares with error bars) and expected (solid line) depolarization between $\lambda20~{\rm cm}$ and $\lambda6~{\rm cm}$ , DP_20/6, due to RM gradients and differential Faraday rotation using Eqs. (8), (11) and (12), for the ring $10<r<12~{\rm kpc}$ and with different ratios of the synchrotron to thermal disk scale heights, $q=h_{{\rm syn}}/h_{{\rm th}}$ .
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