9. RECONCILING MAGNETIC FIELD VALUES

From the results presented in the previous sections, it is derived that cluster magnetic field strengths obtained from RM arguments (Sec. 6) are about an order of magnitude higher than the estimates obtained from both the diffuse synchrotron radio halo emission under equipartition conditions (Sec. 4.1) and the inverse Compton hard X-ray emission (Sec. 5). Relatively high magnetic fields could be present in regions of radio relics at the cluster periphery (Sec. 4.2), and in cold fronts (Sec. 7). However, it is important to note that magnetic field estimates derived in relics and in cold fronts may be not representative of the overall cluster magnetic field strengths because they have been likely enhanced by compression.

The clusters Coma and A3667 [86] are unique in that they allow the field to be estimated by using different techniques. Examples of the variation of magnetic field strength estimates from various methods and in various locations of these clusters are given in Table 3.

Table 3. Magnetic field estimates derived from various methods in the clusters Coma and A3667.

	Name	Method	Field strength	Location	Reference
			(µG)
	Coma	Equipartition	0.45	radio halo	34
		Equipartition	0.55	radio relic	77
		Faraday Rotation	7	cluster center	16
		Faraday Rotation	0.2	cluster center(large scale)	16
		Inverse Compton	0.2	cluster average	114
	A3667	Equipartition	1.5-2.5	NW relic	86
		Inverse Compton	0.4	cluster average	120
		Faraday Rotation	1-2	cluster center	86
		Faraday Rotation	3-5	NW relic	86
		Cold front	10	along the cold fronts	153
Column 2 gives the method used to estimate the field strength, Column 3 the value of the magnetic field in µG, Column 4 describes the location in the cluster at which this estimation is made, Column 5 gives the reference.

Several arguments can be invoked to alleviate the discrepancies between different methods of analysis. First, we remind that the equipartition values rely on several assumptions (Sec. 3.2). Moreover, the radio synchrotron and IC emissions originate from large cluster volumes, and the corresponding magnetic field estimates are averaged over the whole cluster, whereas the RM gives an average of the field along the line of sight, weighted by the thermal gas distribution. Taking into account the radial profile of the cluster magnetic field and of the gas density, Goldshmidt and Rephaeli [163] first showed that the field strength estimated with the IC method is expected to be smaller than that measured with the RM observations. Beck et al. [164] pointed out that field estimates derived from RM may be too large in the case of a turbulent medium where small-scale fluctuations in the magnetic field and the electron density are highly correlated. Finally, more realistic electron spectra should be considered in the analysis of synchrotron and IC emission. It has been shown that IC models which include both the expected radial profile of the magnetic field, and anisotropies in the pitch angle distribution for the electrons allow higher values of the ICM magnetic field in better agreement with the Faraday rotation measurements [46, 127]. Moreover, as shown in Table 3, the magnetic field strength may vary depending on the dynamical history and the location within the cluster.

In some cases a radio source could compress the gas and fields in the ICM to produce local RM enhancements [147, 148] (see also Sec. 6), thus leading to overestimates of the derived ICM magnetic field strength.

The magnetic field may show complex structure, as filaments and/or substructure with a range of coherence scales, therefore the interpretation of RM data as given in Sec. 3.5.2 would be too simplified. Indeed, Newman et al. [149] demonstrated that the assumption of a single-scale magnetic field leads to an overestimation of the magnetic field strength calculated through RM studies. In the next section we show that the use of a numerical approach can significantly improve our interpretation of the data and thus the knowledge of the strength and structure of magnetic fields.