Annu. Rev. Astron. Astrophys. 1980. 18: 165-218 Copyright © 1980 by Annual Reviews. All rights reserved

3.2. Polarization Distributions

In the absence of a thermal plasma, synchrotron radiation is polarized orthogonally to the magnetic field direction by a percentage 100[(3 - 3) / (5 - 3)] [(B_u² / (B_u² + B_r²)] where is the spectral index (S ), and B_u and B_r are the respective field strengths of the uniform and random components of the magnetic field (Gardner & Whiteoak 1966, Moffet 1975). Hence simple inspection of the polarization distributions should apparently give information about the direction and turbulence of the magnetic field in a source. However, at all but the shortest wavelengths the interpretation of polarization distributions is made both more complicated and more informative by Faraday rotation. In a magnetoionic medium the plane of polarization of linearly polarized radiation is rotated through an angle proportional to the square of the wavelength: = 5.73 × 10^-3 R² deg, where R = 812 n_t(s)B_||(s) ds ~ 8.1 × 10⁸ n_tB_|| s rad m^-2 is the "rotation measure", (cm) is the wavelength, s (kpc) is the path length through the medium, n_t(s) (cm^-3) is the density of thermal electrons, and B_||(s) is the component of magnetic field parallel to the path, i.e. in the line of sight.

Integrated rotation measures have been published for several hundred sources (Vallée & Kronberg 1975, Haves 1975). Although a small proportion of sources such as 3C123 and 3C427.1 have intrinsic rotation measures that probably exceed several hundred rad m^-2 (Kronberg & Strom 1977, Riley & Pooley 1978), more than three quarters have absolute values smaller than 50 rad m^-2. Hence, the directions indicated by the measured polarization distributions for 6 cm will usually be within 10° of the unrotated angles (i.e. perpendicular to the uniform magnetic field projected on the plane of the sky). Maps of projected magnetic field directions B are therefore relatively easy to produce from polarization distributions measured at short wavelengths.

In principle, a comparison of polarization distributions at several wavelengths gives the distribution of R across a source and hence information about variations in n_tB_||. Unfortunately, matters are complicated by (a) ambiguities in R due to the limited number of observing frequencies, (b) difficulties in separating the in-source contribution to R from foreground rotation, (c) uncertainties in the path length s through the source, (d) effects of nonuniformity of the magnetic field within the source, including field reversals, (e) smearing due to the presence of several independent "cells" within the same observing beam.

Until now most multifrequency polarization mapping has been carried out using two (or, in a few special cases, three) observing wavelengths ₁, ₂. There is consequently an ambiguity of n in chi and a corresponding ambiguity in R of 3.14 × 10⁴ / (₂² - ₁²) rad m^-2.

Separation of the rotation that occurs within the source from that in the foreground has been attempted in two ways. First, the foreground rotation is obtained as the average of integrated rotation measures of several nearby sources. Second, one ignores the component of rotation that is constant across the source and examines only point-to-point variations in R, on the assumption that all these relatively small-scale changes in R are produced within the source.

One can guess at the path length s using symmetry arguments and assumptions about the distance, as in the preceding discussion of the derivation of the minimum-energy conditions.

A proper treatment of the effects of a tangled magnetic field and of beam smearing must take into account the distribution of the percentage polarization as a function of wavelength. The complex polarization as a function of wavelength can be evaluated in terms of source parameters (Burn 1966, Gardner & Whiteoak 1966). To obtain meaningful information from available data, an unduly large number of assumptions must be made about the statistical properties of the inhomogeneities and the source geometry. Such analyses have been carried out for 3C465 (van Breugel 1980c) and Virgo A (Forster 1980). A slightly different (Monte Carlo) model fitting procedure has been used by Burch (1979b) in a study of 3C47, 3C79, 3C219, 3C234, 3C300, 3C382, and 3C430. The results (electron densities of from 10^-5 to 10^-3 cm^-3 corresponding to total masses of ~ 10¹¹ M) are all quite similar to those given by back of the envelope calculation using the simple formulas in this section. However, the more sophisticated analyses illustrate that multifrequency polarization comparisons made, using many more frequencies than now available, could furnish useful information about the intrinsic distribution of n_t and B along the line of sight.