Abstract
In this paper we derive new expected scaling relations for clusters with giant radio haloes in the framework of the reacceleration scenario in a simplified, but physically motivated, form, namely: radio power (P_{R}) versus size of the radio emitting region (R_{H}), and P_{R} versus total cluster mass (M_{H}) contained in the emitting region and cluster velocity dispersion (σ_{H}) in this region.
We search for these correlations by analysing the most recent radio and Xray data available in the literature for a wellknown sample of clusters with giant radio haloes. In particular we find a good correlation between P_{R} and R_{H} and a very tight ‘geometrical’ scaling between M_{H} and R_{H}. From these correlations P_{R} is also expected to scale with M_{H} and σ_{H} and this is confirmed by our analysis. We show that all the observed trends can be well reconciled with expectations in the case of a slight variation of the mean magnetic field strength in the radio halo volume with M_{H}. A byproduct correlation between R_{H} and σ_{H} is also found, and can be further tested by optical studies. In addition, we find that observationally R_{H} scales nonlinearly with the virial radius of the host cluster, and this immediately means that the fraction of the cluster volume which is radio emitting increases with cluster mass and thus that the nonthermal component in clusters is not selfsimilar.
1 INTRODUCTION
Radio haloes are diffuse Mpc scales synchrotron radio sources observed at the centre of a growing number (∼20) of massive galaxy clusters (see e.g. Feretti 2005, for a review). Radio haloes are always found in merging clusters (e.g. Buote 2001; Schuecker et al. 2001) thus suggesting a link between the dynamical status of clusters and the radio haloes. Observations show that radio haloes are rare; however present data suggest that their detection rate increases with increasing the Xray luminosity of the host clusters and reaches 30–35 per cent for galaxy clusters at z≤ 0.2 and with Xray luminosity larger than 10^{45} h^{−1}_{50} erg s^{−1} (Giovannini, Tordi & Feretti 1999, hereafter GTF99).
Two main possibilities have been so far investigated to explain the radio haloes: (i) the socalled reacceleration models, whereby relativistic electrons injected in the intracluster medium (ICM) are reenergized in situ by various mechanisms associated with the turbulence generated by massive merger events (e.g. Brunetti et al. 2001; Petrosian 2001); (ii) the secondary electron models, whereby the relativistic electrons are secondary products of the hadronic interactions of cosmic rays with the ICM (e.g. Dennison 1980; Blasi & Colafrancesco 1999).
Recently, calculations in the framework of the reacceleration scenario have modelled the connection between radio haloes and cosmological cluster mergers, and investigated the observed correlations between the synchrotron radio power and the Xray properties of the hosting clusters (Cassano & Brunetti 2005, hereafter CB05; Cassano, Brunetti & Setti 2006, hereafter CBS06). Observed correlations relate the radio power at 1.4 GHz (P_{1.4}) with the Xray luminosity (L_{X}), temperature (T) and cluster mass (Colafrancesco 1999; Liang 1999; Feretti 2000; Govoni et al. 2001a; Enßlin & Röttgering 2002; Feretti 2003; CBS06); also a trend between the largest linear size (LLS) of radio haloes and the Xray luminosities of the hosting clusters is found (Feretti 2000). In particular, CBS06 found a correlation between P_{1.4} and the virial mass M_{v} of the hosting clusters, P_{1.4}∝M^{2.9±0.4}_{v}, by combining the P_{1.4}–L_{X} correlation derived from a sample of 17 giant radio haloes with the M_{v}–L_{X} correlation obtained for a large sample of galaxy cluster compiled by Reiprich & Böhringer (2002). However, this correlation, which has been discussed in the particle reacceleration scenario by CBS06, relates quantities which pertain to very different spatial regions: the observed radio emission comes from a radial size R_{H}∼ 3–6 times smaller than the virial radius R_{v}.
In this paper we discuss expected scaling relations for radio haloes in the framework of the reacceleration scenario in its simplest form. Then, we derive a novel observed correlation between the radio power of radio haloes and their extension and a tight ‘geometrical’ correlation between the size of radio haloes and the mass of the cluster within the emitting region. We also present additional correlations which are expected on the basis of these two scalings. Finally, we compare all these observed correlations with the model expectations.
A ΛCDM (H_{0}= 70 km s^{−1}) Mpc^{−1}, Ω_{m}= 0.3, Ω_{Λ}= 0.7) cosmology is adopted.
2 PARTICLE ACCELERATION SCENARIO
2.1 Main features and implications of the reacceleration model
The particle reacceleration model is designed to explain the origin of the synchrotron radio emission diffused on scales larger than that of the cluster cores (giant radio haloes), while the socalled minihaloes and other smaller scale diffuse sources at the cluster centre (e.g. core halo sources) might have a different origin (e.g. Gitti, Brunetti & Setti 2002; Pfrommer & Enßlin 2004, and references therein).
In the conventional particle reacceleration scenario the lower energy electrons (γ∼ 100–300), relic of past activities in the clusters, are reenergized due to resonant and/or nonresonant interactions with the turbulence developed during cluster–cluster mergers. Turbulence and shear flows are expected to amplify the magnetic field in galaxy clusters (e.g. Dolag, Bartelmann & Lesch 2002; Brüggen et al. 2005; Dolag et al. 2005) however, the decay timescale of the magnetic field is expected to be larger than several Gyr (e.g. Subramanian, Shukurov & Haugen 2006) and thus the particle reacceleration process can be thought as occurring in a stationary magnetic field amplified during the previous merging history of the cluster.
The basic features of this model can be briefly summarized as follows.
The average synchrotron spectrum of radio haloes is curved and can be approximated by a relatively steep quasipower law which further steepens at higher frequencies up to a cutoff frequency.
The curved, cutoff spectrum is a unique feature of the reacceleration model, which well represents the typical observed radio halo spectrum, due to the existence of a maximum energy of the radiating electrons (at γ_{max} < 10^{5}) determined by the balance between the energy gains (reacceleration processes) and synchrotron and inverse Compton (IC) losses (e.g. Brunetti et al. 2001; Ohno, Takizawa & Shibata 2002; Kuo, Hwang & Ip 2003; Brunetti et al. 2004). Accordingly, the detection of a radio halo critically depends on cutoff frequency which should be sufficiently larger than the observing frequency. As a consequence, there is a threshold in the efficiency which should be overcome by the reacceleration processes in order to accelerate the electrons at the energies necessary to produce radio emission at the observed frequency in the clusters' magnetic fields. In the mergerrelated scenario it is expected that only mergers between massive galaxy clusters may be able to generate enough turbulence on large scales to power giant radio haloes at GHz frequencies, thus not all clusters which show some merger activity are expected to possess a giant radio halo. In particular, CB05 show that the expected fraction of clusters with radio haloes increases with cluster mass due to a more efficient particle reacceleration process in more massive galaxy clusters, and this is in line with the increase of the fraction of radio haloes with cluster mass which is claimed from the analysis of present radio surveys (e.g. GTF99).
In the reacceleration model radio haloes should be transient phenomena in dynamically disturbed clusters. The timescale of the radio halo phenomena comes from the combination of the time necessary for the cascading of the turbulence from cluster scales to the smaller scales relevant for particle acceleration, of the timescale for dissipation of the turbulence and of the cluster–cluster crossing time.
Present observations suggest that radio haloes are preferentially found in dynamically disturbed systems (e.g. Buote 2001; Govoni et al. 2004). Under the hypothesis that radio haloes form in merging clusters in the hierarchical scenario, Kuo, Hwang & Ip (2004) found that the lifetime of these radio haloes should be ≲1 Gyr to not overproduce the observed occurrence of these sources.
2.2 Predicted scalings for giant radio haloes
In this section we derive scaling expectations for giant and powerful radio haloes in the context of the reacceleration scenario in its simplest form.
The most important ingredient is the energy of the turbulence injected in the ICM. Numerical simulations of merging clusters show that infalling subhaloes induce turbulence (e.g. Roettiger, Loken & Burns 1997; Ricker & Sarazin 2001; Tormen, Moscardini & Yoshida 2004). An estimate of the energy of merginginjected turbulence has been recently derived in CB05 by assuming that a fraction of the PdV work done by the infalling subhaloes is injected into compressible turbulence. They show that the turbulent energy is expected to roughly scale with the thermal energy of the ICM, a result in line with recent analysis of numerical simulations (Vazza et al. 2006, hereafter V06).
Once injected this turbulence is damped by transittimedamping resonance with thermal and relativistic particles (at a rate Γ_{th} and Γ_{rel}, respectively). Since the damping time is shorter than the other relevant timescales (dynamical and reacceleration) the energy density of the turbulence reaches a stationary condition given by , where is the turbulence injection rate (CB05). When reacceleration starts, the bulk of the energy density of compressible modes which is damped by the relativistic electrons goes into the reenergization of these electrons. On the other hand, after a few reacceleration times, in a timescale of the order of the typical age of radio haloes, electrons are boosted at high energies at which radiative losses are severe (∝E^{2}) and the effect of particle reacceleration (∝E) is balanced by that of radiative losses. The electron spectrum gradually approaches a quasistationary condition and it can be assumed that the energy flux of the turbulent modes which goes into relativistic electrons is essentially reradiated via synchrotron and IC mechanisms:
where and are the synchrotron and IC emissivities (and Γ_{th}≫Γ_{rel}, CB05; Brunetti & Lazarian 2007).The ratio simply depends on (B_{cmb}/B_{H})^{2}, where B_{cmb}= 3.2(1 +z)^{2} μG is the equivalent magnetic field strength of the CMB (z, the redshift) and B_{H} the mean magnetic field strength in the radio halo volume, which can be parametrized as with M_{H} the total cluster mass within R_{H} (the average radius of the radio emitting region).
Based on CB05, the injection rate of the turbulence in the radio halo volume can be estimated as , where is the mean density of the ICM in the radio halo volume, v_{i} is the cluster–cluster impact velocity, v^{2}_{i}∝M_{v}/R_{v} and τ_{cros}∝ (R^{3}_{v}/M_{v})^{0.5} is the cluster–cluster crossing time (see CB05) and is constant by definition of virial mass in the cosmological hierarchical model (e.g. Borgani 2006, for a review). In the case R_{H} is larger than the cluster core radius it is v^{2}_{i}∝M_{v}/R_{v}∝M_{H}/R_{H} and σ_{H}, the velocity dispersion inside R_{H}, is σ_{H}≡G M_{H}/R_{H}≈σ^{2}_{v} (for the sake of clarity in Fig. 1 we report a comparison between σ_{H} and σ_{v} for our sample of clusters with radio haloes). Thus we shall simply assume that the injection rate of turbulence in the radio halo volume is . The term Γ_{rel}/Γ_{th} is (Brunetti 2006; Brunetti & Lazarian 2007), where T is the temperature of the cluster gas, and ε_{rel}/ε_{th} is the ratio between the energy densities in relativistic particles and in the thermal plasma. Although this ratio might reasonably vary from cluster to cluster, we shall assume that it does not appreciably change in any systematic way with cluster mass (or temperature), at least if one restricts to the relatively narrow range in cluster mass spanned by clusters with giant radio haloes (see also the results from numerical simulations for cosmic rays in Jubelgas et al. 2006). Then from equation (1) the total emitted radio power is
where we have taken and . The expression (Fig. 2) is constant in the asymptotic limit B^{2}_{H}≫B^{2}_{cmb} or in the simple case in which the rms magnetic field in the radio halo region is independent of the cluster mass. For B^{2}_{H}≪B^{2}_{cmb} one has that , thus in the general case the expected scaling is steeper (slightly for B_{H} of the order of a few μG) than that obtained by assuming a constant .It is important to stress here that the expression in equation (2) is a general theoretical trend which implies simple scaling relations. Indeed, by taking and under the assumption that the mass scales with R_{H} as M_{H}∝R^{α}_{H} (see also Section 3.2), equation (2) (with constant) entails the correlations:
the effect of a nonconstant is a steepening (although not substantial for ∼μ G fields) of these scalings.3 OBSERVED SCALING RELATIONS IN CLUSTERS WITH RADIO HALO
Motivated by the theoretical expectations outlined in the previous section, we have searched for the predicted scaling relations in the available data set for giant radio haloes. Operatively, we will first discuss the case of the P_{R}–R_{H} scaling expected in equation (3), which will allow us to address the tricky point of the measure of R_{H} in radio haloes, and then we will show that a tight observational R_{H}–M_{H} scaling exists for radio haloes. Then, we will discuss and verify the byproduct observational scalings between P_{R}–M_{H} and P_{R}–σ_{H}.
We consider a sample of 15 clusters with known giant radio haloes (R_{H}≳ 300 kpc) already analysed in CBS06, with the exclusion of CL0016+16, due to the lack of good radio images to measure R_{H}, and of A754, due to very complex radio structure. References for 14 giant radio haloes are given in CBS06, while for A2256 we use the more recent radio data from Clarke & Enßlin (2006). In Table 1 we report the relevant observed and derived quantities for our sample.
Cluster's name  z  log(P_{1.4}) (W Hz^{−1})  log(R_{H}) (kpc h^{−1}_{70})  log(M_{H}) (M_{⊙} h^{−1}_{70})  log(σ^{2}_{H}) (km^{2} s^{−2}) 
1E50657−558  0.2994  25.45 ± 0.03  2.84 ± 0.04  14.83 ± 0.07  6.63 ± 0.08 
A2163  0.2030  25.27 ± 0.01  3.01 ± 0.04  15.02 ± 0.05  6.65 ± 0.07 
A2744  0.3080  25.23 ± 0.04  2.90 ± 0.06  14.76 ± 0.10  6.49 ± 0.11 
A2219  0.2280  25.09 ± 0.02  2.84 ± 0.05  14.66 ± 0.08  6.46 ± 0.09 
A1914  0.1712  24.72 ± 0.02  2.77 ± 0.04  14.68 ± 0.05  6.54 ± 0.06 
A665  0.1816  24.60 ± 0.04  2.84 ± 0.04  14.57 ± 0.09  6.37 ± 0.10 
A520  0.2010  24.59 ± 0.04  2.61 ± 0.04  14.21 ± 0.10  6.24 ± 0.11 
A2254  0.1780  24.47 ± 0.04  2.61 ± 0.03  –  – 
A2256  0.0581  23.91 ± 0.08  2.63 ± 0.04  14.17 ± 0.09  6.18 ± 0.11 
A773  0.2170  24.24 ± 0.04  2.71 ± 0.03  14.43 ± 0.05  6.36 ± 0.06 
A545  0.1530  24.17 ± 0.02  2.58 ± 0.03  14.08 ± 0.30  6.13 ± 0.30 
A2319  0.0559  24.05 ± 0.04  2.63 ± 0.02  14.30 ± 0.03  6.30 ± 0.03 
A1300  0.3071  24.78 ± 0.04  2.76 ± 0.14  14.54 ± 0.17  6.42 ± 0.22 
Coma (A1656)  0.0231  23.86 ± 0.04  2.53 ± 0.01  14.12 ± 0.03  6.22 ± 0.03 
A2255  0.0808  23.95 ± 0.02  2.65 ± 0.03  14.16 ± 0.07  6.14 ± 0.07 
Cluster's name  z  log(P_{1.4}) (W Hz^{−1})  log(R_{H}) (kpc h^{−1}_{70})  log(M_{H}) (M_{⊙} h^{−1}_{70})  log(σ^{2}_{H}) (km^{2} s^{−2}) 
1E50657−558  0.2994  25.45 ± 0.03  2.84 ± 0.04  14.83 ± 0.07  6.63 ± 0.08 
A2163  0.2030  25.27 ± 0.01  3.01 ± 0.04  15.02 ± 0.05  6.65 ± 0.07 
A2744  0.3080  25.23 ± 0.04  2.90 ± 0.06  14.76 ± 0.10  6.49 ± 0.11 
A2219  0.2280  25.09 ± 0.02  2.84 ± 0.05  14.66 ± 0.08  6.46 ± 0.09 
A1914  0.1712  24.72 ± 0.02  2.77 ± 0.04  14.68 ± 0.05  6.54 ± 0.06 
A665  0.1816  24.60 ± 0.04  2.84 ± 0.04  14.57 ± 0.09  6.37 ± 0.10 
A520  0.2010  24.59 ± 0.04  2.61 ± 0.04  14.21 ± 0.10  6.24 ± 0.11 
A2254  0.1780  24.47 ± 0.04  2.61 ± 0.03  –  – 
A2256  0.0581  23.91 ± 0.08  2.63 ± 0.04  14.17 ± 0.09  6.18 ± 0.11 
A773  0.2170  24.24 ± 0.04  2.71 ± 0.03  14.43 ± 0.05  6.36 ± 0.06 
A545  0.1530  24.17 ± 0.02  2.58 ± 0.03  14.08 ± 0.30  6.13 ± 0.30 
A2319  0.0559  24.05 ± 0.04  2.63 ± 0.02  14.30 ± 0.03  6.30 ± 0.03 
A1300  0.3071  24.78 ± 0.04  2.76 ± 0.14  14.54 ± 0.17  6.42 ± 0.22 
Coma (A1656)  0.0231  23.86 ± 0.04  2.53 ± 0.01  14.12 ± 0.03  6.22 ± 0.03 
A2255  0.0808  23.95 ± 0.02  2.65 ± 0.03  14.16 ± 0.07  6.14 ± 0.07 
3.1 Radio power versus sizes of radio haloes
A direct scaling between P_{R} and R_{H} for radio haloes is not reported in the literature. We want to check the existence of a P_{R}–R_{H} correlation by making use of directly measurable quantities, such as the power and the radius at 1.4 GHz. In the present literature it is customary to use the LLS, obtained from the largest angular size measured on the radio images as the largest extension of the 2σ or 3σ contour level, as a measure of the radio emitting region (e.g. Giovannini & Feretti 2000; Kempner & Sarazin 2001). Since a fraction of radio haloes in our sample is characterized by a nonspherical morphology, meaning a noncircular projection on the plane of the sky, an adequate measure of a radio halo's size can be obtained by modelling the emitting volume with a spherical region of radius and R_{max} being the minimum and maximum radii measured on the 3σ radio isophotes. In this way we have derived the R_{H} values for all 15 radio haloes, as reported in Table 1, by making use of the most recent radio maps available in literature. In Fig. 3 we report P_{1.4} versus R_{H} for our sample. We find a clear trend with R_{H} increasing with P_{1.4}, that is, the more extended radio haloes are also the most powerful. The best fit of this correlation is given by
A Spearman test yields a correlation coefficient of ∼0.84 and a s= 0.000 11 significance, indicative of a relatively strong correlation.3.1.1 Uncertainties in the measure of the size of radio haloes
The dispersion of the P_{1.4}–R_{H} correlation is relatively large, a factor of ∼2 in R_{H}, and this may be due to the errors associated with the measure of R_{H}. Indeed, radio haloes are low brightness diffuse radio sources which fade away gradually, until they are lost below the noise level of a given observation. Thus, the measure of a physical size is not obvious and, in any case, it needs to be explored with great care. However, what is important here is not so much the precise measure of R_{H} for each radio halo, but rather the avoidance of selection effects which might force a correlation.
In principle the sensitivity in the different maps may play a role because the most powerful radio haloes are also the most bright ones (Feretti 2005), and thus they might appear more extended then the less powerful radio haloes in the radio maps. To check if this effect is present, in Fig. 4 we plot the ratios between the average surface brightness of each radio halo in our sample and the rms of each map used to get R_{H}. It is clear that there is some scattering in the distribution which would yield a corresponding dispersion in the accuracy of R_{H}, however, and most importantly, the ratios are randomly scattered, and there is no trend with R_{H}, that is, fainter radio haloes are usually imaged with a higher sensitivity and thus the P_{1.4}–R_{H} correlation cannot be forced by the maps used to derive R_{H}.
An additional effort in assessing the reliability of R_{H} (and of the P_{1.4}–R_{H} correlation) would be to measure the radial brightness profile of regular radio haloes which are not severely affected by powerful and extended radio sources. In our sample it is feasible to obtain accurate radial profiles from available data for the following radio haloes: A2163, A2255, A2744, A545 and A2319. We take the data at 1.4 GHz (Feretti, Giovannini & Bohringer 1997; Feretti et al. 2001; Govoni et al. 2001a; Bacchi et al. 2003; Govoni et al. 2005, respectively), and use the software package synage++ (Murgia 2001) to extract the radial brightness profiles, after subtraction of the embedded radio sources.
In Fig. 5 we report the integrated brightness profiles of these radio haloes. It is seen that the profiles flatten with distance from the respective cluster centres, indicating that basically all the extended radio emission is caught and that it is possible to extract an accurate physical size. In Fig. 6 we report for these five radio haloes the comparison between R_{H}, estimated directly from 3σ radio isophotes (see the above definition), and R_{85} and R_{75}, that is, the radii, respectively, containing the 85 and 75 per cent of the flux of the radio haloes. We apply the same procedure also to the case of the Coma cluster at 330 MHz for which a brightness profile and radio map were already presented in the literature (Govoni et al. 2001b). For Coma at 330 MHz we find R_{H}∼ 520 h^{−1}_{70} kpc and R_{85}∼ 610 h^{−1}_{70} kpc, which set Coma in a configuration similar to that of the other clusters in Fig. 6.
The linear, almost onetoone correlation between R_{H} and R_{85} and the relatively small dispersion, consistent with the uncertainties in the profiles due to source subtraction, prove that our definition of R_{H} is a simple but representative estimate of the physical size of radio haloes.
We note that the sensitivities of the radio maps, the physical sizes R_{85} and powers P_{1.4} of the five regular haloes are representatives of the values encompassed by the full radio halo sample. Moreover, for these five radio haloes alone we find P_{1.4}∝R^{4.25±0.63}_{85}, fully consistent with the P_{1.4}–R_{H} correlation obtained for the total sample.
3.1.2 Possible biases in the selection of the sample
Since the P_{1.4}–R_{H} correlation is the driving correlation, one has to check whether this correlation may not be forced by observational biases due to the selection of the radio halo population itself. Indeed the great majority of these radio haloes have been discovered by followups of candidates, mostly identified from the NVSS which is surface brightness limited for resolved sources1 and this may introduce biases in the selected sample.
The upper bound of the correlation is likely to be solid: objects as powerful as those at the upper end of the correlation (log P_{1.4}≥ 25) but with small R_{H} (similar to that of radio haloes in the lower end of the correlation) should appear in the NVSS up to the largest redshifts of the sample, since, even at z∼ 0.3, they should be ≥10 times brighter than the lowpower radio haloes in the correlation and extended (∼2.5 arcmin). As a matter of fact A545 (z= 0.15) and A520 (z= 0.2), which are among the smaller radio haloes in our sample, are already detected in the NVSS up to a redshift 0.2 and there is no reason why objects with similar extension, but ∼8–10 times brighter than A545 and A520, should not have been detected at z≤ 0.3.
The lower bound of the correlation deserves much care since the brightness limit of the NVSS may play some role. It is clear that present surveys may significantly affect the selection of the faint end of the radio halo population. However, Feretti (2005) and Clarke (2005) have already concluded that the typical brightness of the powerful and giant radio haloes are well above the detection limit.
In any case, a brightness limit should drive a P_{1.4}∝R^{2}_{H} correlation, much flatter then the observed one. In order to provide a further compelling argument against observational biases, we have run Monte Carlo simulations. To this end we have randomly extracted brightness values of hypothetical radio haloes within a factor of ∼5 interval (consistently with the range spanned in our sample) above a given minimum brightness and each time randomly assigned R_{H} and z among the observed values. In Fig. 7 we report the distribution of the P_{1.4}–R_{H} slopes obtained with our Monte Carlo procedure and note that this distribution is peaked around ∼2.5 with a dispersion of ±0.4 (this is somewhat steeper than the expected P_{1.4}∝R^{2}_{H} due to the wellknown redshift effect, however small given the small redshift range of our sample). The values of the slopes from the Monte Carlo procedure are far from the observed value (Fig. 7) and a statistical test allows us to conclude that the probability that the observed P_{1.4}–R_{H} correlation is forced by observational biases is ≲ 0.05 per cent.
3.2 Geometrical M_{H}–R_{H} scaling for radio haloes
The existence of a possible tight scaling between the size of radio haloes and the cluster mass within the emitting region is not reported in the literature. Yet an observational M_{H}–R_{H} scaling may be important to relate virial quantities σ^{2}_{v}=G M_{v}/R_{v}(≈G M_{H}/R_{H}=σ^{2}_{H}) with quantities (R_{H} and M_{H}) which refer to the emitting region, and to test simple model expectations (Section 2.2).
At this stage of the paper, the main difficulty concerns the measure of the cluster mass inside a volume of size R_{H}. Here the only possibility is to use the Xray mass determination based on the assumption of hydrostatic equilibrium. Nevertheless, radio halo clusters are not wellrelaxed systems and thus the assumption of hydrostatic equilibrium and spherical symmetry may introduce sizeable errors in the mass determination. Several numerical simulation studies, which have been undertaken in order to determine whether the above assumptions introduce significant uncertainties in the mass estimates, indicate that in the case of merging clusters the hydrostatic equilibrium method might lead to errors up to 40 per cent of the true mass, which can be either overestimated or underestimated (e.g. Evrard, Metzler & Navarro 1996; Roettiger, Burns & Loken 1996; Schindler 1996; Rasia et al. 2006). This would cause an unavoidable scattering in the determination of the mass in our sample, although there are indications that a better agreement between the gravitational lensing, Xray and optically determined cluster masses is achieved on scales larger than the Xray core radii (e.g. Wu 1994; Allen 1998; Wu et al. 1998), which is the case under consideration (R_{H} > r_{c}).
However, what is important here is that the mass determination does not introduce systematic errors which depend on the mass itself and which may thus affect the real trend of the P_{1.4}–M_{H} correlation. We thus compute the total gravitational cluster mass within the radius R_{H} as
where r_{c} is the core radius, T the isothermal gas temperature and β the ratio between the kinetic energy of the dark matter and that of the gas (βmodel; e.g. Sarazin 1986). We have excluded from our analysis A2254 for which no information on the βmodel is available. For the remaining 14 cluster references are given in CBS06. From equation (7) one has that M_{H}∝R_{H} for R_{H}≫r_{c} and M_{H}∝R^{3}_{H} for R_{H}≪r_{c}. In Fig. 8 we plot R_{H} versus M_{H} for our sample: we find M_{H}∝R^{2.17±0.19}_{H}, which falls in between the above asymptotic expectations.3.3 Radio power versus mass and velocity dispersion
In principle, the two correlations discussed so far for giant radio haloes, P_{1.4}–R_{H} and M_{H}–R_{H}, imply the existence of correlations between P_{1.4}–M_{H} and P_{1.4}–σ_{H}. In particular P_{1.4} is expected to roughly scale as M^{1.9–2}_{H}. In Fig. 9 we report P_{1.4} versus M_{H} for our sample together with the best fit: P_{1.4}∝M^{1.99±0.22}_{H}, which is indeed in line with the above expectation. A Spearman test of this correlation yields a correlation coefficient of ∼0.91 and s= 7.3 × 10^{−6} significance, indicative of a very strong correlation.
P_{1.4} is expected to scale with σ_{H} and we found for our sample a bestfitting correlation: P_{1.4}∝ (σ^{2}_{H})^{4.64±1.07}; a Spearman test yields a correlation coefficient of ∼0.89 and to s= 2 × 10^{−5} significance, indicative of a very strong correlation.
Finally, as a byproduct of all the derived scalings, it is worth noticing that also a trend between R_{H} and σ_{H} is expected (Fig. 10). This finding might also be tested by observations in the optical domain which can directly constrain the velocity dispersion.
4 IMPLICATIONS OF THE DERIVED SCALINGS
Given that the larger radio haloes are also the most powerful ones and are hosted in the most massive clusters, we expect that the size of a giant radio halo should scale with the size of the hosting cluster. We estimate for each cluster of our sample the virial radius (R_{v}) by combining the virial mass–Xray correlation (M_{v}–L_{X}; CBS06) and the virial radius–virial mass relation (e.g. Kitayama & Suto 1996). This method allows to reduce the effect of scattering due to the uncertainties in the mass measurements (and thus in the R_{v}) of merging galaxy clusters (see discussion in CBS06). In Fig. 11 we plot R_{H} versus R_{v} for our sample. The best fit gives R_{H}∝R^{2.63±0.50}_{v}, that is, a pronounced nonlinear increase of the size of the radio emitting region with the virial radius. A Spearman test yields a correlation coefficient of ∼0.74 and s= 0.0023 significance, indicative of a relatively strong correlation, albeit less strong than the other correlations found in this paper.
Given that massive clusters are almost selfsimilar (e.g. Rosati, Borgani & Norman 2002) one might have expected that R_{H} scales with R_{v} and that the radial profiles of the radio emission are selfsimilar. On the contrary, our results prove that selfsimilarity is broken in the case of the nonthermal cluster components. This property of radio haloes was also noticed by Kempner & Sarazin (2001), which used a sample of radio haloes taken from Feretti (2000) and found evidence for a trend of the LLS with the Xray luminosity in the 0.1–2.4 keV band, LLS ∝L^{1/2}_{X}, while a flatter scaling, LLS ∝R_{v}∝L^{1/6}_{X} is expected in the case of a selfsimilarity. Their results imply R_{H}∝R^{3}_{v}; if one takes R_{H}≈ LLS, this is substantially in line with our findings. It is also worth noticing that Xray–radio comparison studies of a few radio haloes indicate that the profile of the radio emission is typically broader than that of the thermal emission (e.g. Govoni et al. 2001b). The two ingredients which should be responsible for the break of the selfsimilarity are the distributions of relativistic electrons and magnetic fields. In magnetohydrodynamic (MHD) cosmological simulations (Dolag et al. 2002, 2005) it is found that the magnetic field strength in cluster cores increases nonlinearly with cluster mass (temperature). This implies that the radio emitting volume should increase with cluster mass because the magnetic field at a given distance from the centre increases with increasing the mass. A detailed analysis of the magnetic field profiles of massive clusters from MHD simulations could be of help in testing if the magnetic field is the principal cause of the break of the selfsimilarity.
5 PARTICLE REACCELERATION MODEL AND OBSERVED SCALINGS
Although we have been guided by the analysis of equation (2) to predict the existence of scaling relationships, the observed correlations derived in Section 3 are actually independent from the form of this equation. To test equation (2) against the observed quantities of our sample of radio haloes we make use of the monochromatic P_{1.4} instead of the unavailable bolometric P_{R}. This is possible because the typical spectral shape of radio haloes is α_{r}≈ 1.1–1.2[] and thus the Kcorrection is not important (CBS06).
In Fig. 12 we report P_{1.4} versus M_{H} σ^{3}_{H}. The best fit gives P_{1.4}∝ (M_{H} σ^{3}_{H})^{1.24±0.19}. The observed scaling is slightly steeper, but still in line with the linear scaling expected from equation (2) for constant (dashed line). As already discussed in Section 2.2, is constant for B^{2}_{H}≫B^{2}_{cmb} or in the case in which the rms magnetic field in the radio halo region is quite independent from the cluster mass (small b_{H}), while formally a nonconstant always implies a steepening of the P_{1.4}–M_{H}σ^{3}_{H} scaling. Namely, in the case of ∼μ G magnetic fields, by combining equation (2) with the observed M_{H}–R_{H} correlation (Section 3.2, Fig. 8), one has that the best fit in Fig. 12 is fulfilled by the model expectations for 0.05 ≤b_{H}≤ 0.39.
In principle the fit can be used to set constraints on the values of the theoretical parameters entering the normalization of equation (2), (namely ε_{CR}/ε_{th}, and the fraction of the P d V work which goes into turbulence), but we will not pursue this any further here (see CB05 for a discussion).
It is important to stress that not only the trend in Fig. 12, but also the existence of the correlations found in Section 3 could have been predicted on the basis of the reacceleration model (Section 2, equations 3–5) under the very reasonable assumption that M_{H}∝R^{α}_{H}. Indeed, if one uses the observed scaling M_{H}∝R^{2.17±0.19}_{H} to fix the parameter α, from equation (2), and assuming the most simple case in which is constant, one finds P_{1.4}∝R^{3.9}_{H} and P_{1.4}∝M^{1.8}_{H}, which are actually consistent (within the dispersion) with the observed correlations (Section 3); as in the case of the trend in Fig. 12, an even better fulfilment of all these correlations is obtained for a slightly nonconstant .
A relevant point which derives from the comparison of model expectation and observed correlations (unless B^{2}_{H}≫B^{2}_{cmb}) is that, at least under our simplified approach (Section 2.2), B_{H} does not critically depend on cluster mass inside R_{H} and that radio haloes might essentially select the regions of the cluster volume in which the magnetic field strength is above some minimum value (say ∼ μ G level). It is important to note that a roughly constant B_{H} with cluster mass does not contradict the scaling of B, averaged in a fixed volume, with cluster mass (or temperature) found in the MHD simulations (B within the cluster core radius, r_{c}∼ 300 h^{−1}_{70} kpc), and also found in CBS06 (B averaged within a fixed region of ∼720 h^{−1}_{70} kpc size), because the magnetic field B_{H} is averaged over a volume of radius R_{H} that becomes substantially larger than the core radius with increasing the cluster mass (R_{H}/r_{c} goes from ∼1.1 to ∼3 with increasing cluster mass in our sample).
6 SUMMARY AND CONCLUSIONS
The particle reacceleration model is a promising possibility to explain the origin and properties of the giant radio haloes (e.g. Blasi 2004; Brunetti 2004; Hwang 2004; Feretti 2005, for recent reviews).

In its simplest form, as assumed here (Section 2), it predicts a very simple relationship (equation 2) between the total radio power P_{R}, the total mass M_{H} within the radio halo, the gas velocity dispersion σ_{H} and the average magnetic field B_{H}. Under the assumption of a tight scaling between M_{H} and the size R_{H}, and that the gas is in gravitational equilibrium, equation (2) naturally translates into simple scaling relations: P_{R}–R_{H}, P_{R}–M_{H} and P_{R}–σ_{H} (equations 3–5).
Motivated by the above theoretical considerations, we have searched for the existence of this type of correlations by analysing a sample of 15 galaxy clusters with giant radio haloes. A most important point here is the measure of the size R_{H}, in itself a nontrivial matter, since the brightest radio haloes may appear more extended in the radio maps and this might force artificial correlations with radio power. A careful analysis of published 15GHz radio maps of the radio haloes of our sample shows that this effect is not present (Section 3.1.1). From the same data set we derive a meaningful estimate of the radius for each radio haloes. We also show that our procedure leads to estimates fully consistent with the measurements from the brightness profiles worked out from the data for the five most regular radio haloes; this consistency holds over the total range spanned by R_{H} in our sample (Section 3.1.1).

We obtain a good, new correlation (correlation coefficient ∼0.84) between the observed radio power at 1.4 GHz and the measured size of the radio haloes in the form P_{1.4}∝R^{4.18±0.68}_{H} (Section 3.1). In Section 3.1.2 we discuss in detail several selection effects which might affect this correlation and conclude that it is unlikely that the observed correlation is driven by observational biases.

We address observationally also the presence of a tight scaling between M_{H} and R_{H} and this allows us to relate virial quantities to quantities in the emitting region.

The presence of the P_{R}–R_{H} and M_{H}–R_{H} correlations implies also other correlations. We derive relatively strong correlations (Section 3.3) in the form: P_{1.4}∝M^{1.99±0.22}_{H} and P_{1.4}∝ (σ^{2}_{H})^{4.64±1.07}, and, as a byproduct, also σ^{2}_{H}∝R^{0.90±0.25}_{H}.
A correlation between the size R_{H} and the cluster virial radius, R_{v}, is qualitatively expected in the framework of the particle reacceleration model.

In Section 4 we compare R_{H} versus R_{v} for our sample of clusters with giant radio haloes, obtaining the nonlinear trend R_{H}∝R^{2.63±0.50}_{v}, that is, the fraction of the cluster volume that is radio emitting significantly increases with the cluster mass. This break of the selfsimilarity, in line with previous suggestions (e.g. Kempner & Sarazin 2001), points to the changing distributions of the magnetic fields and relativistic electrons with cluster mass and, as such, is potentially important in constraining the physical parameters entering the hierarchical formation scenario, such as the turbulence injection scale and the magnetic field strength and profile. Finally, we note that, by combining the R_{H}–R_{v} and P_{1.4}–R_{H} correlations, one easily derives P_{1.4}∝M^{3}_{v}, which is consistent with previous findings (P_{1.4}∝M^{2.9±0.4}_{v}; CBS06).

These observed correlations are well understood in the framework of the particle reacceleration model. Indeed, we show that the theoretical expectation (equation 2) is consistent with the data (see Fig. 12). Assuming a simple constant form for in equation (2) and the observed M_{H}–R_{H} scaling, which is necessary to fix the model parameter α (Section 2), the model expectations (equations 4, 3 and 5) naturally translates into P_{1.4}∝R^{3.9}_{H}, P_{1.4}∝M^{1.8}_{H} and P_{1.4}∝ (σ^{2}_{H})^{3.4} correlations, all consistent (within the dispersion) with the observed correlations; an even better fulfilment of all these correlations is obtained for a slightly nonconstant , which corresponds to ≈μG field in the radio halo region. Unless it is B^{2}_{H}≫B^{2}_{cmb}, from the comparison of model expectations and observations we conclude that B_{H} should not strongly depend on M_{H}, and thus in our simplified scenario (Section 2.2) radio haloes essentially trace the regions of ≈ μG fields in galaxy clusters in which particle acceleration is powered by turbulence.
To conclude, the particle reacceleration model, closely linked to the development of the turbulence in the hierarchical formation scenario, appears to provide a viable and basic physical interpretation for all the correlations obtained so far with the available data for giant radio haloes. Future deep radio surveys and upcoming data from LOFAR and LWA will be crucial to improve the statistics and to provide further constraints on the origin of radio haloes.
RC acknowledges the MPA in Garching for the hospitality during the preparation of this paper. We thank Matteo Murgia for the use of the synage++ program. We thank Luigina Feretti for providing the data for A2163, A545 and A2319, and Marco Bondi for useful discussions. This work is partially supported by MIUR and INAF under grants PRIN2004, PRIN2005 and PRININAF2005.