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Astron. Astrophys. 332, 395-409 (1998)

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3. Field compression and radio polarization

3.1. General considerations

The high degree of polarization of relic sources should result from the compression, which aligns unordered magnetic fields with the shock plane. If the shock processed relic is seen at some angle [FORMULA] between the line-of-sight and the normal of the shock front, the field structure projected onto the plane of sky shows a preferential direction. Thus, the resulting radio polarization depends on [FORMULA] and R. The direction of the magnetic field should appear to be perpendicular to the line connecting the relic and the cluster center, because of this compression and projection effects. This is the case for 1273+275 (Andernach et al. 1984).

In the following we calculate the magnetic field compression and alignment, and the resulting radio polarization of an unordered magnetized plasma blob in an unmagnetized gas flow going through a shock. The (complex) polarization of synchrotron radiation of an isotropic distribution of electrons within a magnetic field seen from the z -direction is given as (Burn 1966)


The spectral index of the electrons is [FORMULA], where [FORMULA] is the spectral index of the radio emission. In order to average over regions with different field orientations and strength, we have to take into account the different emissivities


Burn has argued that the case [FORMULA] is a valid simplification, which is justified especially in our case, where [FORMULA]. Averaging the polarization (denoted by [FORMULA]) over a (point symmetric) distribution of fields, weighted with the relative emissivity [FORMULA], gives the observed integral polarization:


A magnetic flux tube with field strength [FORMULA] oriented at some angle [FORMULA] to the normal of the shock front will be bent and amplified by a compression [FORMULA]. From flux conservation it follows that


Observing this field, which is given by its spherical coordinates [FORMULA], at the viewing angle [FORMULA] results in


Eq. 18can now be integrated, assuming an isotropic distribution of the unshocked fields [FORMULA] and choosing the compression ratio of the fields [FORMULA], which depends on the shock compression ratio R and the field orientation. We discuss two complementary extreme cases, which give similar polarizations.

3.2. Weak fields

If the magnetic pressure of the relic is small compared to the internal gas pressure the compression of the magnetized regions is equal to the compression of the accretion shock: [FORMULA]. The calculation of the integrals of Eq. 18is straightforward if the ensemble average is done over the isotropic, unshocked fields, transformed via Eqs. 19, 20, and 21to the shocked fields in the observer's coordinate system:


Inserting numbers of the relic 1253+275 ([FORMULA], [FORMULA], and [FORMULA], derived from [FORMULA]) gives a polarization of [FORMULA]. The observed polarization of [FORMULA] (Giovannini et al. 1991) is higher and might indicate an intrinsically ordered field structure of the relic as could be left behind from ordered field structures within an unshocked radio lobe or tail as a progenitor. Or it is due to a larger viewing angle of [FORMULA], corresponding to an accretion shock radius of [FORMULA] Mpc instead of [FORMULA] Mpc, as given by Eq. 1. Regarding all the simplifying assumptions made to derive both estimates (Eqs. 1and 22) of the shock radius, especially the assumed spherical symmetry, we feel that the derived values are quite consistent.

3.3. Strong fields

If the relic is supported by magnetic pressure only, then the compression of flux tubes differs from the compression of the unmagnetized surrounding medium: The progenitor of the cluster relic, an extended radio lobe, was in pressure equilibrium with its surrounding gas in the upstream region. After the radio plasma has passed the shock, it expands or contracts rapidly, depending on the field orientation, in order to achieve pressure equilibrium with its environment again, since the downstream flow (in the rest frame of the shock) is subsonic. This relates the upstream and downstream fields by [FORMULA]. Eqs. 4and 20then give the field compression factor as a function of the field orientation:


The polarization in the strong field case is then


Inserting the numbers of 1253+275 gives [FORMULA]. This is similar to the weak field case, also demonstrated by Fig. 3. Magnetic fields of [FORMULA] G are necessary within 1253+275 in order to support this structure against the thermal pressure at a cluster radius of [FORMULA] Mpc, assuming a density and temperature as given in Sect. 2.2and a magnetic pressure of [FORMULA]. This rough estimate depends on an extrapolation of the central density profile of the cluster, and has therefore a large error. The magnetic field strength could be lower if nonmagnetic pressure such as thermal gas and/or relativistic particles also support the radio structure. This number is higher than the magnetic field strength derived from minimum energy arguments of the synchrotron emitting plasma of [FORMULA] G measured by Giovannini et al. (1991). Since these authors use a source depth of [FORMULA] kpc along the line-of-sight, while we expect only [FORMULA] kpc due to the compression of the source by the shock, we can correct this number for our model to [FORMULA] G. Using the rise of the thermal and (here in the strong field case) of the magnetic pressure given by Eq. 4we find that the field strength of the unshocked relic plasma needs to be [FORMULA] G, if the equipartition value gives the relic field strength, or [FORMULA] G if only magnetic pressure has to support the relic. Field strengths of the pre- and after-shock region should be within these ranges.

[FIGURE] Fig. 3. Polarization of synchrotron emission of shocked magnetic fields as a function of the viewing angle [FORMULA]. The strong field case is given by solid lines, the weak field case by dashed lines. The upper pair of curves (solid & dashed) corresponds to a radio spectral index of [FORMULA], and therefore [FORMULA] and [FORMULA]. The lower lines correspond to [FORMULA], [FORMULA] and [FORMULA]. Observed polarizations of radio relics are plotted above the viewing angle predicted from the accretion shock theory (1253+257, 1712+64), simulations of a cluster merger (1706+78), or - if no data is available - at their best-fit position (0917+75). The large projected radius of 0917+75 indicates a large viewing angle, consistent with the best-fit position. Uncertainties in the angles are large.

3.4. Depolarization

A somewhat stronger difference between the polarization in the strong and weak field case might arise from internal Faraday depolarization, which should only appear in the weak field case, since the relic has to be filled with a sufficient amount of thermal gas. Internal depolarization should be strongest for large viewing angle [FORMULA], due to the larger number of field lines aligned with the line-of-sight.

We believe that the strong field case is a better description of the remnant of a radio galaxy's lobe, since we find rough equivalence between the measured equipartition field strength and predicted magnetic fields from pressure equilibrium with the surrounding gas. Therefore, internal depolarization should be weak. We do not attempt any estimate because of the large uncertainties entering the calculation such as coherence scale of the fields, the true field strength, and gas density in the relic.

External Faraday depolarization can lower the observed polarization if the relic is seen through the central magnetized regions of the cluster. The degree of depolarization would depend on the field strength and reversal scale within the intra-cluster medium, where both quantities have large uncertainties (Kim et al. 1986; Feretti et al. 1995; Enßlin et al. 1997; Enßlin & Biermann 1997). In order for external depolarization to occur the relic has to be on the back side of the cluster, and its viewing angle [FORMULA] must be low. Since then the polarization of the relic would be very small anyway, we do not need to correct for this.

We note that, since the observed polarization of 1275+273 is high, nearly all of the synchrotron emission has to come from the ordered fields of the post-shock region as assumed in Sect. 2.4, otherwise the polarization would be lower.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998