Relics and Halos: homogeneous CRIs, evolving magnetic fields

Common origin for radio relics and halos: galaxy cluster-wide, homogeneous cosmic-ray distribution, and evolving magnetic fields

Abstract

Some galaxy clusters show diffuse radio emission in the form of peripheral relics (so far attributed to primary, shock-(re)accelerated electrons) or central halos. Analysing radio and X-ray data from the literature, we find new connections between halos and relics, such as a universal linear relation between their peak radio brightness and the gas column density. Our results indicate that halos, relics, and halo–relic bridges in a cluster, all arise from the same, homogeneous cosmic ray (CR) ion (CRI) distribution. We analytically derive the signature of synchrotron emission from secondary electrons and positrons (CREs) produced in hadronic CRI collisions, for an arbitrary magnetic field evolution. In our model, flat spectrum halos (both giant and minihalos) arise from steady-state magnetic fields, whereas relics and steep halos reflect recent or irregular magnetic growth. This naturally explains the properties of halos, relics, and the connections between them, without invoking particle (re)acceleration in weak shocks or turbulence. We find CRI energy densities in the range , with a spectral index , and identify an magnetic fraction in some halos and behind relics, as far as from the cluster’s centre. The CRI homogeneity suggests strong CR diffusion, . The strong magnetisation imposes strict upper limits on CRE (re)acceleration in weak shocks (efficiency ) and turbulence; indeed, each weak shock slightly lowers the energy fraction of flat CRs.

keywords:
galaxies: clusters: general — galaxies: clusters: intracluster medium — X-rays: galaxies: clusters — radio continuum: general — magnetic fields
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1 Introduction

As the largest gravitationally bound structures in the Universe, galaxy clusters are the focus of intense cosmological and astrophysical research. Nonthermal radiation from clusters was observed in the radio band (for review, see Feretti 2005) and in hard X-rays (for review, see Rephaeli et al. 2008), and is expected to be observed in -rays in the near future (by the 5-year Fermi mission; see Keshet et al. 2003).

Such nonthermal signals trace the cosmic rays (CRs) and magnetic fields permeating the intracluster medium (ICM). These nonthermal components play an important role in the evolution of clusters on multiple scales, affecting their dynamical and thermal structure, for example by modifying the transport and dissipation processes. The distributions of CRs and magnetic fields in the ICM hold a unique record of past dynamical processes, such as the history of merger-induced shocks and turbulence in the cluster. Modeling these component also constrains the poorly understood processes of particle acceleration and plasma magnetisation.

A fair fraction of the hot galaxy clusters ( of the clusters with X-ray luminosty ; Giovannini et al. 2002) show extended, nonthermal radio emission with low surface brightness, which is not associated with any particular member galaxy. This is believed to be synchrotron radiation emitted by CR electrons or positrons (CREs), injected locally into the ICM and gyrating in its pervasive magnetic fields. Arguably, such radio observations hold more information regarding the nonthermal components of the ICM than presently available in any other band.

1.1 Source classification: halos and relics

ICM radio sources are broadly classified, according to their location, morphology, and polarisation, as giant halos (GHs; also known as a cluster-wide halos), minihalos (MHs; or core halos), or relics (Feretti & Giovannini 1996). In general, halos (both GHs and MHs) are regular, unpolarised emission around the cluster’s centre, whereas relics are peripheral, polarised, typically elongated, and thought to be associated with shocks. For a recent review, see Ferrari et al. (2008). (We use the conventional term “relic”, although the recently suggested terms “flotsam” or “gischt” may be more appropriate; see Kempner et al. 2004).

GHs are found in the centres of merger, non-cool core clusters. They are typically unpolarised, and show a regular morphology which follows the thermal plasma. Their spectral indices lie in the range (flat halos) or (steep halos), where is the specific radio power and is the frequency. GHs extend over large, scales, farther than the distance a CRE can cross before cooling. Therefore, CREs must be injected locally and continuously into the ICM. Two types of models have been proposed for CRE injection in GHs: (i) secondary production by hadronic collisions between CR ions (CRI) and the ambient plasma (Dennison 1980; Blasi & Colafrancesco 1999); and (ii) in-situ turbulent acceleration or reacceleration of primary CREs (Enßlin et al. 1999; Brunetti et al. 2001; Petrosian 2001). It was recently shown (Kushnir et al. 2009; Keshet & Loeb 2010) that the radio–X-ray correlations in GH luminosity (Brunetti et al. 2007) and in surface brightness (Govoni et al. 2001; Keshet & Loeb 2010) strongly support the first, secondary CRE model, and imply that the defining property of GHs is a strongly magnetised, ICM. (For a different view, see Brunetti et al. (2009).) This model reproduces the spectral, morphological and energetic properties of flat GHs (Keshet & Loeb 2010, henceforth KL10). Independent measurements of within halos are presently not sufficiently precise to test this connection; low-significance evidence for higher magnetisation in halo clusters was reviewed in KL10.

MHs are found in the centres of more relaxed, cool-core clusters (CCs). They extend roughly over the cooling region (Gitti et al. 2002), encompassing up to a few percent of the typical GH volume, and often overlap the radio emission from an active galactic nucleus (AGN). They resemble miniature versions of flat GHs, typically being unpolarised, regular, and spectrally flat with . They show radio–X-ray correlations consistent with those of GHs, and a similar ratio between the radio and X-ray surface brightness (KL10). This indicates that they arise from the same mechanism as GHs: secondary CREs losing most of their energy to synchrotron radiation in highly magnetised cores (KL10). This conclusion is supported by the morphological association between MH edges and cold fronts (CFs), reported by Mazzotta & Giacintucci (2008). Such CFs, present in most CCs (Markevitch & Vikhlinin 2007), were identified as tangential discontinuities lying above (i.e. at larger distances from the cluster’s centre) regions magnetised by bulk shear flow (Keshet et al. 2010). We do not focus on MHs here; for a discussion of their properties, see KL10.

As opposed to halos, radio relics are typically polarised (at levels), irregular (often elongated), and far from the cluster’s centre (up to a few Mpc). Different classification schemes have been proposed for relics (Giovannini & Feretti 2004; Kempner et al. 2004); here we focus on relics which are true ICM emission, not associated with any galaxy or AGN. All relic clusters which have been carefully analysed in the optical or X-ray bands show evidence of a recent merger (Giovannini & Feretti 2004). These properties, and the absence of nearby CRE sources, suggest that relics are associated with merger shocks propagating through the ICM. Indeed, in some cases (e.g., in A521; Giacintucci et al. 2008), a relic was found to coincide with an X-ray brightness edge consistent with a shock front. Thus far, it has been thought that CREs in relics must be primary particles, injected by the weak collisionless shock. Two mechanisms were proposed for such injection: (i) diffusive shock acceleration (DSA) in merger or accretion shocks (Ensslin et al. 1998); and (ii) adiabatic compression of fossil radio plasma caught by a shock (a “radio phoenix”; Enßlin & Gopal-Krishna 2001). However, a secondary CRE model for relics was never ruled out (D. Kushnir 2008, private communications).

1.2 Motivation: observational inconsistencies and unconstrained assumptions

Accumulating evidence indicates that the present modeling of these ICM radio sources is, at best, incomplete. Observationally, there are several results that are peculiar or inconsistent with the present models. This includes multiple similarities between halos and relics, radio bridges sometimes observed to connect a halo and a relic, halos with a steep spectrum, the remarkably similar spectrum at the edges of all relics, and the selective appearance of spectral steepening inward of relics. Interpreting these results under the assumption that halos are produced by secondary CREs while relics arise from primary CREs requires fine tuning and implausible assumptions, as we show in §2. The problem becomes worse if one assumes that halos arise from turbulent-accelerated primary CREs, as this leads to additional inconsistencies and unnatural assumptions, as discussed in KL10 and below.

On the theoretical side, the present models rely on some questionable, unconstrained assumptions. Primary CRE models are sensitive to the poorly understood processes of particle acceleration and magnetisation in weak collisionless shocks and in turbulence. Halo primary CRE models thus make multiple assumptions which are not independently tested or constrained (see, for example Brunetti & Lazarian 2007). Models assuming pristine particle acceleration at merger shocks typically compute the spectrum from diffusive shock acceleration (DSA) theory, but the predicted spectrum was not confirmed for weak shocks and the acceleration efficiency is unknown; only loose upper limits on the efficiency are available. Analogously, due to the poor understanding of CRI diffusion through the ICM, secondary CRI models typically make some simplifying assumptions regarding the distribution of CRIs or magnetic fields within the cluster.

In addition, previous secondary CRE models assume a steady-state magnetic field, in the sense that the magnetic energy density evolves on a timescale much longer than the CRE cooling time, . These models assume, in addition, a steady-state CRE injection rate. These assumptions are violated for example in the vicinity of shocks and during the onset of turbulence. Halo models are sensitive to these assumptions, in particular when the halos are young, near the edge of halos, and near shocks embedded in the halo or at its edge, as observed in several cases.

1.3 A unified halo–relic, secondary CRE model

We find that a single secondary CRE model simultaneously accounts for all types of diffusive radio emission from the ICM, including GHs, MHs, relics, and halo–relic bridges. We begin in §3 by studying radio data, extracted from the literature, for all known relic clusters and for a sample of halo clusters. We parameterise the distributions of CREs and magnetic fields as unknown power-law functions of the bulk plasma (for brevity: gas) density, thus avoiding unnecessary assumptions regarding the nature of the CREs and the distributions of CREs, magnetic fields, and gas.

A useful diagnostic of diffuse radio sources is the ratio between radio and X-ray emission, as it relates the nonthermal and thermal components of the plasma. The X-ray emissivity, dominated by thermal bremsstrahlung, is proportional to the gas density squared, , where is the electron number density. The radio emissivity is roughly proportional to the product of the CRE energy density and the magnetic energy density, . As clusters are approximately isothermal, the ratio gauges the energy fractions of the CREs and of the magnetic field, measured with respect to the thermal energy density . Here, is the temperature, is Boltzmann’s constant, and the factor accounts for the thermal contribution of ions, with being the average particle mass.

The radio to X-ray ratio is particularly useful in regions where exceeds the energy density of the cosmic microwave background (CMB), . Here, CRE cooling is regulated by the magnetic field with little Compton losses, so in a steady state , and is independent of the magnetic field. This appears to be the case near the centres of halos, according to the correlation between radio power and X-ray luminosity in GHs (Kushnir et al. 2009), and the linear correlation between radio and X-ray surface brightness near the centres of both GHs and MHs (KL10).

Therefore, we begin by modeling the radio to X-ray brightness ratio near the centres of well studied halos. We find that in these clusters, valid out to distances . Equivalently, this may be written as , where is the gas column density. As in strongly magnetised regions, this is our first direct indication that the CRI energy density is homogeneous. We identify a radial break in in some halos, which we interpret as the transition from strong to weak fields (see KL10), complicating any estimate of at larger distances based on halo data alone.

Next, we examine in relics, using models of the clusters based on ASCA data (Fukazawa et al. 2004). In particular, we choose the location along the relic where radio brightness is maximal, presumably corresponding to the highest magnetic field and a favourable projection. We find that these relics values lie close to, but slightly above, the curve normalised by halos, where subscripts denote a quantity measured at the centre of the cluster, , and here is the projected density in the plane of the cluster. Therefore, relics too approximately satisfy , and with a proportionality coefficient similar but slightly higher than in halos.

This surprising result is very unnatural in the context of present models, because all model variants attribute halos and relics to different CRE populations. It strongly suggests that relics, like halos, arise from secondary CREs, produced by the same population of CRIs. This would imply that the distribution of CRIs remains homogeneous out to , close to the virial shock of the cluster.

1.4 Incorporating deviations from a CRE steady state: essential for relics and spectral analyses

In order to test the applicability of one secondary CRE model for both halos and relics, we examine the morphological and spectral properties of relics, and the halo–relic regions in clusters that harbour both. Recognising that the rapid changes in the magnetic field and in CRE injection at the relic discontinuity are responsible for the elevated (with respect to ) relic brightness and the spectral steepening observed inward of several relics, indicates that the space-time evolution of the CRE population must be incorporated in the model.

Therefore, in §4 we compute the CRE evolution and the resulting synchrotron signal, for an arbitrary temporal evolution of the magnetic field and of the CRE injection rate, and examine the effects of CRE diffusion across magnetic irregularities. In particular, we study the structure of a weak shock, and derive the properties of radio emission arising from weak shocks and turbulence. We show that a weak shock of Mach number raises the pressure of flat spectrum CRIs and CREs by a factor , thus lowering the CR energy fraction with respect to the (shocked) gas.

Quite generally, synchrotron emission brightens, and subsequently steepens, in regions that experienced strong recent magnetic growth which exceeds the gas compression. This explains the steepening observed downstream of several (but not all) relics, provided that these relics are strongly magnetised, as confirmed in some cases by independent estimates of the relic magnetic field. This effect also provides an alternative explanation for the spectral steepening observed near the edges of some halos, interpreted by KL10 as evidence for a steep cosmic ray proton (CRP) spectrum, and explains the very steep spectrum of a subset of GHs.

1.5 Model calibration, tests, and implications

Our time-dependent model is applied to halo and relic observations in §5, in order to test the model and calibrate its parameters. We find that the model reproduces the observations, provided that the CRI distribution is homogeneous, and that a fraction of the thermal energy density downstream is deposited in magnetic fields. We then show that the model naturally explains the multiple connections inferred between halos and relics, such as the halo–relic bridges, which all arise because the same CRIs are involved. The model also explains the universally flat spectrum observed at the edges of relics, provided that the CRP spectrum is flat out to from the centre.

The brightening (dimming) and spectral steepening (flattening) of the radio signal reflect recent magnetic growth (decay), gauging the dynamical state of the cluster. The possibility that steep GHs are young mergers associated with recent or irregular magnetic growth is tested, by showing that they are preferentially associated with nearby relics.

The homogeneous CRI distribution we infer on cluster scales, and the strong, evolving magnetic fields required to explain the various radio sources, bear several implications for the energy budget in clusters, their nonthermal emission in radio and other bands, and the physics of magnetisation and particle acceleration and evolution. After briefly reviewing the model in §6, we estimate the CRI energy density and spectrum , and show that the magnetic fields in several halos are consistent with , on the order of . We show that assuming that and are universal constants in GHs, approximately reproduces the and correlations observed, where is the halo size.

We show that the CRI distribution can be explained by particle acceleration in strong shocks: SNe shocks, the virial shock, or a combination of both. The homogeneous CRI distribution then requires that either the diffusion of CRIs is sufficiently strong and their escape from the cluster is quenched, or that gas mixing is highly efficient. Various implications of our results are discussed, in particular the connections between the different radio sources, and additional hadronic signals. As the magnetic fields are found to be strong in both halos and relics, we impose strict upper limits on the efficiency of particle acceleration in weak shocks and turbulence.

1.6 Central argument

The diffuse (not associated with any local source) radio emission observed from the ICM, in its difference forms (flat and steep GHs, MHs, relics, and halo–relic bridges), can be explained as synchrotron emission from secondary CREs, produced by hadronic collisions involving CRIs with a flat spectrum () and with a homogeneous distribution, with energy density in the range , provided that the spatial and temporal variations of the magnetic field are taken into account.

This model resolves the present puzzles outlined in §2, reproduces the morphology, spectra and energetics of flat spectrum halos (for example in Figs. 47), and explains the spectral and morphological properties of relics and halo–relics bridges (see §5). Taking into account the magnetic amplification by shocks, the model also reproduces the brightness of relics (compare Figs. 12 and 26, before and after accounting for shock magnetisation). Interpreting steep spectrum GHs as young mergers, based on their association with nearby relics (see Fig. 28), explains their spectral steepening as arising from an increasing level of magnetic turbulence, in particular if CRE diffusion is strong (see §5.5).

1.7 Paper layout and definitions

The paper is organised as follows. In §2 we discuss some peculiarities and inconsistencies of present halo and relic models. In §3 we analyse halo and relic observations, present the phenomenological evidence for a homogeneous CRI distribution, and show that a time-dependent model is required in order to explain the morphologies and spectra observed. A time-dependent model is derived in §4, generalising secondary CRE emission for the case where the magnetic field and CRE injection evolve rapidly, in particular near a weak shock and under turbulent conditions. The model is then tested against observations in §5, both among different sources and within well-studied halo clusters that harbour shocks or relics. Here we derive for relics, discuss the spectral steepening and curvature, and show the association between steep GHs and nearby relics. In §6, we discuss the model and its implications. In particular, we compute the energy density and the spectral index of the CRIs, and outline the implications of their homogeneous distribution. Also discussed are the implied constraints on CRI diffusion, primary CRE acceleration in shocks and turbulence, and additional hadronic signals. Finally, our analysis and results are briefly summarised in §7. Supporting computations for the model are provided in Appendices AD.

Considering the observationally-driven structure of the paper, the reader may wish to skip the observational motivation for the model in §2 and §3, and the analysis of variable fields and injection in §4, and proceed directly to the description of the model and its application to observations in §5, or to its review and discussion in §6. Alternatively, the summary in §7 provides references to all the results derived in the text.

We assume a concordance CDM model with dark matter fraction , baryon fraction , and a Hubble constant . Error bars are confidence intervals, unless otherwise stated. The main parameters used in the study are defined in Table 5. We use the term cosmic-ray proton (CRP) instead of CRI when discussing processes in which a proton plays an individual role, for example when describing the high energy spectrum.

2 Present halo and relic models: puzzles and inconsistencies

Several observations are inconsistent with, or unexplained by, the present halo and relic models. These discrepancies and coincidences — manifestations of the same physical process, as we shall see — indicate that the present models are, at best, incomplete. In order to better understand the present models and their limitations, and to motivate the search for a more successful model, we now review these observational clues.

2.1 Giant halos with a steep spectrum

In the secondary CRE model for halos (both GHs and MHs), the CREs are produced through hadronic collisions, and their energy spectrum closely reflects the spectrum of their parent CRPs. This primary CRP spectrum is uncertain, but thought to be well approximated by a power-law of index in the relevant, few– proton energy range (see KL10 for a discussion). For slowly evolving magnetic fields, the corresponding radio spectrum is roughly , as observed in flat halos. Here, () implies equal energy per logarithmic interval in photon (proton) energy.

Recent years saw increasing evidence for the existence of GHs with a steep spectrum, where . An extreme example is the halo in A521, where the spectral index in the frequency range was recently shown to be (Dallacasa et al. 2009). (We shall henceforth use this notation to represent the spectrum between frequencies and measured in GHz.) There are currently six known steep halos — in A521, A697, A754, A1300, A1914, and A2256; their parameters are summarised in Table 2. This sample of steep GHs constitutes a minority — less than  — of all GHs, as at least GHs (see Giovannini et al. 2009) are currently known. However, halo observations are usually selected based on high frequencies maps, so steep halos could in principle be much more common than revealed by the present data. Future low frequency studies with MWA3, LOFAR4, and SKA5 are expected to discover many more halos, and would better estimate the steep fraction.

It was recently claimed that such steep GHs cannot arise from secondary CREs, because this would require a primary CRI population with unrealistically large energy and steep spectrum (Brunetti et al. 2008; Dallacasa et al. 2009; Brunetti 2009). Indeed, present secondary CRE models must be revised if they are to explain halos with .

A closely related phenomenon, which challenges secondary CRE models, involves the strong spectral steepening observed as a function of frequency in some GHs. Examples include Coma (A1656; and ; Giovannini et al. 2009), A2319 ( and ; Feretti et al. 1997), and A3562 ( and ; Venturi et al. 2003; Giacintucci et al. 2005). Such steepening exceeds the weak spectral variations expected in secondary CRE models due to the energy dependence of the cross section for secondary production, unless (see KL10). The thermal Sunyaev-Zeldovich (SZ) effect (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1980) contributes to such steepening, and was suggested as a possible explanation for the spectrum of halos such as Coma (Enßlin 2002). However, it was recently argued that the SZ effect is not sufficiently strong to account for the substantial steepening observed (Brunetti 2004; Donnert et al. 2010).

2.2 Spectral peculiarities of relics

Detailed spectral maps obtained recently pose a challenge for relic models, as they do for halos. One challenge here is to explain why some relics show substantial spectral steepening inward of the relic, while others do not. Another difficulty involves several relics, found in different clusters and radii (i.e. distance from the cluster centre), all showing a nearly identical, flat spectrum, which is unnatural in a primary CRE model.

All relic models so far agree that due to CRE cooling, the radio spectrum should gradually steepen away from the shock (e.g., Giacintucci et al. 2008); in most cases this implies steepening with decreasing radius. Indeed, in several cases (A521, A3667, A1240, A2256, A2345), the radio spectrum was found to vary significantly across the relic, being flatter () at the outer rim and steeper () towards the cluster’s centre. If the spectrum at the outer rim is a pure power-law with index , one expects the spectrum of the integrated radio signal to be steeper, , due to CRE cooling. This appears to be the case for example in A521, where , but inward of the rim the spectrum dramatically steepens to values , leading to an average spectrum (Giacintucci et al. 2008).

However, some relics show only a modest (e.g., A2744; Orrú et al. 2007) or no (e.g., A2163; Feretti et al. 2004) steepening. For example, the textbook relic A1253+275 in Coma shows mild steepening, with at the outer rim and at smaller radii (Giovannini et al. 1991). This leads to an integrated, pure power-law spectrum with index (Thierbach et al. 2003), flatter than the spectrum anticipated from the above arguments. A clue to the difference between these spectral behaviours is the distance of the relics from the centre of their clusters; all steepening relics are found at radii , whereas little or no steepening is found in relics beyond .

Considering the observational difficulty of integrating the entire diffuse, weak radio emission, a useful measure of the relic spectra is the (usually) flatter spectrum at the outer edge. All models associate relics with shocks, so is the most pristine measure of the spectrum just behind (i.e. downstream of) the shock. It was measured for a handful of relics, by binning the radio maps radially. The spectra of the outermost bins are summarised in Table 1; all the relic agree with the range . These measurement can be supplemented by the edge spectrum in relics that show little spatial variations, such that can be estimated from the uniform spectral map or from the computed . Examples include Coma ( typically found by Giovannini et al. 1991), A2163, and A2744 (see Table 1).

The narrow distribution of the edge spectral indices among the different relics shown in Table 1 is unnatural or inconsistent with the present relic models, because they either cannot explain the pure power-law spectrum observed (radio Phoenix model), or attribute no special significance to the value (DSA models), as we now show.

Cluster Relic position Outer spectrum Reference Shock Mach number
name (or name) according to DSA
A521 Southeast Giacintucci et al. (2008, figure 7)
A1240 North (A1240-1) Bonafede et al. (2009, figure 8)
A1240 South (A1240-2) Bonafede et al. (2009, figure 8)
A2345 East (A2345-2) Bonafede et al. (2009, figure 3)
A2256 North Clarke & Ensslin (2006, figure 5)
A1656 West Giovannini et al. (1991)
A2163 Northeast Feretti et al. (2004)
A2744 Northeast Orrú et al. (2007)

The estimated outer spectral index is shown for each relic, along with the corresponding shock Mach number according to DSA theory. Quoted values of give the spectrum of the outermost, flattest bin extracted from published spectral profiles (upper rows), or otherwise estimated for relics with little spectral variability (bottom rows). We exclude the exceptionally flat, non-power-law spectrum of the relic in A2255 (Pizzo & de Bruyn 2009), which cannot be modeled in a DSA context, and the West relic in A2345 (A2345-1; Bonafede et al. 2009), which has an unclear projected geometry and shows inward flattening, rather than steepening.

Table 1: Spectrum of radio emission from the outer rim of relics.

In some relics, was found to be very well fit by a pure power-law, spanning a wide frequency range, with no evidence for spectral curvature. The best examples are Coma (; Thierbach et al. 2003) and A521 (; Giacintucci et al. 2008). The lack of spectral curvature rules out the radio Phoenix model as a plausible explanation for such relics (see Giacintucci et al. 2008). Henceforth we focus on the shock acceleration model.

In the DSA model, the outer rim of the relic is identified with a shock, where CREs with a power-law energy distribution, , are injected into the downstream plasma. The injection spectrum is determined solely by the shock compression ratio, or — henceforth assuming an adiabatic index  — by the Mach number of the shock (Krymskii 1977; Axford et al. 1977; Bell 1978; Blandford & Ostriker 1978),

(2.1)

This fixes the radio spectrum at the outer rim of the relic, where CREs had no time to cool, through (e.g., Rybicki & Lightman 1986). Consequently,

(2.2)

CREs farther inward of the rim had progressively more time to cool, so the spectrum steepens. If all the radio emission is accounted for, the integrated spectrum becomes

(2.3)

One can invert equation (2.2) and find the shock Mach numbers corresponding to the measurements. As shown in Table 1, this yields Mach numbers in the range for the five relics with measured (and similarly for A2163, A2744, and Coma). This range is alarmingly narrower than expected. Numerical simulations show a wide distribution of shock Mach numbers, in the range at the radii characteristic of relics, with rare shocks as strong as (Vazza et al. 2010, and references therein). Moreover, simulations predict a radial dependence of the Mach number distribution (e.g., Vazza et al. 2010). The above relics span a wide range of radii, , and yet show very similar according to the DSA model. Finally, X-ray data suggests that some of these relics are associated with shocks stronger (e.g., A521, see Giacintucci et al. 2008) or weaker (e.g., Coma, see Feretti & Neumann 2006) than the values inferred under the DSA assumption.

Finally, we stress that in addition to the observational discrepancies demonstrated above, the DSA relic model is of limited use here because particle acceleration in weak shocks is neither theoretically understood, nor observationally constrained. DSA theory, while successfully reproducing the spectrum in strong shocks, does not self-consistently compute the acceleration efficiency. The efficiency is well constrained from observations only for strong shocks, for example in supernova remnants (for a discussion, see Keshet et al. 2003). Only loose upper limits on the acceleration efficiency of weak shocks are available, on the order of of the thermal downstream energy (Nakar et al. 2008). In the absence of a calibrated model for weak shocks, DSA relic models utilise ad-hoc prescriptions for the efficiency, which probably depends non-trivially on the plasma parameters and the shock strength. Note that similar comments can be made regarding particle acceleration in turbulence, for example as the putative source of CREs in radio halos.

2.3 Halo–relic connections

There is still no widely accepted agreement regarding the origin of CREs in neither halos (secondary vs. primary CREs) nor relics (acceleration vs. reacceleration of primary CREs, as so far believed). Nevertheless, so far all models agree that the two phenomena involve CREs generated in different physical processes. Here we point out several connections between GHs and relics, indicating that the two phenomena are more intimately related than is plausible in a framework with different CRE injection mechanisms. These connections suggest, as we confirm in §3, that the CREs in halos and relics in fact share a common origin.

  1. In clusters that harbour both a halo and a relic, a faint radio bridge is sometimes observed to connects the two, for example in Coma, A2255, and A2744 (Giovannini & Feretti 2004). The spectrum of the bridge is similar or somewhat steeper than in the halo and in the relic (Kim et al. 1989; Pizzo & de Bruyn 2009; Orrú et al. 2007). The low surface brightness of the radio sources combined with anecdotal evidence, such as an alignment between the relic position and the halo elongation axis in A2163 (see Feretti et al. 2004), suggest that observational limits and projection effects hide many such bridges. The origin of the bridges is unknown; in present models they would require a fine-tuned spatial interpolation between the halo and relic CRE populations.

  2. Although relics — and not halos — are thought to be generated by shocks, weak shocks were discovered at the edges of several GHs. (In fact, so far no relic shock has been confirmed, due to the low ambient density). This includes confirmed shocks in 1E 0657–56 (the bullet cluster; Markevitch et al. 2002), A520 (Markevitch et al. 2005) and A754 (Krivonos et al. 2003), and suspected shocks in A665 (Markevitch & Vikhlinin 2001), A2219 (Million & Allen 2009), and Coma (Brown & Rudnick 2010).

    Assuming that as these weak shocks travel outward and disconnect from the halo they become relics, further exacerbates the problem in interpreting the two phenomena with different CRE injection mechanisms. With the present data, this would also require a fine-tuned temporal interpolation between the two phenomena.

    Incidentally, notice that while a weak outgoing shock at the very edge of a GH is consistent with the secondary CRE model (where shock magnetisation is sufficient to generate the halo), it is inconsistent with primary CRE models: strong turbulence is not expected in the shock region (Govoni et al. 2004), and even if it was, turbulent acceleration would not be effective immediately behind the shock.

  3. A telling and somewhat surprising halo-relic connection is the apparent coincidence between the presence of a steep spectrum GH in a cluster, and the presence of a nearby relic. All six steep GHs discussed in §2.1 either have a nearby relic (A521, A754, A1300, A2256; see Giovannini et al. 2009), or have an irregular morphology containing a relic-like filamentary protrusion (filaments West of A697 and Southwest of A1914; Venturi et al. 2008; Bacchi et al. 2003). This incidence rate of relics is significantly higher than found amongst flat spectrum GHs: only five () out of the 25 flat halo clusters reviewed in Giovannini et al. (2009) harbour a relic. As we show in §5, there appears to be a bimodality in the distance of halo-cluster relics: relics in steep GH clusters are found at , whereas relics in flat GH clusters are more distant. This behaviour is reminiscent — and indirectly related — to the excessive steepening behind central relics mentioned in §2.2.

  4. Halos and relics share many traits. To some extent, the observationally-driven classification of diffuse radio emission into halos and relics is artificial and blurred, with some halos showing relic features and vice versa.

    1. Halos and relics have a similar spectrum, in particular if we restrict ourselves to the central part of the halo and the external edge of the relic. This can be seen, for example, in the spectral maps of A2744 (Orrú et al. 2007) and A2163 (Feretti et al. 2004). In the context of present halo and relic models, this is a highly peculiar coincidence.

    2. There are telling exceptions to the polarisation classification scheme described in §1. So far, strong polarisation was detected in one GH (at a level, in A2255; see Govoni et al. (2005); intermediate polarisation, on average, was found in MACS J0717.5 +3745; see Bonafede et al. (2009)), and in one MH (at a level, in A2390; Bacchi et al. 2003). Some relics show a low polarisation level, for example was reported in A133 (Slee et al. 2001); however this may be an AGN relic (Kempner et al. 2004). These exceptions suggest a continuous distribution of polarisation properties among halos and relics.

    3. Some GHs are irregular, showing a clumpy or filamentary morphology. Examples include RXC J2003.5–-2323 (Giacintucci et al. 2009), A2255, and A2319 (Murgia et al. 2009). Morphologically, they could be interpreted as an ensemble of relics, as recently suggested for A2255 (Pizzo et al. 2010).

    4. Some GHs show spectral steepening with increasing radius, resembling that observed in relics. This can be seen, for example, in the spectral maps of A665, A2163 (Feretti et al. 2004), A2219, and A2744 (Orrú et al. 2007).

  5. A correlation has been reported (Giovannini & Feretti 2004) between the total radio power of relics and the bolometric X-ray luminosity of their host clusters, somewhat reminiscent of the radio–X-ray correlation found in GHs. Recall that the strong correlation observed in halos provides a strong evidence that they arise from secondary CREs (Kushnir et al. 2009, KL10).

The above connections between halos and relics suggest that the two phenomena arise from the same population of CREs. However, no unified model has been proposed thus far for halos and relics. There is evidence, preliminary in relics but strong in GHs and in MHs (and so, by proxy, also in relics), that secondary CREs are involved.

In spite of the halo-relic connection, we do not find significant evidence for an enhanced incidence rate of halo/relic detection in relic/halo clusters. For example, 11 out of the 31 () GHs summarised in Giovannini et al. (2009) also harbour at least one relic, whereas about 7 out of the 30 () relics reported in Giovannini & Feretti (2004) are found in halo clusters. These incidence rates are similar to the unconditional halo and relic detection rates in X-ray bright clusters (Giovannini et al. 2002). This behaviour could arise, for example, if halos are much more long-lived than the time scale during which a relic is detectable. A careful analysis of the selection effects involved is necessary in order to quantify a correlation between the presence of halos and relics in a cluster, or the lack thereof.

3 Relic and halo phenomenology: both arise from the same, homogeneous CRI distribution

The preceding discussion, in particular the connections between radio halos and relics outlined in §2.3, motivates a unified exposition of cluster radio sources. We thus begin by showing in Figure 1 a sample of various types of halos and relics, presented in the phase space of maximal radio brightness vs. coincident X-ray brightness .

Roughly speaking, is proportional to the product of the energy densities of CREs and magnetic fields, whereas is proportional to the square of the plasma density. The data used to produce this and the following figures were extracted from the literature, as summarised in Table 2. The data preparation and source selection and classification are described below in §3.1 and §3.2.

Figure 1: Peak radio brightness vs. coincident X-ray brightness of radio halos (black disks) and relics.
The radio-to-X-ray brightness ratio of most halos (all but A2219 found in relic clusters) is somewhat larger than the best fit derived for halos by KL10 (in a partly overlapping sample of GHs, mostly residing in non-relic clusters; dashed line with yellow shaded band showing the dispersion). Relics show much higher ; of them lie within (cyan shaded region). Note that the classification of the relics in A754 and A2034 is highly uncertain (see §3.2), and A2219 is strongly contaminated by point sources (see §3.1); the bimodality in between halos and relics is significant without these three sources.
Following Giovannini & Feretti (2004), relics are classified as found in halo clusters (red filled squares), found in double relic systems (blue diamonds), found near the first-rank galaxy (brown up-triangles), circular peripheral relics (orange double arrows), and classical elongated relics (magenta five-pointed stars).
The radio data (vertical error bars) show the values of the two brightness contours bracketing the radio peak in published, contour maps, and their mean. The X-ray brightness is based on -models from the literature. In order to avoid excessive error propagation, the (horizontal) error bars reflect the standard deviation of , but not of or (see §3.1). The -model is supplemented by some measurements (red empty squares) of (in A2163 and A2744; see §3.4.3), or an upper limit (in Coma; Feretti & Neumann 2006). The data used to produce the figure are summarised in Table 2.
Also shown is a possible shock region in A2219, reported by Million & Allen (2009), with measured upstream and downstream (green right and left triangles), and as inferred from the -model (green six-pointed star).
Inset: particularly bright radio sources.

Notice the similarity — to order of magnitude, at least — in the peak values of the relics and halos shown in Fig. 1. In particular, the peak brightness of halos and relics found in the same cluster appear to be somewhat correlated. Could these subtle connections reflect a unified CR origin in these clusters?

We begin answering this question by showing in §3.3 that the dimensionless ratio between and provides a useful diagnostic of the nonthermal plasma. In GHs, where the radio emission is associated with secondary CREs and strong magnetic fields, is a direct measure of the local CRI fraction (in regions of slowly varying magnetic fields).

Next, we present in §3.4 evidence for a universal, radially rising profile in GHs, and reconcile it with previous reports of uniform in some clusters. The scaling we derive indicates that the CRIs are uniformly distributed in GH clusters, even beyond the scales illuminated by strong magnetic fields.

Finally, in §3.5 we combine the data of halos and relics, and derive a universal profile extending out to large, radii. The data motivate a unified model, in which halos and relics both arise from secondary CREs, produced from the same homogeneous distribution of primary CRIs. This model resolves several of the previous model discrepancies outlined in §2. Addressing the remaining discrepancies and the spectral properties of relics requires a generalised model, involving time-dependent CRE injection and dynamic magnetic fields, derived in §4.

3.1 Data reduction

For a handful of halo and relic clusters, detailed radio and X-ray brightness maps can be found in the literature. In a small subset of these clusters, detailed maps of the radio spectrum are also available. This includes the GH in A665, and the clusters A2163 and A2744 which harbour both a halo and a relic. The combination of surface brightness and spectral maps provides strong constraints on the radio model and on the plasma parameters. We examine and simultaneously model the brightness and spectral profiles in these clusters; here we focus on the brightness maps of the GHs, and defer modeling the relics and the spectral variations to §5, where the description of time-dependent CRs and magnetic fields developed in §4 is incorporated. We do not discuss the published radio maps of A2219 (Orrú et al. 2007) and A2255 (Pizzo & de Bruyn 2009), because the halo in A2219 is strongly contaminated in the central by a blend of radio galaxies (Orrú et al. 2007), and the GH in A2255 is highly irregular and filamentary, and could be identified as an ensemble of relics rather than a halo (Pizzo et al. 2010).

Unfortunately, in most clusters, only low resolution contour maps and the integrated properties of the radio sources (power, spectrum, polarisation) are available. We choose to examine the surface brightness, rather than the integrated luminosity of each source, in order to obtain a local measure of the plasma which is less sensitive to background and resolution effects. However, the radio sources are diffuse and extend over large, sometimes scales, thus spanning a wide range of surface brightness. In order to have a simple, yet well-defined prescription for assigning a brightness to each source, we identify it with the position (line of sight) of maximal radio brightness. This choice has additional advantages. For example, in relics, it is likely that the magnetic field is maximal or the projection is most favourable at peak radio brightness, simplifying the analysis considerably, as discussed in §3.5.3 and §6.

For each radio source, we define an uncertainty range of peak radio brightness (vertical error bars in Fig. 1) such that is the value of the brightest contour found in the radio map of the source, and is the value of the next, putative contour (not found in the map). The value assigned to each source (shown by symbols) is the arithmetic mean of these lower and upper limits. This introduces some nonphysical error; in all cases it is less than a factor of . We use as a measure of the radio brightness (as the ordinate in Fig. 1) because it varies weakly with frequency when the spectrum is nearly flat, . Nevertheless, due to some frequency dependence and for consistency with previous work, we only use data in frequencies around (within in ); see Table 2 for details.

Radio relics are often found at large distances from the cluster’s centre, where the X-ray emission of the cluster becomes too faint to be distinguished from the background. Therefore, we compute by extrapolating the bright, central X-ray emission out to the radio source position, using an isothermal -model (Cavaliere & Fusco-Femiano 1976) for each cluster taken from the literature. In these models, the electron number density varies as a function of radius according to

(3.1)

where , and are the free parameters of the model (see Table 2 for individual model references), and . We identify the X-ray peak as the centre of the cluster. A distance range is assigned to each radio source according to the minimal, maximal distance of the contour from the cluster’s centre. The value associated with each symbol in Fig. 1 is taken as the arithmetic mean of these lower and upper limits.

The -model extrapolation introduces inevitable errors, as illustrated in Fig. 1 by comparing the extrapolated and the measured values of in a few relics. In nonspherical clusters where the -model fits poorly, it may produce significant errors in . Another source of error is an enhanced brightness observed in several relics, thought to be caused by shock compression of the plasma. In such cases, the -model tends to underestimate the X-ray signal, for example in A2163 and A2744 (see Fig. 1). To illustrate this effect, Fig. 1 also shows (as empty triangles) both upstream and downstream of a suspected shock front in A2219 (within the GH and outside the central, contaminated region; see Million & Allen 2009). Nevertheless, the -model extrapolation errors are much smaller than the three orders of magnitude in spanned by the data; in most cases where is known, the error is less than a factor of 3.

We estimate the confidence intervals arising from the -model uncertainties by adopting the largest propagated uncertainty due to any single one of the three parameters of the model, , , or . As the correlations between the uncertainties of these parameter are highly correlated, this appears to be the most reliable estimator of the error in the absence of covariance matrices. Notice that our estimate could be either larger or smaller than the true error. Propagating all three parameter uncertainties by assuming that they are, for example, uncorrelated, would probably spuriously increase the confidence intervals.

For consistency with previous work, we use exclusively in the energy range . Emission in this band is dominated by thermal bremsstrahlung, and has the advantage (for our present purpose) of being weakly dependent upon temperature and metallicity in the relevant parameter range. The X-ray emissivity in these energies, calculated using the MEKAL model (Mewe et al. 1985; Mewe et al. 1986; Kaastra 1992; Liedahl et al. 1995) in XSPEC v.12.5 (Arnaud 1996), is well-fit by

(3.2)

where is the electron number density in units of , with being Boltzmann’s constant, and is the metallicity in units of . We use equation (3.2) to compute from the -models, assuming uniform temperature and metallicity in each cluster. In this approximation

(3.3)

where the argument is a two-dimensional angular vector with the centre of the cluster at , , and

(3.4)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Name Type -ref -ref
A85R1 RG 0.056 (F04a) (F04a) (F04a) F04a 1.4 444 451 8.14 11.40 S01
A85R2 RG 0.056 (F04a) (F04a) (F04a) F04a 1.4 483 488 8.14 11.40 S01
A115 RI 0.197 (G01) (F04a) (F04a) G01 1.4 401 512 4.44 8.89 G01
A133 RG 0.060 (F04a) (F04a) (F04a) F04a 1.4 42.0 45.0 254 396 S01
A521H H 0.247 (B08) (F04a) (F04a) F04a 1.4 0 0 0.24 0.48 D09
A521R RH 0.247 (B08) (F04a) (F04a) F04a 1.4 846 910 1.92 3.85 D09
A548N RC 0.042 (F04a) (F04a) (F04a) F04a 1.4 416 549 1.33 2.22 F06
A548NW RC 0.042 (F04a) (F04a) (F04a) F04a 1.4 426 597 1.33 2.22 F06
A754E H 0.056 (F04a) (F04a) (F04a) F04a 1.5 61.0 167 0.81 1.62 B03
A754W RH 0.056 (F04a) (F04a) (F04a) F04a 1.5 578 709 0.57 0.81 B03
A1300H H 0.306 (R99) (F04a) F04a 1.4 0 1.0 1.89 3.77 R99
A1300R RH 0.306 (R99) (F04a) F04a 1.4 458 663 3.77 6.29 R99
A1664 RC 0.128 (G01) (G01) G01 1.4 956 1037 2.96 5.92 G01
A2034H H 0.151 (F04a) (F04a) (F04a) F04a 1.4 76.0 240 0.65 0.92 G09
A2034R RH 0.151 (F04a) (F04a) (F04a) F04a 1.4 404 518 0.32 0.46 G09
A2163H H 0.170 (F04a) (F04a) (F04a) F04a 1.4 0 231 2.59 3.33 F04b
A2163R RH 0.170 (F04a) (F04a) (F04a) F04a 1.4 1258 1345 2.59 3.33 F04b
A2219H H 0.226 (O07) (C06) (W00) F04a 1.4 0 10.0 55.40 78.40 O07
A2219S S 0.226 (O07) (C06) (W00) F04a 1.4 430 430 0.43 0.61 O07
A2255H H 0.080 (F04a) (F04a) (F04a) F04a 1.2 0 0 1.14 2.29 P09
A2255R RH 0.080 (F04a) (F04a) (F04a) F04a 1.2 907 1110 0.57 1.14 P09
A2256H H 0.046 (F04a) (F04a) (F04a) F04a 1.4 0 219 0.47 0.66 CE06
A2256R RH 0.046 (F04a) (F04a) (F04a) F04a 1.4 438 543 5.28 7.47 CE06
A2744H H 0.308 (O07) (C06) (A09) F04a 1.4 0 140 1.92 2.72 O07
A2744R RH 0.308 (O07) (C06) (A09) F04a 1.4 1479 1641 1.36 1.92 O07
A3376E RD 0.046 (B06) (F04a) (F04a) F04a 1.4 471 592 1.20 2.50 B06
A3376W RD 0.046 (B06) (F04a) (F04a) F04a 1.4 1352 1447 1.20 2.50 B06
A3667NW RD 0.055 (F10) (F04a) (F04a) F04a 1.4 1986 2068 1.98 2.79 J04
A3667SE RD 0.055 (F10) (F04a) (F04a) F04a 1.4 999 1090 1.98 2.79 J04
A4038 RG 0.026 (F04a) (F04a) (F04a) F04a 1.4 29.0 39.0 46.50 93.00 S01
ComaH H 0.023 (F04a) (F04a) (F04a) V02 1.4 0 10.0 0.66 0.70 K00
ComaR RH 0.023 (F04a) (F04a) (F04a) V02 1.5 2080 2090 0.80 1.00 G91; B10
RXJ1314 RI 0.247 (G01) (V02) V02 1.4 1072 1121 6.70 8.94 V02
S573 RI 0.014 (F04a) (F04a) (F04a) F04a 1.4 121 194 2.54 3.81 S03
Table 2: Parameters of the radio sources in our sample.

Columns: (1) source name (composed of the host cluster’s name and optional suffix letters designating the source type or position); (2) source type: H–halo; RI–isolated, elongated relic; RH–relic in a halo cluster; RD–relic in a double relic cluster; RC-circular peripheral relic; and RG–relic near the first-rank galaxy; (3) redshift (with reference); (4) cluster temperature in keV (with reference); (5) cluster metallicity in solar units (with reference); (6) reference for the -model of the cluster; (7) radio frequency in GHz; (8) and (9) possible distance range between the X-ray peak of the cluster and the radio peak of the source, in kpc; (10) and (11) possible range of the peak radio brightness, , in units of ; and (12) reference for the radio data.

Reference abbreviations: A09 – Andersson et al. (2009); B03 – Bacchi et al. (2003); B06 – Bagchi et al. (2006); B08 – Brunetti et al. (2008); C06 – Cassano et al. (2006); CE06 – Clarke & Ensslin (2006); D09 – Dallacasa et al. (2009); F04a – Fukazawa et al. (2004); F04b – Feretti et al. (2004); F06 – Feretti et al. (2006); F10 – Finoguenov et al. (2010); G91 – Giovannini et al. (1991); G01 – Govoni et al. (2001); G09 – Giovannini et al. (2009); J04 – Johnston-Hollitt (2004); K00 – Kim et al. (1990); O07 – Orrú et al. (2007); P09 – Pizzo & de Bruyn (2009); R99 – Reid et al. (1999); S01 – Slee et al. (2001); S03 – Subrahmanyan et al. (2003); V02 – Valtchanov et al. (2002); W00 – White (2000).

3.2 Source selection and classification

Our sample consists of all the radio relics reported in the literature in clusters that satisfy the criteria outlined in §3.1, and all the halos found in those relic clusters. This corresponds to all the known diffuse radio sources in clusters that (i) harbour a relic; (ii) have a published radio map at some frequency in the range ; and (iii) have a published X-ray-based -model of the cluster. In total, our sample consists of 23 relics and 9 halos. These halos are all GHs in relic clusters; GHs in non-relic clusters and MHs were analysed separately in KL10. In addition, we include here the non-relic cluster A2219, where a suspected shock is embedded within the GH (source denoted A2219S; Million & Allen 2009), as mentioned in §3.1.

Different classification schemes have been proposed for relics. We follow the classification of Giovannini & Feretti (2004), which is based solely on the position and morphology of the relic. The relics in our sample are thus divided into the following five categories:

  1. Isolated, classical elongated relics — shown as magenta five-stars in Fig. 1;

  2. Relics in halo-relic systems — red squares;

  3. Relics in double relic systems — blue diamonds;

  4. Circular peripheral relics — ostensibly, face-on relics; shown as orange double arrows because the coincident X-ray emission may have been overestimated due to projection;

  5. Relics near, but not coincident with, the first ranked galaxy — brown, filled triangles.

Due to the lack of coincident X-ray data, our sample does not include two other classes of relics proposed by Giovannini & Feretti (2004): relics at a large distance () from the centre of a cluster, and large-scale filaments.

Most of the known relics are found in rich, merger Abell clusters. However, S0573 is a poor cluster that harbours a relic (Subrahmanyan et al. 2003), and some of the relic clusters have a cool core: A115 (classical relic), A85, and A133 (both relics near the first ranked galaxy); see Giovannini & Feretti (2004).

The classification scheme we adopt, unlike the scheme proposed for example by Kempner et al. (2004), avoids committing to any specific relic model. This is advantageous, as the models have not yet converged. For example, the relics in A85, A133, and A4038 are classified here as relics near the first rank galaxy, whereas van Weeren et al. (2009) classify them as radio phoenixes, and Kempner et al. (2004) classify A133 as an AGN relic.

Nevertheless, the present classification scheme is by no means unambiguous. For example, the northern emission in A2034, coincident with a CF (Kempner & Sarazin 2001), was originally identified as a relic (Kempner & Sarazin 2001; Kempner et al. 2003), and we adopt this classification in Fig. 1. It was later suggested that the source be identified as an irregular GH, because its centre coincides with the X-ray peak (Rudnick & Lemmerman 2009; Giovannini et al. 2009). However, it seems most reasonable to identify the source as a (possibly disrupted) MH, considering its central position, its relatively small scale (; Giovannini et al. 2009), the evidence for a cool core (see discussion in Kempner et al. 2003), and the association with a CF. Such an association has not been reported so far for GHs or for relics. (One exception is the CF near the GH in A2319, but this cluster appears to be in an intermediate state between a MH and a GH; see KL10.) Note that a putative relic-CF association would imply the existence of a new class of non-shock relics.

Another example of uncertain source classification is A754. The diffuse radio emission here is quite irregular, showing two main components with a West-East orientation (Bacchi et al. 2003), and a confirmed shock at the Eastern edge of the East component (Krivonos et al. 2003). Following Kale & Dwarakanath (2009), we identify the East component (labeled A754E) as a halo and the West component (A754W) as a relic, based mostly on the location of the X-ray peak. An opposite interpretation has been given by Bacchi et al. (2003), who also suggest that the emission may be classified as either two halos or two relics.

Source names are abbreviated as the cluster name with suffix letters identifying the source type or location. For halos, we use suffix H. For relics, we use suffix R if they are found in halo clusters, the source’s cardinal position initials (W for West, NW for Northwest, etc.) in multiple relic clusters, and no suffix otherwise. In A85, the (Southwest) relic is composed of two disconnected regions, radially separated by ; we denote them by A85R1 and A85R2. In A2219, emission from the region suspected as a shock (Million & Allen 2009) is denoted by A2219S.

3.3 ICM diagnostic: the ratio between radio and X-ray surface brightness

A useful dimensionless quantity in the study of diffuse emission from galaxy clusters is the ratio between the radio and the X-ray surface brightness,

(3.5)

We use (henceforth) radio frequencies and the X-ray energy range . One advantage of using the brightness ratio is that it is not sensitive to errors in estimating the redshift of the cluster. In most models, it is less sensitive than is to local variations in density.

The brightness ratio is particularly useful if the radio emissivity and the X-ray emissivity are either proportional to each other, or are strongly peaked along the line of sight. In such cases, the brightness ratio can be approximated by the emissivity ratio

(3.6)

in the densest position along the line of sight . The observed distribution around a cluster thus translates, approximately, to the emissivity ratio in the plane of the cluster’s centre, approximately perpendicular to the line of sight.

Note that the arguments , of the functions , are in general two-, three-dimensional vectors. Vector notations are omitted where spherical symmetry is assumed.

The relation between , and can be stated more precisely in the context of a density model. The X-ray emissivity is proportional to the plasma density squared, and is approximately independent of other plasma parameters (see Equation (3.2)), . We shall parameterise the synchrotron emissivity as

(3.7)

corresponding to an scaling of CRIs in the framework of a secondary CRE model. The radio–X-ray emissivity ratio then scales as

(3.8)

where .

In the -model, if we assume that applies locally with some constant , then necessarily also . Note that this property does not occur, in general, for more complicated density distributions. It is useful because it allows us to relate the line-of-sight integrated, observed behaviour, to the local relation. Another benefit of adopting a model is that the relevant properties can be expressed as power-law functions of the observed X-ray brightness .

Adopting the parametrisation equation (3.8) in the context of a -model, we find that

(3.9)

The index depends weakly on in the relevant, range, for which . For clusters well fit by , equation (3.3) becomes

(3.10)

In §3.4 below we use equation (3.10) as a proxy of in clusters even where the -model fails, for example in cases where reveals an underlying nonspherical gas distribution.

Notice that in the -model, the column density and the projected density are simply related to , through

(3.11)

where the expressions to the right of the arrows correspond to the case .

For steady-state CREs and static magnetic fields, is proportional to the product of the CRE energy density injection rate , and the magnetic energy density , weighted by the CRE cooling parameter . CRE cooling is dominated by inverse-Compton scattering off CMB photons or by synchrotron emission due to the magnetic field. Therefore, , where

(3.12)

is the putative magnetic field amplitude for which . Combining these relations, we obtain

(3.13)

Kushnir et al. (2009) have argued that the tight correlation observed between the radio power and the X-ray luminosity of GH clusters, and the bimodality of the GH distribution (Brunetti et al. 2007, and references therein), strongly suggest that the magnetic fields within GHs are strong, (for a different opinion, see Brunetti et al. 2009). If so, the second factor in equation (3.13) is approximately unity, such that the varying magnetisation levels among different GHs introduce only little dispersion in the correlation. Furthermore, the tight GH correlation observed can be reproduced if the CREs are secondary particles, produced in hadronic collisions between primary CRIs and the ambient plasma. This is most evident if the CRI distribution follows the bulk plasma, . In such a case, , resulting in a nearly constant in regions. Its value provides a direct measure of the CRI fraction in the ICM, .

An equivalent, but stronger argument can be made regarding the radio–X-ray correlation in central surface brightness. The correlation is remarkably tight near the centres of halos — both GHs and MHs (see KL10). We illustrate the GH correlation in Fig. 2 by reproducing Fig. 2 of KL10, showing for the halos with published radial profiles from both Very Large Array (VLA; Murgia et al. 2009) and XMM-Newton (Snowden et al. 2008). Here we plot not as a function of , but rather as a function of , determined using the cluster -models summarised in Table 1 of KL10. The nearly identical values of found towards the centres () of these clusters, in regions spanning more than an order of magnitude in density, confirm both the presence of strong magnetic fields () and suggest that .

Figure 2: Radial profiles of the radio-to-X-ray ratio in GHs with overlapping profiles from VLA (Murgia et al. 2009) and from XMM-Newton (Snowden et al. 2008, k-corrected). The data is the same as is Fig. 2 of KL10, but here is plotted as a function of (so the distance from the centre of the cluster increases to the left). We determine using individual cluster -models, detailed in Table 1 of KL10. Vertical error bars are the confidence intervals of the radio normalisation (Murgia et al. 2009), horizontal error bars are the propagated uncertainties, and the solid lines serve to guide the eye. The best fit of KL10 is shown as a horizontal dashed line, embedded in a shaded region showing the confidence level. Note that in A2163, scales roughly as (dot dashed red line) down to densities.

It is important to notice that the radio–X-ray correlations observed in luminosity and in central brightness do not imply that an scaling must hold throughout each halo. A linear correlation between the central CRI density and the central plasma density must exist among different halo clusters in order to reproduce the tight correlation observed. However, CRI distributions that are not proportional to away from the centre are possible, and in fact better reproduce the observed correlations and morphologies. In particular, one must take into account the weakly magnetised region, and the location of the transition between the two regimes. Different models for the CRI distribution are discussed in §3.4.1, and the luminosity correlations are revisited in §6.

3.4 Rising distribution in GHs: homogeneous CRI distribution

Before comparing (in §3.5) the values of in relics and in halos, it is useful to examine the distribution within GHs, where the X-ray emission is more constrained and the radio model is better understood. Distinct profiles are predicted by different variants of the secondary CRE model, depending mainly on the properties of CRI diffusion. Observational evidence is now sufficient to distinguish between these models.

After briefly reviewing different CRI distribution models (in §3.4.1) and previous evidence for a rising profile in halos (in §3.4.2), we analyse the radio and X-ray maps of a few well-studied GHs. The method of analysis is presented in §3.4.3, applied to the GHs in A2163, A665 and A2744 in §3.4.43.4.6, and summarised in §3.4.7. In §3.4.8 we show that our results are consistent with previous studies that found linear or mildly sublinear radio–X-ray correlations in GHs.

Different distributions in a secondary CRE model

The distribution of CRIs in a cluster depends on the nature of the CR sources, the properties of CR diffusion through the magnetised ICM, the escape of CRIs beyond the virial shock, and the mixing of the gas. The most plausible sources of cluster CRIs are either supernovae (SNe; see KL) or the virial shock of the cluster (see Kushnir & Waxman 2009; Kushnir et al. 2009); weak shocks are unable to produce the flat CRI spectra necessary to explain the radio observations. This leads to the following main possibilities:

  1. Negligible diffusion and mixing: CRIs are mostly due to the virial shock and subsequent adiabatic compression. Compression leads to and , assuming an isothermal distribution. Equivalently (see §3.3), and . Here, is probably proportional to (Kushnir et al. 2009).

  2. Significant escape, unsaturated diffusion and mixing: CRIs are mostly produced by SNe, and are distributed roughly as the gas, , so (); could depend on the local temperature (KL).

  3. Saturated diffusion or mixing: CRIs in the halo are produced by some combination of SNe and the virial shock, and are uniformly distributed throughout the cluster, , so ().

For brevity, we refer to the these models below as different diffusion models. The different outcomes of strong diffusion and gas mixing are discussed in §6.7.2.

As mentioned in §3.3, the tight radio–X-ray correlations observed in the luminosity () and in the central brightness () of halos imply that the central density of the CRIs is linearly correlated with the central plasma density . These correlations — in particular in luminosity — are also sensitive to the CRI distribution away from the centre. However, the interpretation of the data is complicated by effects such as a possible temperature dependence of (Kushnir et al. 2009, although such a dependence cannot be strong, see KL10) and different scalings of the radio and X-ray bright volumes (see KL10). Therefore, these correlations, by themselves, do not clearly distinguish between the possible CRI distributions outlined above.

The small, factor dispersion in the correlations also does not, by itself, fix the CRI distribution, because the different models outlined above entail a similar dispersion, as we show below and in Fig. 3. In particular, the small dispersion does not necessitate a spatial linear correlation between the CRIs and the ambient plasma, , as assumed to hold in the models of Kushnir et al. (2009) and KL10.

To see this, consider first the correlation. Assuming spherical symmetry, let denote the radius out to which the halo is observed, and denote the break radius where the magnetic field becomes weak (; see KL10 for a discussion of the magnetic break). For a CRI distribution that does not satisfy , the radio emissivity does not scale as inside , so is not uniform. Consequently, the luminosity ratio depends on the halo size, introducing some spurious dispersion in the correlation. However, this dispersion is not large for the typical outer radii of halos, (e.g., Murgia et al. 2009), even if the CRI distribution is flat (i.e. homogeneous). Moreover, beyond , the rapid radial decline of introduces an additional, -dependent dispersion in the relation for all models, in particular for the steeper, distributions. As a result, the dispersion introduced by the varying halo sizes and magnetisation levels among different clusters is similar in the different models outlined above, and is comparable in all cases to the (factor ) dispersion observed. Analogous arguments can be made regarding the dispersion in the central brightness correlation.

To illustrate the dispersion corresponding to the different models, we show in Fig. 3 the ratio as a function of halo size and magnetisation level, for different CRI distributions. For simplicity, we use the same cutoff radius for radio and X-ray emission, adopt an isothermal -model with , and assume that the magnetic energy density scales linearly with that of the plasma, , as often inferred from observations (e.g., Murgia et al. 2009; Bonafede et al. 2010). The figure shows that for the typical ranges of halo size () and central halo magnetic field (), the dispersion is rather similar for the different CRI distributions outlined above.

Figure 3: The ratio between radio power and X-ray luminosity from the central, halo region, assuming an isothermal -model with , and a magnetic scaling. To highlight the dispersion due to and (which together determine the magnetic break radius ), we normalise to unity as . We assume a power-law CRI distribution , with (homogeneous CRIs; dashed curves), (dotted curves), and (CRI density proportional to the plasma density ; dot-dashed curves). Shaded, labeled regions correspond to a range of central magnetic fields between (lower ; short dashing) and (long dashing). The shaded yellow region between vertical dotted lines shows the typical halo size range. Also shown is the relation inferred from the observed , , and relations (solid black and short-dashed brown curves; equation (3.4.2)), with two arbitrary choices of normalisation (note that the full curve must begin at the origin).

Previous constraints on

Inspection of the azimuthally averaged radial profiles of in the halos shown in Fig. 2 reveals a substantially different behaviour in each cluster. We ignore the dispersion seen in the central value, ; in an SNe model this could be attributed to different star formation histories (KL10). The radial profile is quite flat () in A665 and in A773, scales roughly as in the centre of A2163 and as in the centre of A2218, and is irregular in A2319. No universal CRI spatial distribution can be identified based solely on these radial profiles. The dependence of on the local temperature within these clusters was shown to be weak, (KL).

Notice that only one of the GH clusters in the KL10 sample — A2163 — harbours a relic. This is also the most extended and the most regular amongst the halos (see Murgia et al. 2009) in the figure, suggesting that it may be the most representative of the universal CRI distribution, if such exists. The profiles in the other halos could be contaminated by asymmetry and irregularities induced by substructure or regions of low magnetic field (see KL).

Evidence for a radially rising (i.e. increasing with declining ) profile in some other clusters is inferred, for example, from the comparison of the radio and X-ray morphologies in four GHs studied by Govoni et al. (2001). They find that two of these GHs (A2255 and A2744) show a linear correlation between radio and X-ray brightness (so ), whereas two other GHs (Coma and A2319) show a sublinear relation between and . Using the large radii scaling of the ASCA-based -models of Fukazawa et al. (2004), the results of Govoni et al. (2001) crudely translate to in Coma, and in A2319. In comparison, the sublinear relation found in A2163 (Feretti et al. 2001) translates to .

Indirect evidence for a radially rising profile stems from the different scalings of the integrated luminosities and with the respective radiating volumes. Cassano et al. (2007) found that the radio power increases rapidly with the halo size, . A weaker radial dependence is found in X-rays, , according to the relations (Zhang et al. 2008) and (Markevitch 1998, with referring, as usual, to ). (Here we assumed that the X-rays are integrated within a radius proportional to the virial radius .) These different scalings could arise, in part, from some dependence of upon the global parameters of the cluster. However, in the absence of evidence for such a global dependence (see KL10), the results suggest that is monotonically rising.

We may combine the above scaling with the relations (Cassano et al. 2007) and (Kushnir et al. 2009, KL10), in order to compare the profile of the different models in Fig. 3 to the observed correlations. The profile based on these phenomenological relations,

(3.14)

is illustrated in Fig. 3 with two different choices of normalisation (as solid black and dashed brown curves). These curves are quite crude, as the normalisation is arbitrary (it depends on the unspecified or uncertain normalisation of the above relations), we have not incorporated the substantial dispersions of the underlying phenomenological relations, and no dependence of upon global cluster parameters was allowed. The combined uncertainty in is thus sufficiently large to allow even the scaling (see KL10). Nevertheless, the agreement with a homogeneous CRI distribution is better, as illustrated by the figure, and favours strong magnetic fields.

More work is necessary in order to resolve the processes leading to the statistical brightness and luminosity correlations observed, if they are to be used to measured the CRI distribution. A more direct approach is to study the radio and X-ray maps of individual halo clusters.

Evaluating in well-studied GHs: method

In order to investigate the possible existence of a universal distribution in halos, we analyse the radio profiles of a sample of well studied, flat spectrum GHs. As explained in §3.1, we select A665, A2163, and A2744, where detailed spectral maps are available. The profiles of and in these clusters are presented in Figs. 49 below.

Avoiding the assumption of spherical symmetry when possible, we compute the radio profiles along two perpendicular directions in each cluster, without performing an azimuthal average. We choose these two axes such that they intersect at the X-ray peak of the cluster (denoted as ), and one of them crosses the relic (in A2163 and A2744) or shock (in A665) found in the cluster. (The analysis of these relics and shocks is deferred to §5.) The profiles are shown in the bottom panels of Figs. 49; the coincident profiles of the spectral index are shown in the upper panels. The orientation examined in each figure is specified in the upper right corner box.

We compute by combining or radio maps with X-ray data from ROSAT (the resulting profile is shown in each figure as a blue solid curve) and, when possible, also from Chandra (black short-dashed curves, with orange shaded band showing the X-ray uncertainty estimated by Million & Allen (2009)). In addition, we derive somewhat model-dependent profiles, using published -models to estimate (red long-dashed curves, with pink shaded band showing the uncertainty). Model parameters are take from the ASCA-based analysis of Fukazawa et al. (2004).

In order to test the flat, model (second model in §3.4.1), we plot the central GH fit of KL10,

(3.15)

as a horizontal dotted line with yellow shaded region showing the dispersion. Although lies within this range in the very central parts of A665 and A2163, it exceeds it in their outer parts, and throughout A2744. Figs. 47 show clear evidence for rising in A665 and A2163. As the figures show, this rise is stronger than the behaviour anticipated from adiabatic CRI evolution with no diffusion (first model in §3.4.1).

The data shown in Figs. 410 reveal a rapidly radially rising profiles in A665 and A2163, and a striking radio similarity between A2744 (where the X-ray morphology is highly irregular) and A2163 (see §3.4.6). This suggests that among the three models of §3.4.1, observations agree best with the distribution resulting from homogeneous CRIs (first model in §3.4.1; saturated diffusion). This can be tested by comparing the profiles computed from the -models with the hypothesis

(3.16)

shown as red long-dotted curves, where we use the measured values of for normalisation.

The model can also be tested more directly, using the observed , by utilising the relation in equation (3.3). The three clusters at hand are consistent with (Fukazawa et al. 2004), so we may use equation (3.10), whereby the scaling leads to . The resulting hypothesis,