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Astron. Astrophys. 318, 700-720 (1997)

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4. Recognition of magnetic field patterns

4.1. The parametrization of the magnetic structure

When fitting the observed distribution of the polarization angle [FORMULA] at a given wavelength [FORMULA] we adopt for each layer the following truncated Fourier representation for the cylindrical components of the regular magnetic field [FORMULA]:

[EQUATION]

where [FORMULA] and [FORMULA] are the strengths of the horizontal (parallel to the galactic plane) and vertical (perpendicular to the plane) components of the [FORMULA] mode, respectively, [FORMULA] and [FORMULA] are those of the [FORMULA] mode, [FORMULA] and [FORMULA] are the pitch angles, and [FORMULA] and [FORMULA] are the azimuthal angles at which the corresponding non-axisymmetric components are maximum. In Eq. (8) only the two lowest modes have been retained; this proves to be sufficient to fit the available data. The magnetic pitch angle is the small angle measured from the magnetic field vector to the tangent of the local circumference. It is positive (negative) if the magnetic field spiral opens counterclockwise (clockwise). We note that the magnetic field direction can be either inwards or outwards along the magnetic spiral. In the case of M 51, a negative pitch angle corresponds to a trailing spiral.

The intrinsic polarization angle [FORMULA] in Eq. (5) is determined by the transverse component of the magnetic field. A suitable expression relating [FORMULA] to the magnetic field of the form given in Eq. (8) was given by Sokoloff et al. (1992); its detailed derivation can be found in Appendix A - see Eq. (A3). Insofar as the synchrotron emissivity in the halo is significantly weaker than in the disk, we assume that the intrinsic polarization angle depends solely on the field in the disk.

We should emphasize that we analyze simultaneously and consistently the longitudinal and transverse (with respect to the line of sight) components of the magnetic field which manifest themselves through RM and [FORMULA], respectively. Previous work either considered solely the Faraday rotation measures between pairs of wavelengths or simplified the model by supposing that [FORMULA] (see Sokoloff et al. 1992). In both cases only the line-of-sight magnetic field could be recovered from observations. Attempts to extract additional information about the transverse component of the magnetic field from an independent analysis of "magnetic pitch angles," or [FORMULA], from total and polarized intensity, etc. often led to results that were inconsistent with those obtained from the RM analysis. Here for the first time we propose a consistent three-dimensional model of the regular magnetic field observed in a galaxy.

4.2. The fitting procedure

Having represented the galactic magnetic field in terms of the Fourier series (8), we calculate the corresponding intrinsic polarization angle using Eq. (A3) and the model Faraday rotation measure using Eqs. (6) and (A2) with [FORMULA] and [FORMULA] given in Table 3. This leads to the model azimuthal distributions of the polarization angle via Eq. (5). The model parameters are then determined from fits to the observed azimuthal distributions of polarization angles at the four wavelenghts for each ring as described in Appendix B. Since the galactic magnetic field reveals itself mainly through Faraday rotation, this analysis yields an estimate of the products [FORMULA] rather than [FORMULA]. Therefore, we have to use independent information on the volume density of thermal electrons [FORMULA] and its distribution, and on the length of the line-of-sight through the magneto-ionic region L in order to estimate [FORMULA] ; these parameters are discussed in Sect. 3.2.

The results of the fits discussed below are given in terms of the Fourier coefficients [FORMULA] defined by

[EQUATION]

for each part of the regular magnetic field [FORMULA] in the disk; for the halo, we adopt a similar definition with [FORMULA] replaced by Z.

We emphasize that the quantities [FORMULA], albeit having the dimension of Faraday rotation measure, have an entirely different physical meaning: if [FORMULA] and h are constants, they characterize the amplitudes of the individual Fourier components of the magnetic field in a given layer. Unlike Faraday rotation measure, the coefficients [FORMULA] do not add algebraically in a system consisting of several layers with distinct magnetic structures. The relation of the coefficients [FORMULA], or [FORMULA], to the line-of-sight and transverse components of the magnetic field (and thus to the Faraday rotation measure and the intrinsic polarization angle) is provided in Appendix A. We also use the quantity [FORMULA] which characterizes in a similar manner the total regular magnetic field, [FORMULA].

Representations of the form (9) are introduced for the disk and the halo separately. For example, [FORMULA] denotes the amplitude of the [FORMULA] mode in the disk, [FORMULA] is the pitch angle of the [FORMULA] mode in the halo, etc. For the radial range 9-15 kpc (see Sect. 4.4.3 and 4.4.4) the fits were performed for a one-layer magnetic field model without distinguishing the disk and halo contributions to [FORMULA] (that is, we put [FORMULA] there).

In order to obtain satisfactory fits to the data, we calculated the residual S defined in Eq. (B1), that characterizes the deviation of the fit from the measured points, found its minimum with respect to the above fit parameters, and then employed the [FORMULA] and Fisher statistical tests briefly discussed in Appendix B to assess the reliability of the fit. The [FORMULA] statistical test, Eq. (B2), ensures that the fit is close enough to the measured points with allowance for their weights equal to [FORMULA]. The Fisher test, Eq. (B3), was then applied to verify that the quality of the fit is the same at all individual wavelengths (see Sokoloff et al. 1992 for details).

In earlier studies [FORMULA], and hence the pitch angle of the magnetic field, were assumed to be independent of [FORMULA] so that the residual S defined by Eq. (B1) had a unique minimum in the parameter space (Ruzmaikin et al. 1990; Sokoloff et al. 1992). It was then possible to apply a linear least-squares method to find this minimum. The corresponding values of the parameters were considered the most probable ones. In our model S is a strongly nonlinear function of the parameters of the model, mainly because of the nonlinear nature of [FORMULA]. Therefore, S can have several local minima and, generally, many of them may satisfy the [FORMULA] test. In order to choose the best fit one should apply (i) the Fisher test and (ii) use all a priori information about the galaxy itself and the structure of the magnetic field. For instance, it is obvious that the resulting values of [FORMULA] in different rings must agree within the errors. However, for [FORMULA] -15 kpc this does not occur unless we apply special restrictions. Then we have to consider a conditional minimum of S, restricted by the requirement that [FORMULA] resides within a certain range obtained for other rings.

A method of estimating the errors of the model parameters is also discussed in Appendix B. The errors given in Table 4 below are the estimates of the uncertainties corresponding to a [FORMULA] Gaussian deviation.


[TABLE]

Table 4. Model for the global magnetic field in [FORMULA]


Even though our model of magnetic field is advanced enough to provide generally good fits, it can happen that a few measurements deviate very strongly from both the fit curves and the neighboring measured points. This can be due to many various reasons, including strong local distortions of the magnetic field (that are beyond the scope of the present analysis focussed on a large-scale structure), underestimated errors of these measurements or systematic errors. In such cases the only reasonable remedy is to exclude the strongly deviating measurements from the analysis. Of course, this is the last resort and we did this only when other ways to reach a good fit failed. In our analysis below only one measurement in the ring 9-12 kpc was omitted.

4.3. Statistical evidence for a magneto-ionic halo in M 51

Fits for all wavelengths obtained with one transparent magneto-ionic layer were generally inconsistent with the Fisher test at [FORMULA]. Therefore, we tried fits to the polarization angle distributions for [FORMULA] 2.8/6.2 cm and [FORMULA] 18.0/20.5 cm separately. The magnetic patterns revealed in the two wavelength ranges in the inner two rings turned out to be very different from each other. For example, the values of [FORMULA] obtained at the shorter and longer wavelengths had different signs. Apparently the only plausible explanation for this is that different regions are sampled in the two wavelength ranges and that the magnetic field has completely different configurations in them. Moreover, [FORMULA] changes sign either in the upper part of the disk [FORMULA] or in a layer above [FORMULA]. Since it is very unlikely that the regular magnetic field may have such a complicated vertical structure within the upper part of the disk, we concluded that the change of direction of [FORMULA] suggests the existence of one more extended component in the galaxy. Thus, at [FORMULA], at least two extended components of the magneto-ionic medium in M 51 are present, namely the disk and the halo.

The measurements of depolarization discussed above also indicate that the galaxy is not transparent to polarized emission in the wavelength range [FORMULA]. This again implies that different regions are sampled at the two wavelength ranges: at [FORMULA] 2.8/6.2 cm the whole galaxy is transparent to polarized radio waves, whereas at [FORMULA] 18.0/20.5 cm polarized emission originating from only the upper part of the galaxy is visible. Then the observed Faraday rotation at the short wavelengths is dominated by the magnetic field in the disk whereas that at the long wavelengths is mainly determined by the field in the halo.

Interestingly, recent observations of M 51 in X-rays (Ehle et al. 1995) provide an independent confirmation of the presence of a hot thermal halo of about 10 kpc in radius. We also note that the distribution of thermal emission at [FORMULA] (Klein et al. 1984) implies that the thermal electron density in the disk abruptly decreases at [FORMULA] (see Table 2) indicating that the star formation rate decreases steeply at this radius. We adopted an elliptical shape for the halo with a height of 6 kpc above the midplane near the center (see Fig. 2) and assumed the Milky Way value for the electron density of [FORMULA].

It is clear that separate fits for different wavelength ranges cannot be physically satisfactory. Therefore all the fits discussed below were obtained from all four wavelengths simultaneously using the values of [FORMULA] in Table 3. Thus we used a two-layer model of the magneto-ionic medium for the interval 3-9 kpc and a one-layer model for 9-15 kpc.

4.4. Results of the fitting

In this Section we discuss the results of the fitting procedure described in Sect. 4.2. In Figs. 3-6 we present the variations of the measured polarization angles with the azimuthal angle [FORMULA] and the fits. The results of the fitting are given in Table 4. For each ring we compile the model parameters and their uncertainties at the significance level [FORMULA] as well as the value of the residual S with the contributions from individual wavelengths and the value of the [FORMULA] test.

[FIGURE] Fig. 3. The polarization angles (dots with error bars, measured from the local radial direction in the plane of M 51) as a function of azimuthal angle in the galactic plane and fits (solid lines) for the ring [FORMULA] at a [FORMULA], b [FORMULA], c [FORMULA] and d [FORMULA] cm. Error bars show the 1 [FORMULA] errors of the measurements. Dashed lines indicate the foreground levels, [FORMULA]
[FIGURE] Fig. 4. The polarization angles (dots with error bars, measured from the local radial direction in the plane of M 51) as a function of azimuth and fits (solid lines) for the ring [FORMULA] at a [FORMULA], b [FORMULA], c [FORMULA] and d [FORMULA] cm. Error bars show the 1 [FORMULA] errors of the measurements. Dashed lines indicate the foreground levels, [FORMULA]
[FIGURE] Fig. 5. The polarization angles (dots, measured from the local radial direction in the plane of M 51) as a function of azimuth and fits (solid lines) for the ring [FORMULA] at a [FORMULA], b [FORMULA] and c [FORMULA] cm. Error bars show the 1 [FORMULA] errors of the measurements. Dashed lines indicate the foreground levels, [FORMULA]. The point which is omitted from the analysis is shown without error bar
[FIGURE] Fig. 6. The polarization angles (dots with error bars, measured from the local radial direction in the plane of M 51) as a function of azimuth and fits (solid lines) for the ring [FORMULA] at a [FORMULA] and b [FORMULA] cm. Error bars show the 1 [FORMULA] errors of the measurements. Dashed lines indicate the foreground levels, [FORMULA]. Observations at [FORMULA], [FORMULA] and [FORMULA] are not available

4.4.1. The radial range [FORMULA] -6 kpc

A satisfactory fit for 3-6 kpc (i.e., with both the [FORMULA] and Fisher tests satisfied) is achieved without a vertical component of the magnetic field. As shown by the fits given in Table 4 and in Fig. 3, the magnetic field in this ring represents a superposition of horizontal axisymmetric ([FORMULA]) and bisymmetric ([FORMULA]) components in the disk and an axisymmetric field ([FORMULA]) in the halo. The axisymmetric and bisymmetric components in the disk have comparable amplitudes with a moderate dominance of the latter. Since the axisymmetric components in the disk and halo have opposite signs, the fields in the disk and the halo are oppositely directed in most of the ring. In both the disk and the halo the pitch angles are negative, therefore the magnetic pattern is trailing like the optical spiral arms.

There is one more fit which satisfies the statistical criteria equally well and has the same number of parameters. This is a fit with an axisymmetric field in the disk and a combination of axisymmetric and bisymmetric fields in the halo. 1 However, in the next outer ring there is only one fit satisfying the statistical criteria (discussed in Sect. 4.4.2), which has a combination of an axisymmetric and a bisymmetric field in the disk and an axisymmetric field in the halo. Since it should be expected that the global configuration of the magnetic field in the galaxy cannot change strongly over the radial distance of about 3 kpc, we consider the fit shown in Fig. 3 and Table 4 as the more plausible one.

4.4.2. The radial range [FORMULA] -9 kpc

In Table 4 and in Fig. 4 we show the results of the fit for this ring. The modes [FORMULA] and [FORMULA] are of almost equal amplitudes in the disk and the [FORMULA] mode dominates in the halo. The direction of the magnetic field in the halo is opposite to that in the disk as in the ring 3-6 kpc. The field is predominantly horizontal and the field pattern represents a trailing spiral.

The configurations of magnetic field in the two inner rings are very similar to each other as evidenced by the similar combinations of azimuthal modes, close values of the pitch angles, and the decrease of [FORMULA] with radius as expected for a spiral of this pitch angle (cf. Appendix in Krause et al. 1989b). The amplitude of each mode decreases with radius between 3-6 and 6-9 kpc.

The very large positive error of [FORMULA] as given in Table 4 can be explained as follows. There is another fit (corresponding to shallower minima in S than the above one) which satisfies the [FORMULA] criterion but fails to meet the Fisher criterion. It has an axisymmetric field in the disk and a combination of [FORMULA] and 1 modes in the halo with the ratio on the left-hand side of Eq. (B3) being 3.13 and [FORMULA]. The minimum in S corresponding to the latter fit and that given in Table 4 are connected by a long, narrow 'valley' extended along the [FORMULA] -axis which is mostly below the [FORMULA] level. Since our estimate of errors is sensitive only to the relative values of S and [FORMULA] (see Appendix B) the resulting error in [FORMULA] turns out to be that large. In other words, the admissible region in the parameter space corresponding to the above fit is more like a dumbbell than an ellipsoid; the error of [FORMULA] given corresponds to the larger dimension of the region. 2

4.4.3. The radial range [FORMULA] -12 kpc

As the measurements at [FORMULA] are unreliable beyond [FORMULA] (see Sect. 2) we consider only the wavelengths [FORMULA], 18.0 and 20.5 cm in this ring.

The data at the short and long wavelengths do not exhibit the strong difference typical of the two inner rings (see Sect. 4.3) and a one-layer model is consistent with observations. We therefore conclude that there is no halo visible in this ring.

The [FORMULA] and Fisher tests cannot be satisfied if all the measurements are included into the fit. In Fig. 5 and Table 4 we give the fit parameters with a combination of the [FORMULA] and 1 modes obtained after one measurement at [FORMULA], [FORMULA], was omitted from the analysis. This point strongly deviates from both the general trend and the corresponding measurement at the close wavelength [FORMULA]. We should note that one might prefer to include this point and to employ a more complicated model with a vertical magnetic field in order to meet the [FORMULA] test. However, then the result that there is a vertical magnetic field in this ring would rely on only a single measurement.

The magnetic field inferred for this ring also represents a superposition of [FORMULA] and 1 modes with almost equal weights. Magnetic lines again form a spiral opening clockwise, even though the spiral is more tightly wound than in the inner rings. The amplitudes of the two modes are smaller than in the inner regions, thus following the trend of the two inner rings. However, the decrease of [FORMULA] with radius by about [FORMULA] between two adjacent rings is not continued between the rings at [FORMULA] -9 and 9-12 kpc.

We should note that if we omit two more measurements strongly deviating from other points, at [FORMULA] at [FORMULA] and 20.5 cm, then a model with a purely axisymmetric field provides a good fit.

4.4.4. The radial range [FORMULA] -15 kpc

Results of the fit to the data at [FORMULA] and 20.5 cm are shown in Table 4 and Fig. 6. Again measurements at [FORMULA] are not available for this ring. In addition, in this ring the galaxy is completely transparent for polarized emission at all the wavelengths used (that is, [FORMULA] - see Tables 2 and 3), and we applied a one-layer model.

Only a few measurements are available at [FORMULA] which were also neglected; we stress, however, that the available points at [FORMULA] agree very well with the fit obtained. The data at [FORMULA] and [FORMULA], [FORMULA], are not available because the polarized intensity in these sectors is too low.

A model with a purely horizontal magnetic field does not satisfy the [FORMULA] test and we included the vertical field into the model. This leads to the following difficulty. An axisymmetric vertical magnetic field affects the polarization angles in almost the same way as the foreground Faraday rotation with the only difference that it not only produces a uniform Faraday rotation but also affects the intrinsic polarization angle, [FORMULA]. If [FORMULA] is weak compared with horizontal components its effect on [FORMULA] is similarly weak, so it is difficult to separate its effect from the foreground Faraday rotation. 3 Thus, we present in Table 4 and Fig. 6 the fit obtained with [FORMULA] fixed to be [FORMULA], approximately the median value obtained for the other rings. Horellou et al. (1992) estimated [FORMULA] using 9 sources located within [FORMULA] from M 51, which agrees within errors with our estimates.

4.4.5. Sensitivity to the model parameters

The adopted values of [FORMULA] (and, hence, [FORMULA]) are the lower estimates (see Sect. 3.3.2). Therefore, we checked how sensitive our results are to this parameter. The fits turned out to be quite stable under the variation of [FORMULA] showing an ordered, slow variation of the fit parameters. For example, in the ring 3-6 kpc [FORMULA] ([FORMULA]) varies between -201 and [FORMULA] (-290 and [FORMULA]) for [FORMULA] varying between 0.16 and 1.00. The other parameters vary very weakly: for instance, the corresponding range for [FORMULA] is 37-47 [FORMULA], and those for the pitch angles and [FORMULA] are as narrow as [FORMULA] and [FORMULA], respectively. For [FORMULA] the fit fails to satisfy the [FORMULA] test. 4

Since our rings were chosen more or less arbitrarily with the only requirement that they should be wide enough to match the resolution of the observations, we also tried other rings to test the stability of the results. In particular, we considered the rings [FORMULA] -6 kpc and [FORMULA] -14 kpc. The results are only weakly sensitive to this change.

Figures 3 and 4 show that our fits do not exactly follow the variations in the polarization angles at [FORMULA] and [FORMULA]. Although at these wavelengths the errors in [FORMULA] are larger than at [FORMULA] and [FORMULA], we stress that the quality of the fit is uniformly good at all the wavelengths as ensured by the Fisher test.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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