## 4. Recognition of magnetic field patterns## 4.1. The parametrization of the magnetic structureWhen fitting the observed distribution of the polarization angle at a given wavelength we adopt for each layer the following truncated Fourier representation for the cylindrical components of the regular magnetic field : where and are the strengths of the horizontal (parallel to the galactic plane) and vertical (perpendicular to the plane) components of the mode, respectively, and are those of the mode, and are the pitch angles, and and 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 in Eq. (5) is determined by the transverse component of the magnetic field. A suitable expression relating 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 , respectively. Previous work either
considered solely the Faraday rotation measures between pairs of
wavelengths or simplified the model by supposing that
(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 , 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 ## 4.2. The fitting procedureHaving 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 and
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
rather than . Therefore, we have to use
independent information on the volume density of thermal electrons
and its distribution, and on the length of the
line-of-sight through the magneto-ionic region The results of the fits discussed below are given in terms of the Fourier coefficients defined by for each part of the regular magnetic field
in the disk; for the halo, we adopt a similar definition with
replaced by We emphasize that the quantities , albeit
having the dimension of Faraday rotation measure, have an entirely
different physical meaning: if and Representations of the form (9) are introduced for the disk and the halo separately. For example, denotes the amplitude of the mode in the disk, is the pitch angle of the 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 (that is, we put there). In order to obtain satisfactory fits to the data, we calculated the
residual In earlier studies , and hence the pitch
angle of the magnetic field, were assumed to be independent of
so that the residual 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 Gaussian deviation.
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 51Fits for all wavelengths obtained with one transparent
magneto-ionic layer were generally inconsistent with the Fisher test
at . Therefore, we tried fits to the
polarization angle distributions for
2.8/6.2 cm and 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
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,
changes sign either in the upper part of the
disk or in a layer above
. 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 suggests the existence of one more extended
component in the galaxy. Thus, at , at least
two The measurements of depolarization discussed above also indicate that the galaxy is not transparent to polarized emission in the wavelength range . This again implies that different regions are sampled at the two wavelength ranges: at 2.8/6.2 cm the whole galaxy is transparent to polarized radio waves, whereas at 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 (Klein et al. 1984) implies that the thermal electron density in the disk abruptly decreases at (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 . 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 ## 4.4. Results of the fittingIn 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 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
as well as the value of the residual
## 4.4.1. The radial range -6 kpcA satisfactory fit for 3-6 kpc (i.e., with both the
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
() and bisymmetric ()
components in the disk and an axisymmetric field
() 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 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.
## 4.4.2. The radial range -9 kpcIn Table 4 and in Fig. 4 we show the results of the fit
for this ring. The modes and
are of almost equal amplitudes in the disk and
the mode dominates in the halo. The direction
of the magnetic field in the halo is 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 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 as given
in Table 4 can be explained as follows. There is another fit
(corresponding to shallower minima in ## 4.4.3. The radial range -12 kpcAs the measurements at are unreliable beyond (see Sect. 2) we consider only the wavelengths , 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 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 and 1 modes obtained after one measurement at , , was omitted from the analysis. This point strongly deviates from both the general trend and the corresponding measurement at the close wavelength . 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 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 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 with radius by about between two adjacent rings is not continued between the rings at -9 and 9-12 kpc. We should note that if we omit ## 4.4.4. The radial range -15 kpcResults of the fit to the data at and 20.5 cm are shown in Table 4 and Fig. 6. Again measurements at 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, - see Tables 2 and 3), and we applied a one-layer model. Only a few measurements are available at which were also neglected; we stress, however, that the available points at agree very well with the fit obtained. The data at and , , 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 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,
. If is weak compared
with horizontal components its effect on is
similarly weak, so it is difficult to separate its effect from the
foreground Faraday rotation.
## 4.4.5. Sensitivity to the model parametersThe adopted values of (and, hence,
) 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 showing an ordered, slow
variation of the fit parameters. For example, in the ring 3-6 kpc
() varies between -201
and (-290 and ) for
varying between 0.16 and 1.00. The other
parameters vary very weakly: for instance, the corresponding range for
is 37-47 , and those
for the pitch angles and are as narrow as
and , respectively. For
the fit fails to satisfy the
test.
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 -6 kpc and -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 and
. Although at these wavelengths the errors in
are larger than at and
, we stress that the quality of the fit is
uniformly good at © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |