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

Appendix A: the intrinsic polarization angle

Below we give a detailed derivation of the intrinsic polarization angle as a function of and parameters of the magnetic field. The magnetic field is specified by Eq. (1), the azimuthal angle is measured counterclockwise in the galaxy plane from the northern end of the major axis of the galaxy, is measured in the sky plane from the local outward radial direction in the galaxy.

Let us introduce a Cartesian reference frame lying in the galaxy plane with the origin at the galaxy's center and another one, lying in the sky plane; the x -axes of both frames point to the northern end of the major axis. Correspondingly, we introduce also the azimuthal angle in the sky plane, .

It can be easily seen that and are related through since and , with i being the galaxy inclination angle ( corresponds to the face-on view). Insofar as the transverse component of the magnetic field (with respect to the line of sight) is represented by and the intrinsic polarization angle is orthogonal to , we have

or

Now we have

for the transverse field and

for the longitudinal field, where the direction to the observer is adopted as a positive direction of in accordance with the standard definition of the Faraday rotation measure. Here the inclination angle i is measured from the galaxy's rotation axis to the line of sight. As the left-hand side of the image of M 51 is closer to the observer, this direction is clockwise when seen from the northern end of the major axis (i.e., the point from which is measured). We thus have with our definition.

Using the above expressions for and in terms of the cylindrical polar components and , we obtain from Eq. (A1):

where is the total horizontal magnetic field, is the pitch angle of the total horizontal magnetic field with and defined in Eq. (10). The dependence of on is illustrated in Figs. 1 and 2 of Sokoloff et al. (1992) for various magnetic field configurations and inclination angles.

Fig. 9. Contour lines of are shown on the plane (). The minimum corresponding to the best fit is marked by a cross. The boundary of the region specified by the test, Eq. (B2), is shown by a hatched line. The two thick lines with arrows are the projections of the trajectories followed by the iterative procedure of minimization of the residual for two different initial conditions. The two basic types of the uncertainty estimates for the resulting values of and are also indicated, for and for .

Appendix B: statistical tests and errors

The criterion of quality, or reliability of a fit to the observational data is provided by the following dimensionless sum known as the residual:

where n enumerates the wavelengths at which observations have been carried out, of the total number , and i refers to individual sectors, of them per ring; is the set of polarization angles measured at the wavelength , is the value of the fitting function of the form specified by Eqs. (4)-(7), (A2) and (A3) for ; and are the uncertainties of the polarization angle values discussed in Sect. 2.

The representation (5) for the Faraday rotation measure was applied only for -12 kpc, where the disk is not transparent for the polarized emission at the longer wavelengths. For -15 kpc the fitting was performed for a one-layer model.

The fit is considered to be satisfactory if the following two conditions are fulfilled: the test

and, for any n and l, the Fisher test

where is the distribution with the total number of measurements, k the number of independent parameters of the model, the confidence level ( corresponds to a error of a Gaussian random variable), and is the Fisher distribution with and being the numbers of measurements at different wavelegths in a given ring.

Since the residual S is a strongly nonlinear function of its arguments, the estimation of the uncertainties of the magnetic field parameters resulting from the fit becomes also more complicated in comparison with the earlier linear models. The inequality Eq. (B2) defines a region in the k -dimensional parameter space where the values reside of the parameters that are considered admissible. For a quadratic function S typical of the linear models, this region is an ellipsoid and the uncertainties of the parameters are determined by the sizes of the ellipsoid along the corresponding axes. They can be expressed through the diagonal terms of the matrix , with the parameters of the model. On the contrary, for the present nonlinear model the above region has a very complicated shape that may differ drastically from an ellipsoid. Therefore, the above estimation of the uncertainties in terms of the second derivative matrix usually leads to strongly underestimated values. In the case presented in Fig. B1, this happens because the minimum point, marked by a cross, is far from the "centre" of the admissible region marked by the hatched line.

We thus used two additional estimates to characterize the uncertainties (see Fig. B1). The first one is the distance , from the minimum point, to the border of the region defined by Eq. (B2) as measured along the axis corresponding to a given parameter. The second estimate of the uncertainty has been obtained as follows. When searching for the minimum of S, we apply an iterative procedure starting with certain initial conditions which results in a sequence of parameter values, or a trajectory that converges to the final estimate corresponding to the minimum of S. At a certain step of the iterations, the trajectory intersects the border of the region defined by Eq. (B2); after that, the trajectory can be quite complicated and tangled within the region but never leaves it. Thus the second estimate of the uncertainty of the final fit results, , is provided by the lengths of the projections of the trajectory segment within the admissible region onto the corresponding axes (see Fig. B1). In Table 4 we adopted for the uncertainties the maximum of these three estimates. Insofar as the confidence level of the test was adopted as 95%, the resulting uncertainties correspond, in a certain restricted sense, to a 2 deviation of a Gaussian random variable.

Appendix C: basic notation

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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