become observable in this energy range because of non-thermal bremsstrahlung from the γ ∼ 100−500 electrons, which should be present in all massive clusters. Alternatively, some models predict the existence of a population of very high energy electrons (E&100 TeV), which should up-scatter the photons of the CMB up to the TeV range (see Sect.
2.5.2).
The first attempts of detecting high-energyγ-ray emission from clusters of galaxies were made by the EGRET instrument on board CGRO. Reimer et al. (2003) presented the results of EGRET observations of 58 clusters. Although they failed to detect any of them, they computed upper limits for theγ-ray flux of these clusters which have been used to constrain the models. In the best studied cases, such as Coma, the upper limit reported by EGRET allows to show that the relativistic electrons responsible for the radio halo cannot be completely secondary. The EGRET upper limit on theγ-ray flux of the Coma cluster also implies that no more than 10−30% of the thermal energy is under the form of relativistic particles. The Large Array Telescope (LAT) on board theGLAST satellite, which was successfully launched on June 10, 2008, should reach a sensitivity better than EGRET by a factor of 10. It will therefore put strong constrains on the secondary models.
Using cosmological simulations with both primary and secondary electron components, Pfrommer (2007) estimated theγ-ray luminosity through π0 decay of a number of nearby clusters. Figure 2.10 shows theγ-ray (E >100 MeV) flux function of the cluster sample.
Based on these results, the author predicts thatGLAST will detect∼10 nearby clusters.
In conclusion, although there has been no detection yet of high-energyγ-ray emission from a galaxy cluster, observations withGLAST will put strong constrains (even in the case of a non-detection) on the particle acceleration models, especially secondary, and will probe the energy content of relativistic particles in clusters.
2.7 Methods of magnetic field measurement
Observations of diffuse radio sources associated with clusters of galaxies imply the existence of large-scale magnetic fields in the range B ∼ 0.1−10µG throughout the volume of clusters. This value exceeds the magnetic field expected from adiabatic amplification of seed fields. Because the radiative lifetime of high-energy electrons is proportional to 1/B, a good knowledge of the magnetic field is important to constrain the properties of the population of relativistic electrons in clusters. The existence of these relatively strong magnetic fields also raises the question of the origin of the field. In this section, I will present the three different methods which have been used to derive magnetic field values in clusters: Faraday rotation measure, equipartition and synchrotron/IC ratio. This section follows Feretti & Giovannini (2007).
2.7.1 Faraday rotation measure
The synchrotron radiation from cosmic radio sources (e.g. radio galaxies) is well know to be linearly polarized. A linearly polarized electromagnetic wave at a wavelengthλtravelling
Figure 2.10: Predictions for theγ-ray flux from π0 decay from a model considering both primary and secondary electron populations (Pfrommer 2007).
through a magnetized medium of diameter L will see its polarization angle rotate by an angle ∆χ= RMλ2, where RM, the “Faraday rotation measure”, is defined as
RM = e2 2πm2ec4
Z L 0
neB~ ·d~l. (2.41)
Numerically,
RM = 812 Z L
0
neBkdl rad m−2, (2.42)
where ne is in units of cm−3,B in units of µG and the path length l is in kpc. By con-vention, RM is positive (negative) for a magnetic field directed towards (away from) the observer. The values of RM can be obtained by measuring the polarization of a linearly polarized radio source located in the background or within the cluster, at three or more wavelengths. Once the contribution of our Galaxy is subtracted, the dominant contribu-tion should come from the ICM. Assuming a density profilene(r), the mean magnetic field Bk can be determined from Eq. 2.42.
Kim et al. (1991) presented magnetic field estimates through Faraday rotation measure in a sample of 53 clusters. Their analysis revealed that relatively high values of magnetic field,B ∼1−5µG, are widespread in the ICM. More recently, Clarke et al. (2001) analyzed data of a sample of 16 clusters, with the same conclusion. They also noted that the RM decreased when the position of the background source was slightly away from the center
2.7. Methods of magnetic field measurement 23 of the cluster, which indicates that the magnetic field strength could be decreasing with radius.
2.7.2 Equipartition magnetic field
In a synchrotron source, the total energy is given by the energy of the relativistic particle (Ue for electrons andUp for protons) plus the energy in the magnetic field (UB),
Utot=Ue+Up+UB. (2.43)
The total magnetic energyUB is UB=
Z
V
uBdV ≡ B2
8πΦV, (2.44)
where the “filling factor” Φ is the fraction of the source volume occupied by the magnetic field. Estimating the amount of energy in the relativistic particles, one can derive the condition of minimal energy. Assuming that the total energy in protons is proportional to that in electrons,Up =kUe, the condition of minimal energy is obtained when
UB= 3
4(1 +k)Ue. (2.45)
The total energy content is therefore minimal whenUB and Ue are approximately equal.
For this reason, the magnetic field value which minimizes the total energy is known as the equipartition magnetic field. The total minimum energy densityUmin=Umin/VΦ (in erg cm−3) can be expressed in terms of observables as
umin= 1.23×10−12(1 +k)4/7ν04α/7(1 +z)(12+4α)/7I04/7d4/7, (2.46) whereν0is the characteristic synchrotron frequency (see Eq. 2.39) in MHz,I0 is the source brightness which is directly observed at the frequencyν0 in mJy arcsec−2,dis the source depth along the line of sight in kpc, andαis the source spectral index. The magnetic field for which the minimum energy assumption is achieved is derived as
Beq= 24π
7 umin 1/2
. (2.47)
One must be aware that the magnetic field values derived with this method rely on several assumptions. Indeed, since the population of relativistic electrons might not be in equi-librium, the assumption of minimal energy might not be valid. Furthermore, the value of the ratio between the relativistic electron and proton energy, k, is uncertain, as well as the volume filling factor Φ.
The magnetic field estimates through equipartition assumption typically find values for the magnetic field which are slightly lower than those derived through Faraday rotation measure (B ∼0.5−1µG, Thierbach et al. (2003), Feretti et al. (1995), Beck et al. (2003)).
2.7.3 Synchrotron/inverse-Compton flux ratio
In the case of the population of relativistic electrons found in clusters of galaxies, the incoming radiation field is dominated by the CMB radiation, whose properties are very well known. In this case, the ratio between radio synchrotron and hard X-ray IC emission is given by (see Sect. 2.2)
Assuming that the non-thermal HXR emission which is observed in several clusters (see Sect. 2.6.2) is due to IC emission from the same electrons that produce radio halos, one can use Eq.2.48 to determine the mean magnetic field value. This method is the most un-biased for the determination of cluster magnetic fields, but it requires a good knowledge of the hard X-ray emission, which is not yet the case.
Using their estimate of the excess flux from the Coma cluster, Fusco-Femiano et al. (1999) estimate a magnetic fieldB ∼0.15µG, which is an order of magnitude less than the value derived from Faraday rotation measure. For comparison, equipartition between the energy emitted through synchrotron and IC is achieved for a magnetic fieldBeq= 3µG. The low values B ≪ Beq obtained with this method indicate that IC energy losses dominate over synchrotron losses.
2.7.4 Reconciling the values obtained with different methods
As shown above, magnetic field estimations obtained with different methods lead to very different results. In the specific case of the Coma cluster, the tentative detection of hard X-ray emission leads to a low magnetic fieldB ∼0.15µG, while measurements of Faraday rotation measure in the central regions of the cluster lead to magnetic field values as high as 6µG. In order to reconcile the values derived using different methods, several mecha-nisms were proposed.
The method of magnetic field measurement through Faraday rotation measure relies on several assumptions. In particular, in order to derive the mean magnetic field value along the line of sight, magnetic field and density profiles must be assumed. Beck et al. (2003) showed that if the magnetic field and density profiles are correlated, which is the case in a turbulent medium, the magnetic field values obtained through Faraday rotation mea-sure can be strongly over-estimated. If this explanation is correct, the low magnetic field values derived through synchrotron/IC flux ratio would be correct, and turbulence in the ICM would induce an over-estimation of the magnetic field. Alternatively, Goldshmidt
& Rephaeli (1993) suggested a radial decrease of the magnetic-field profile, as indicated by Farady rotation measure in some clusters. Given that the measurement of hard X-ray emission was integrated over a wide area because of the large field-of-view of the PDS instrument on boardBeppoSAX, it would mean that the hard X-ray emission would come predominantly from the outer regions of the cluster where the magnetic field is low, while the radio synchrotron emission would be peaked towards the centre.
2.8. Conclusion 25