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As stressed above, the presence of diffuse and extended synchrotron emission in galaxy clusters indicates the existence of weak magnetic fields in the cluster volume. Different possibilities for their origin have been proposed which are reviewed by Dolag et al. (2008) - Chapter 15, this volume. Radio observations of galaxy clusters allow us to measure intracluster magnetic fields and test the different theories on their origin, as reviewed by Carilli & Taylor (2002) and Govoni & Feretti (2004). In the following the main methods to study magnetic field intensity and, eventually, structure are summarised.

4.1. Equipartition magnetic fields

In the optically thin case, the total monochromatic emissivity J(nu) from a set of relativistic electrons in a magnetic field B depends on a) the magnetic field strength, b) the energy distribution of the electrons, which is usually assumed to be a power law (Eq. 1), and c) the pitch angle between the electron velocity and the magnetic field direction (theta)

Equation 3 (3)

where alpha = (delta - 1) / 2 is the spectral index of the synchrotron spectrum 4.

Synchrotron emission from diffuse and extended radio sources can give us a direct measure for the intensity of cluster magnetic fields if the relativistic electron flux is measured or constrained. That can be achieved, for example, if Compton-produced X-ray (and gamma-ray) emission was detected simultaneously (see Sect. 4.2). In the case of polarised radio emission, we can also get an indication of the projected magnetic field orientation and its degree of ordering. To break the degeneracy between magnetic field strength and electron density (Eq. 3), and to obtain a measure for cluster magnetic fields from the observed luminosity of radio sources, it is typically assumed that the energy density of the relativistic plasma within a radio source is minimum

Equation 4 (4)

where UB is the energy density in magnetic fields, and Uel and Upr are the energy in electrons and in protons respectively. The energy in the heavy particles (protons) is considered to be related to the electron energy

Equation 5 (5)

The value of k depends on the mechanism of (re-)acceleration of electrons, whose physical details, as seen above, are still unknown. A typical value of k = 1 is adopted for halo and relic sources. Another important assumption of this method relates to the value of the filling factor, Phi, i.e. the fraction of the source volume V occupied by magnetic field and relativistic particles. The energy density in magnetic field is given by

Equation 6 (6)

It is usually considered that particles and magnetic fields occupy the entire volume, i.e. Phi = 1. It can be derived easily that the condition of minimum energy is obtained when the contributions of cosmic rays and magnetic fields is approximately equal

Equation 7 (7)

This is the so-called classical equipartition assumption, which allows us to estimate the magnetic field of a radio source from its radio luminosity L (see Pacholczyk 1970 for a rigorous derivation)

Equation 8 (8)

In the standard approach presented above, L is the observed synchrotron luminosity between two fixed frequencies nu1 and nu2 (usually nu1 = 10 MHz and nu2 = 100 GHz). In this way, however, the integration limits are variable in terms of the energy of the radiating electrons, since, based on Eq. 3, electron energies corresponding to nu1 and nu2 depend on magnetic field values. This point is particularly relevant for the lower limit, owing to the power-law shape of the electron energy distribution and to the expected presence of low energy electrons in radio halos/relics. Alternatively, it has been suggested to derive equipartition quantities by integrating the electron luminosity over an energy range (gammamin - gammamax) (Brunetti et al. 1997, Beck & Krause 2005). It can be shown that, for gammamin << gammamax and alpha > 0.5, the new expression for the equipartition magnetic field is

Equation 9 (9)

where Beq is the equipartition magnetic field expressed in Gauss derived through Eq. 8. Typically, for Beq ~ µG, gammamin ~ 100 and alpha ~ 0.75 - 1, this new approach gives magnetic field values 2 to 5 times larger than the standard method.

Estimates of equipartition fields on scales as large as ~1 Mpc give magnetic field intensities in the range 0.1-1 µG. As we have seen, these estimates are based on several assumptions both on different physical properties of the radio emitting region (e.g. the filling factor Phi and the ratio between electron and proton energies k), and on the condition of minimum energy of the observed relativistic plasma. Since the validity of these assumptions is not obvious, one has to be aware of the uncertainties and thus of the limits inherent to the equipartition determination of magnetic fields.

4.2. Compton scattering of CMB photons

As reviewed by Rephaeli et al. (2008) - Chapter 5, this volume, 3K microwave background photons can be subject to Compton scattering by electrons in the cluster volume. If the presence of thermal particles in the ICM results in a distortion of the Cosmic Microwave Background (CMB) spectrum well known as "Sunyaev-Zel'dovich effect" (Sunyaev & Zel'dovich 1972), non-thermal hard X-ray (HXR) photons are produced via Compton scattering by the same cosmic rays that are responsible for the synchrotron emission observed at radio wavelengths. Compton scattering increases the frequency of the incoming photon through

Equation 10 (10)

The Planck function of the CMB peaks at nuin ~ 1.6 × 1011 Hz. Based on Eq. 10, for typical energies of relativistic electrons in clusters (gamma ~ 1000 - 5000), the scattered photons fall in the X-ray and gamma-ray domain (~ 2 × 1017 - 5×1018 Hz, i.e. ~ 0.8 - 20.7 keV).

Non-thermal HXR emission from galaxy clusters due to Compton scattering of CMB photons was predicted more than 30 years ago (e.g. Rephaeli 1977) and has now been detected in several systems (Rephaeli et al. 2008 - Chapter 5, this volume; Fusco-Femiano et al. 2007 and references therein). Alternative interpretations to explain the detected non-thermal X-ray emission have been proposed in the literature (Blasi & Colafrancesco 1999, Enßlin et al. 1999, Blasi 2000, Dogiel 2000, Sarazin & Kempner 2000). However, these hypotheses seem to be ruled out by energetic considerations, because of the well known inefficiency of the proposed non-thermal Bremsstrahlung (NTB) mechanism. NTB emission of keV regime photons with some power P immediately imply about 105 times larger power to be dissipated in the plasma that seems to be unrealistic in a quasi-steady model (Petrosian 2001, 2003). For a more detailed treatment of the origin of HXR emission from galaxy clusters, see the review by Petrosian et al. (2008) - Chapter 10, this volume.

The detection of non-thermal HXR and radio emission, produced by the same population of relativistic electrons, allows us to estimate unambiguously the volume-averaged intracluster magnetic field. Following the exact derivations by Blumenthal & Gould (1970), the equations for the synchrotron flux fsyn at the frequency nuR and the Compton X-ray flux fC at the frequency nuX are

Equation 11 (11)

Equation 12 (12)

Here h is the Planck constant, V is the volume of the source and Phi the filling factor, DL is the luminosity distance of the source, B the magnetic field strength, T the radiation temperature of the CMB, r0 the classical electron radius (or Thomson scattering length), N0 and delta are the amplitude and the spectral index of the electron energy distribution (Eq. 1). The values of the functions a(delta) and F(delta) for different values of delta can be found in Blumenthal & Gould (1970). The field B can thus be estimated directly from these equations

Equation 13 (13)

Typical cluster magnetic field values of ~ 0.1 - 0.3 µG are obtained (e.g. Rephaeli et al. 1999, Fusco-Femiano et al. 1999, Fusco-Femiano et al. 2000, Fusco-Femiano et al. 2001, Rephaeli & Gruber 2003, Rephaeli et al. 2006). Compared to equipartition measures, this method has the great advantage of using only observables, assuming only that the spatial factors in the expressions for the synchrotron and Compton fluxes (Rephaeli 1979) are identical.

4.3. Faraday rotation measure

Faraday rotation analysis of radio sources in the background or in the galaxy clusters themselves is one of the key techniques used to obtain information on the cluster magnetic fields. The presence of a magnetised plasma between an observer and a radio source changes the properties of the polarised emission from the radio source. Therefore information on cluster magnetic fields along the line-of-sight can be determined, in conjunction with X-ray observations of the hot gas, through the analysis of the Rotation Measure (RM) of radio sources (e.g. Burn 1966).

The polarised synchrotron radiation coming from radio galaxies undergoes the following rotation of the plane of polarisation as it passes through the magnetised and ionised intracluster medium

Equation 14 (14)

where PsiInt is the intrinsic polarisation angle, and PsiObs(lambda) is the polarisation angle observed at a wavelength lambda. The RM is related to the thermal electron density (ne), the magnetic field along the line-of-sight (B||), and the path-length (L) through the intracluster medium according to

Equation 15 (15)

Polarised radio galaxies can be mapped at several frequencies to produce, by fitting Eq. 14, detailed RM images. Once the contribution of our Galaxy is subtracted, the RM should be dominated by the contribution of the intracluster medium, and therefore it can be used to estimate the cluster magnetic field strength along the line of sight.

The RM observed in radio galaxies may not be all due to the cluster magnetic field if the RM gets locally enhanced by the intracluster medium compression due to the motion of the radio galaxy itself. However a statistical RM investigation of point sources (Clarke et al. 2001, Clarke 2004) shows a clear broadening of the RM distribution toward small projected distances from the cluster centre, indicating that most of the RM contribution comes from the intracluster medium. This study included background sources, which showed similar enhancements as the embedded sources.

We also note that there are inherent uncertainties in the determination of field values from Faraday Rotation measurements, stemming largely from the unknown small-scale tangled morphology of intracluster fields, their large-scale spatial variation across the cluster, and from the uncertainty in modelling the gas density profile (see e.g., Goldshmidt & Rephaeli 1993, Newman et al. 2002, Rudnick & Blundell 2003a, Enßlin et al. 2003, Murgia et al. 2004).

RM studies of radio galaxies have been carried out on both statistical samples (e.g. Lawler & Dennison 1982, Vallee et al. 1986, Kim et al. 1990, Kim et al. 1991, Clarke et al. 2001) and individual clusters by analysing detailed high resolution RM images (e.g. Perley & Taylor 1991, Taylor & Perley 1993, Feretti et al. 1995, Feretti et al. 1999, Govoni et al. 2001a, Taylor et al. 2001, Eilek & Owen 2002, Govoni et al. 2006, Taylor et al. 2007, Guidetti et al. 2007). Both for interacting and relaxed (cooling flow) clusters the RM distribution of radio galaxies is generally patchy, indicating that cluster magnetic fields have structures on scales as low as 10 kpc or less. RM data are usually consistent with central magnetic field strengths of a few µG. But, radio galaxies at the centre of relaxed clusters have extreme RM, with the magnitude of the RM roughly proportional to the cooling flow rate (see Fig. 6). Strong magnetic fields are derived in the high density cooling-core regions of some clusters, with values exceeding ~ 10 µG (e.g., in the inner region of Hydra A, a value of ~ 35 µG was deduced by Taylor et al. 2002). It should be emphasised that such high field values are clearly not representative of the mean fields in large extended regions.

Figure 6

Figure 6. RM magnitudes of a sample of radio galaxies located in cooling flow clusters, plotted as a function of the cooling flow rate X (from [Taylor et al. 2002]).

Dolag et al. (2001b) showed that, in the framework of hierarchical cluster formation, the correlation between two observable parameters, the RM and the cluster X-ray surface brightness in the source location, is expected to reflect a correlation between the cluster magnetic field and gas density. Therefore, from the analysis of the RM versus X-ray brightness it is possible to infer the trend of magnetic field versus gas density.

On the basis of the available high quality RM images, increasing attention is given to the power spectrum of the intracluster magnetic field fluctuations. Several studies (Enßlin & Vogt 2003, Murgia et al. 2004) have shown that detailed RM images of radio galaxies can be used to infer not only the cluster magnetic field strength, but also the cluster magnetic field power spectrum. The analyses of Vogt & Enßlin (2003, 2005) and Guidetti et al. (2007) suggest that the power spectrum is of the Kolmogorov type, if the auto-correlation length of the magnetic fluctuations is of the order of few kpc. However, Murgia et al. (2004) and Govoni et al. (2006) pointed out that shallower magnetic field power spectra are possible if the magnetic field fluctuations extend out to several tens of kpc.

4.4. Comparison of the different methods

As shown in Table 3 of Govoni & Feretti (2004), the different methods available to measure intracluster magnetic fields show quite discrepant results (even more than a factor 10). RM estimates are about an order of magnitude higher than the measures derived both from the synchrotron diffuse radio emission and the non-thermal hard X-ray emission (~ 1 - 5 µG vs. ~ 0.2 - 1 µG).

This can be due to several factors. Firstly, equipartition values are severely affected by the already mentioned physical assumptions of this method. Secondly, while RM estimates give a weighted average of the field along the line of sight, equipartition and Compton scattering measures are made by averaging over larger volumes. Additionally, discrepancies can be due to spatial profiles of both the magnetic field and the gas density not being constant all over the cluster (Goldshmidt & Rephaeli 1993), or due to compressions, fluctuations and inhomogeneities in the gas and in the magnetic field, related to the presence of radio galaxies or to the dynamical history of the cluster (e.g. on-going merging events) (Beck et al. 2003, Rudnick & Blundell 2003b, Johnston-Hollitt 2004). Finally, a proper modelling of the Compton scattering method should include a) the effects of aged electron spectra, b) the expected radial profile of the magnetic field, and c) possible anisotropies in the pitch angle distribution of electrons (Brunetti et al. 2001, Petrosian 2001).

An additional method of estimating cluster magnetic fields comes from the X-ray analysis of cold fronts (Vikhlinin et al. 2001). These X-ray cluster features, discovered by Markevitch et al. (2000) thanks to the exquisite spatial resolution of the Chandra satellite, result from dense cool gas moving with near-sonic velocities through the less dense and hotter ICM. Cold fronts are thus subject to Kelvin-Helmholtz (K-H) instability that, for typical cluster and cold front properties (Mach number, gas temperatures, cluster-scale length), could quickly disturb the front outside a narrow (ltapprox 10°) sector in the direction of the cool cloud motion 5. Through the Chandra observation of A 3667, Vikhlinin et al. (2001) instead revealed a cold front that is stable within a ± 30° sector. They showed that a ~ 10 µG magnetic field oriented nearly parallel to the front is able to suppress K-H instability, thus preserving the front structure, in a ± 30° sector. The estimated magnetic field value, significantly higher than the typical measures given by the other methods outside cluster cooling flows, is likely an upper limit of the absolute field strength. Near the cold front the field is actually amplified by tangential gas motions (see Vikhlinin et al. 2001).

Variations of the magnetic field structure and strength with the cluster radius have been recently pointed out by Govoni et al. (2006). By combining detailed multi-wavelength and numerical studies we will get more insight into the strength and structure of intracluster magnetic fields, and into their connection with the thermodynamical evolution of galaxy clusters. More detailed comparisons of the different approaches for measuring intracluster magnetic fields can be found, for instance, in Petrosian (2003) and Govoni & Feretti (2004).

4 delta is the electron energy index, see Eq. 1. Back.

5 For a more precise treatment, see Vikhlinin et al. (2001) and Markevitch & Vikhlinin (2007). Back.

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