Malcolm Kennett, Don Melrose, PASA, 15 (2), 211
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Next Section: Discussion and conclusions Title/Abstract Page: Propagation-induced circular polarization in Previous Section: Elliptically polarized natural modes | Contents Page: Volume 15, Number 2 |
Relativistic rotation measure (RRM)
We discuss the specific requirements for significant circular polarization to arise due to generalized Faraday rotation by referring to the familiar rotation measure (RM) and defining a relativistic rotation measure (RRM) that is its counterpart due to the contribution of the highly relativistic particles.
The parameter RM is defined for the case where only the cold plasma is present and is such that one has , where the dependence on follows from for a cold plasma. (The factor 2 arises from the plane of polarization rotating through when the relative phase changes by , that is, from for circularly polarized modes.) The conventional RM is defined by
where the integral is along the ray path, with the number density of the cold electrons and the angle between the magnetic field and the direction of propagation. Writing the integral in (3) in the form and expressing L in parsecs, per cubic centimeter and B in gauss, (3) becomes .
By analogy we define the RRM for the case where only the relativistic gas is considered, so that the natural modes are linearly polarized with (Sazonov 1969; Melrose 1997a). In this case we write . The definition of RRM depends on the details of the distribution of relativistic particles. We assume a power-law distribution of particles with energy spectrum
where is the number density of the relativistic particles, is the power law index and , are cutoff values for the Lorentz factor. Assuming and , the difference between the two modes (Melrose 1997a,b) implies
Writing the integral in (5) in the form and expressing L in parsecs, per cubic centimeter and B in gauss, (5) becomes
Returning to the two examples in the previous section, example 1. has two specific requirements: (a) the relativistic plasma must dominate over the cold plasma in determining the wave properties, and (b) the relative phase needs to be either over a small portion of the source or over a substantial portion of the source. Condition (a) requires that be dominated by the relativistic particles rather than the cold plasma, which is satisfied for , with is the lower cutoff energy of a power-law distribution of relativistic electrons (Melrose 1997b). This requirement can be expressed in the form , with
where the subscript on is to emphasize that the relevant RM is that obtained by integrating along only the portion of the ray path within the source where the cold plasma and relativistic electrons coexist. Condition (b) requires over a small portion of the source or over a substantial portion of the source. Inserting (3) and (5) into (7), requires , with . This condition is likely to be satisfied only in relativistic pair-dominated plasmas where there are essentially no cold electrons. In particular this condition may be satisfied in pulsar winds and pulsar nebulae (plerions). It might also be satisfied in pair-dominated relativistic plasmas in compact extragalactic sources. Let us assume approximate equipartition, that is, that the energy density in the relativistic electrons, , is of order the magnetic energy density. Then one has
with L in parsec and B in gauss. For example, in a plerion with and for , (8) requires . On the other hand, if one requires over a large part of the source, then may be adequate to account for the observed values of . It is interesting to note that this estimate of the field strength required for propagation to cause the circular polarization is similar to that estimated by Weiler (1975) for the Crab Nebula based on the intrinsic polarization of synchrotron radiation. For compact extragalactic sources the value of L is larger, requiring a correspondingly smaller value of B; again the values required are of the same order of magnitude as those required for the intrinsic polarization of synchrotron radiation (e.g., Weiler & de Pater 1980). However, the frequency dependence is more like than that () predicted for a uniform, optically thin synchrotron source. We conclude that for both applications, a more detailed investigation of the two alternatives is clearly warranted.
The other possibility 2. in the previous section is when the relativistic gas causes the natural modes to be only very slightly elliptical. The relative phase difference is then determined by the cold plasma, and provided one has , the resulting circular polarization should be of order the eccentricity of the modes, which is . It follows that this possibility implies , as found by Pacholczyk (1973), compared with for the intrinsic circular polarization of synchrotron radiation. The existing data do not appear consistent with with either a=0.5 or a=1. However, further multi-frequency data are required to clarify whether or not this possibility is a viable one.
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