Propagation-induced circular polarization in synchrotron sources

Malcolm Kennett, Don Melrose, PASA, 15 (2), 211
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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 tex2html_wrap_inline524, where the dependence on tex2html_wrap_inline386 follows from tex2html_wrap_inline528 for a cold plasma. (The factor 2 arises from the plane of polarization rotating through tex2html_wrap_inline530 when the relative phase changes by tex2html_wrap_inline532, that is, from tex2html_wrap_inline534 for circularly polarized modes.) The conventional RM is defined by
 equation82

where the integral is along the ray path, with tex2html_wrap_inline536 the number density of the cold electrons and tex2html_wrap_inline538 the angle between the magnetic field and the direction of propagation. Writing the integral in (3) in the form tex2html_wrap_inline540 and expressing L in parsecs, tex2html_wrap_inline536 per cubic centimeter and B in gauss, (3) becomes tex2html_wrap_inline548.

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 tex2html_wrap_inline550 (Sazonov 1969; Melrose 1997a). In this case we write tex2html_wrap_inline552. 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
 eqnarray95

where tex2html_wrap_inline560 is the number density of the relativistic particles, tex2html_wrap_inline562 is the power law index and tex2html_wrap_inline564, tex2html_wrap_inline566 are cutoff values for the Lorentz factor. Assuming tex2html_wrap_inline568 and tex2html_wrap_inline570, the difference between the two modes (Melrose 1997a,b) implies
 equation104

Writing the integral in (5) in the form tex2html_wrap_inline572 and expressing L in parsecs, tex2html_wrap_inline560 per cubic centimeter and B in gauss, (5) becomes
 equation112

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 tex2html_wrap_inline500 needs to be either tex2html_wrap_inline582 over a small portion of the source or tex2html_wrap_inline584 over a substantial portion of the source. Condition (a) requires that tex2html_wrap_inline406 be dominated by the relativistic particles rather than the cold plasma, which is satisfied for tex2html_wrap_inline588, with tex2html_wrap_inline590 is the lower cutoff energy of a power-law distribution of relativistic electrons (Melrose 1997b). This requirement can be expressed in the form tex2html_wrap_inline592, with
 equation119

where the subscript on tex2html_wrap_inline594 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 tex2html_wrap_inline596 over a small portion of the source or tex2html_wrap_inline598 over a substantial portion of the source. Inserting (3) and (5) into (7), tex2html_wrap_inline596 requires tex2html_wrap_inline602, with tex2html_wrap_inline604. 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, tex2html_wrap_inline606, is of order the magnetic energy density. Then one has
 equation131

with L in parsec and B in gauss. For example, in a plerion with tex2html_wrap_inline612 and for tex2html_wrap_inline614, (8) requires tex2html_wrap_inline616. On the other hand, if one requires tex2html_wrap_inline618 over a large part of the source, then tex2html_wrap_inline620 may be adequate to account for the observed values of tex2html_wrap_inline622. 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 tex2html_wrap_inline628 than that (tex2html_wrap_inline630) 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 tex2html_wrap_inline500 is then determined by the cold plasma, and provided one has tex2html_wrap_inline522, the resulting circular polarization should be of order the eccentricity of the modes, which is tex2html_wrap_inline636. It follows that this possibility implies tex2html_wrap_inline638, as found by Pacholczyk (1973), compared with tex2html_wrap_inline630 for the intrinsic circular polarization of synchrotron radiation. The existing data do not appear consistent with tex2html_wrap_inline642 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.


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

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