The synchrotron radiation from a single electron is elliptically polarized, and the degree of polarization is a function of / c. In a uniform magnetic field the linear polarization of an ensemble of electrons with a power law index , the polarization is perpendicular to the magnetic field in the transparent part of the spectrum and is given by
and is of the order of 70% for typical values of . In the opaque part of the spectrum, the polarization is parallel to the magnetic field and is given by
so that P is typically only about 10%.
Since the observed polarization in the transparent sources is typically only a few percent, it may be concluded that the mag netic fields are generally tangled, and so the observed polarization integrated over the source is greatly reduced. This is confirmed by the observations of polarization that indeed approaches the theoretical value in limited regions of some sources, although it is somewhat remarkable that such highly ordered fields can exist over regions extending up to 10 or more kpc.
In the elongated sources the orientation of the polarization vectors indicate that the magnetic field is often aligned perpendicular or parallel to the direction of elongation. In some sources there appears to be a radial magnetic field.
If the pitch-angle distribution is non-isotropic, then there is a net circular polarization since the circularly polarized components of the radiation from the individual electrons do not completely cancel. Even if the distribution is isotropic, there will be a small net circular polarization, since there are more electrons in the solid angle defined by + d than in the one defined by - d. This effect is particularly important if the cone of radiation of a single electron ( ~ E / mc2) is large, which will occur at very low frequencies or in regions of high magnetic field strength.
In a uniform magnetic field of B Gauss, and isotropic distribution of electron pitch angles, the integrated circular polarization is ~ 100(3B / )1/2 percent at a frequency (Sciama and Rees, 1967). In a few sources the degree of circular polarization has been measured to be - 0.01 to 0.1% near 1 GHz. This corresponds to magnetic fields ~ 3 × 10-5±1 gauss - in good apparent agreement with the values derived from the synchrotron self-absorption cut-off frequency and the angular size.