AAstron. Nachr. / AN
XXX , No. XXX, 1 – 17 (2013) /
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Modelling Giant Radio Halos
J.M.F. Donnert ,(cid:63) Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, ER, ItalyReceived xx xxx 2013, accepted xx xxx 2013Published online xxxxx
Key words galaxies: clusters, radio continuum: general, radiation mechanism: non-thermalWe review models for giant radio halos in clusters of galaxies, with a focus on numerical and theoretical work. Aftersummarising the most important observations of these objects, we present an introduction to the theoretical aspects ofhadronic models. We compare these models with observations using simulations and find severe problems for hadronicmodels. We give a short introduction to reacceleration models and show results from the first simulation of CRe reaccel-eration in cluster mergers. We find that in-line with previous theoretical work, reacceleration models are able to elegantlyexplain main observables of giant radio halos. c (cid:13) Galaxy clusters form the knots of the cosmic web throughmerging of smaller structures seeded as density fluctuationsin the early Universe. Despite the name, only a few per-cent of a clusters mass is accounted for by actual galax-ies - most of it is in collisionless dark matter, which dom-inates the gravitational potential. The merging process de-posits baryons in this potential which form the intra-cluster-medium (ICM). During large mergers this infall can dissi-pate up to erg of kinetic energy in the ICM into turbu-lence, shocks, relativistic particles and eventually heat (e.g.Subramanian et al. 2006).Due to its high temperatures of more than K thethermal bremsstrahlung of the ICM plasma is prominentlyobserved in the X-rays (Cavaliere et al. 1971; Meekins et al.1971). However the ICM was first discovered through ra-dio observations of the Coma cluster (Willson 1970). Herethe interaction of relativistic electrons with the magneticfield of the plasma produces synchrotron radiation - a pro-cess well known from lobes of radio galaxies (Ryle &Windram 1968). This proves the presence of non-thermalcomponents in the ICM: cosmic-ray electrons (CRe) andmagnetic fields. Later turbulence was independently esti-mated in Coma through observed temperature fluctuations(Schuecker et al. 2004).Today non-thermal phenomena in galaxy clusters areclassified as radio halos, radio relics and mini halos (Feretti& Giovannini 1996). Radio halos are centered diffuse unpo-larised objects found exclusively in clusters with disturbedX-ray morphology.
Radio relics are thin, elongated, po-larised radio structures offset from the cluster center. Bothprevious classes extend for one to two Mpc in size, while radio mini halos are usually limited to a few hundred kpc (cid:63) e-mail: [email protected] sc a tt e r i ng CR p + t he r m damping of MHD-waves S y n c h r o t r o n b r i g h t a t . G H z CR protons
CR electrons e s c a p e t i m e f r o m c l u s t e r γ-rays Fig. 1
Lifetime of CR electrons (blue) and CR protons(red) in the ICM as well as their escape time (pink). Plot-ted for magnetic field strength of µ G (full) and . µ G (dashed). Synchrotron bright energies are marked blue. Fig-ure adapted and modified from Blasi et al. (2007a).in the center of relaxed clusters. In this paper we focus onradio halos.Independent measurements of the ICM magnetic fieldthrough Faraday rotation of intrinsic polarised radio sources(e.g. Bonafede et al. 2010) reveal field strengths of up to afew µ G in the center of clusters, declining outwards. Thefield is observed to be highly tangled (Kuchar & Enßlin2011), which is expected for a turbulent high beta plasma.This is consistent with simulations, which predict ef-ficient amplification by MHD turbulence as substructuresmerge with a parent cluster (Dolag et al. 2001; Dolag &Stasyszyn 2009; Dolag et al. 2005). However Pfrommer c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J un J.M.F. Donnert: Giant Radio Haloes
Fig. 2
Complex non-thermal emission observed with GBT at 1.41 Ghz in Coma by Brown & Rudnick (2011). The beamsize is roughly 360 kpc. Contours start at 20 mJy/beam and increase by 20 mJy/beam.& Dursi (2010) claim a radial magnetic distribution in theVirgo cluster. The origins of cluster magnetic fields are stilldebated (Widrow 2002).Considering radio halos the observed field strengths im-ply a population of synchrotron bright CRe with energiesof >
100 GeV in the whole cluster volume. It is well es-tablished that CRe are injected in shocks with Mach num-bers larger than 5 through observations of supernova rem-nands (e.g. Eriksen et al. 2011). And on smaller scales AGNand SN shocks indeed contribute to the CRe content ofa cluster. In contrast, accretion shocks during major merg-ers of clusters have low Mach numbers of ≈ . The exactmechanism that might lead to an injection in these shocksis still debated (Blasi 2010; Gargat´e & Spitkovsky 2012;Spitkovsky 2008). These primary CRe’s are believed tocause radio relics, which are associated with merger shocksin observations. However due to their geometry none ofthese mechanisms are volume filling. Therefore CRe wouldhave to diffuse hundreds of kpc to form a radio halo.Due to the low ambient thermal density of n th < − cm − , CRe have a classical mean-free-path of kpcin the ICM (Spitzer 1956). However particle motion fasterthan the Alv´en speed in a plasma is known to cause plasmawaves through the streaming instability (Wentzel 1974).These fluctuations in the plasma e-m-field themselves actas scattering agents in the medium and isotropise CR mo-tions. Therefore CRe diffuse in a random walk through thecluster atmosphere, limiting the effective diffusion speed tothe Alv´en speed of ≈
100 km / s .Relativistic electrons are subject to a number of energylosses in the ICM as well (Longair 1994). The most impor-tant being inverse Compton scattering (IC), synchrotron ra- through galactic outflows diation, bremsstrahlung and Coulomb scattering. Thus onecan define an approximate life-time of these particles at anmomentum p as t life = p (cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12) loss . (1)In figure 1 we show this life-time for CRe and relativisticprotons (CRp) as well as the escape time from the centralICM (modified from Blasi et al. 2007b).It has been realised early on that these losses limit thediffusion length of radio bright CRe in the ICM to a few 10kpc at best. It is unclear how a cluster-wide population ofCRe can be maintained to produce a giant radio halo (Jaffe1977). This is known as the life-time problem.The existence of giant radio halos underlines the incom-pleteness of the physical picture described above. IndeedDennison (1980) pointed out that relativistic protons havesufficiently long life-times to fill the cluster volume (seefigure 1). CRp are likely to be injected alongside CRe intothe ICM and can themselves inject synchrotron bright CRein-situ through a hadronic cascade. Models relying on thismechanism are called hadronic or secondary models (seesection 3).In addition turbulent reacceleration might be able to ex-plain giant radio halos (Brunetti et al. 2001; Jaffe 1977;Petrosian 2001). At low energies the synchrotron dark pop-ulation of CRe with life-times of > yrs can undergostochastic reacceleration through damping of merger in-jected turbulence. This Fermi II like mechanism (Fermi1949) lets CRe diffuse to higher momenta creating volumefilling radio bright CRe. It may act on CRp as well (Brunetti& Blasi 2005; Brunetti & Lazarian 2011a). Effectively thismechanism can boost the radio brightness of a CRe popu-lation without altering the total CRe density by producing c (cid:13) stron. Nachr. / AN (2013) 3 deviations from power-law spectra. This ansatz is realisedin reacceleration models (see section 4).Today the clear distinction between the two modelclasses can be considered a historic artefact as both pro-cesses certainly take place in the ICM to some degree. How-ever, CRp have not been observed directly in the ICM sofar and the efficiency of CRp injection in large scale ICMshocks is unclear. Hybrid particle in cell simulations sug-gest efficient CRp or CRe injection to be depending on thealignment of the magnetic field to the plane of the shock(Gargat´e & Spitkovsky 2012). However, the low densitiesand mach numbers of large shocks in the ICM make thesesimulations just barely feasable. To this end the efficiencyof CRp injection processes has not been firmly established.In general the plasma physics in the exotic regime of theICM presents a serious challenge to theorists and simula-tors. It should be considered that with mean free paths of > and collision times > yrs in the ICM (e.g. Sarazin1999; Spitzer 1956) particle-particle interactions are closeto time and length scales relevant to radio halos. Thereforecollisionality on kpc scales can not be mediated by particle-particle interactions . But observations of cold fronts andshocks confirm that the ICM is indeed a collisional plasmaon smaller scales (Markevitch & Vikhlinin 2007). Colli-sionality can be recovered by plasma waves which resultsin very high Reynolds numbers of the ICM of > or even > (Brunetti & Lazarian 2011b). Interestinglyfield-particle interactions happen on much smaller lengthand time scales in the ICM (debye length ≈ × cm (Schlickeiser 2002)).To date the Coma cluster remains the prototypical clus-ter with non-thermal radio emission (figure 1, left). It hasbeen studied extensively in the past and detailed studies onother clusters have been somewhat limited (Venturi 2011).However the situation will fundamentaly change with thearrival of new instruments like EMU+WODAN, LOFARand later the SKA (Cassano et al. 2012, 2010). At low fre-quencies of less than 100 MHz, radio halos are expected tobe bright and ubiquitous, due to their steep emission spec-trum. As increasingly more LOFAR stations see first light,studies of theoretical models for radio halos are particularytimely.In the past decade theoretical research on halos has beendevided along the lines of the two classical models. Numer-ical approaches used CRp dynamics and hadronic injection,sometimes with shock injection (Br¨uggen et al. 2012; Dolag& Ensslin 2000; Ensslin et al. 2007; Hoeft et al. 2004, 2008;Keshet & Loeb 2010; Miniati et al. 2001; Pfrommer et al.2008; Pinzke & Pfrommer 2010; Vazza et al. 2012). Inde-pendently there were (semi-)analytic approaches on reac-celeration models, sometimes including CRp (Brunetti &Blasi 2005; Brunetti & Lazarian 2007, 2011a; Brunetti et al.2001; Cassano & Brunetti 2005; Petrosian 2001). Turbu-lence and magnetic fields have been studied extensively in It comes at no surprise that hadronic models run into problems withnormalisation as we will see later. the cluster context as well (Dolag et al. 1999; Iapichino &Niemeyer 2008; Iapichino et al. 2011; Maier et al. 2009;Ryu et al. 2008; Subramanian et al. 2006; Vazza et al. 2008,2011, 2009). However hadronic models have only recentlybeen compared in detail to observations, while more com-plete approaches had not been considered in simulations yet(Donnert et al. 2010,?).This report is organised as follows: We start with a gen-eral overview of the important observations of radio ha-los. We then show results from the comparison of hadronicmodels with these observations. We continue with an in-troduction to reacceleration models and conclude with thefirst simulations of CRe reacceleration in direct cluster col-lisions.
Today roughly 50 radio halos are known, mostly at frequen-cies of 600 MHz and up (Feretti et al. 2012; Giovannini et al.2009). These halos are all hosted in massive systems withdisturbed X-ray morphology (Cassano et al. 2010). Thesesystems form a correlation in X-ray luminosity and radiobrightness with a slope of ≈ . (figure 3 left panel). How-ever in a complete X-ray sample these make only 30% of allsystems (Venturi et al. 2007). Therefore for most clustersonly upper limits in the radio are known (arrows in figure3, left). Recently a first detection of these clusters has beenclaimed through a stacking analysis (Brown et al. 2011).In the γ -ray regime, clusters remain not observed (Ack-ermann 2010; Aharonian 2009; Perkins 2008; Veritas Col-laboration et al. 2012).The Coma cluster has beed studied extensively in thepast. Three observations at 1.4 GHz are commonly used toconstrain radio synchrotron profiles of the cluster. It was de-bated (Pfrommer & Enßlin 2004) wether to use the steepersingle-dish profiles from Deiss et al. (1997) or the flatter in-terferometer profiles from Govoni et al. (2001). Howevernew single-dish GBT observations by Brown & Rudnick(2011) confirm the flat profile found previously by Govoniand present the deepest profile currently known.For the Coma cluster (Thierbach et al. 2003) and a smallnumber of other systems the spectrum of the diffuse emis-sion is observed to be a power-law with a spectral index of1.2 to 2.5. In figure 3, middle we show a few known exam-ples from Venturi (2011). Halos with a synchrotron spectralindex α ν > . (e.g. A521) are commonly named ultra-steep spectrum haloes. The Spectrum of the Coma clustershows a break/steepening of the emission, which has im-portant theoretical implications (see section 2.1). Howeverdifferences in instrumentation and analysis might contributeto the total shape of the spectrum. Individual points were es-timated with single-dish instruments as well as interferom-eters. While interferometers might lose diffuse flux becauseof limited UV-coverage, point source subtraction is compli-cated for single-dish data. Therefore the spectrum should betaken with a grain of salt. c (cid:13)(cid:13)
Today roughly 50 radio halos are known, mostly at frequen-cies of 600 MHz and up (Feretti et al. 2012; Giovannini et al.2009). These halos are all hosted in massive systems withdisturbed X-ray morphology (Cassano et al. 2010). Thesesystems form a correlation in X-ray luminosity and radiobrightness with a slope of ≈ . (figure 3 left panel). How-ever in a complete X-ray sample these make only 30% of allsystems (Venturi et al. 2007). Therefore for most clustersonly upper limits in the radio are known (arrows in figure3, left). Recently a first detection of these clusters has beenclaimed through a stacking analysis (Brown et al. 2011).In the γ -ray regime, clusters remain not observed (Ack-ermann 2010; Aharonian 2009; Perkins 2008; Veritas Col-laboration et al. 2012).The Coma cluster has beed studied extensively in thepast. Three observations at 1.4 GHz are commonly used toconstrain radio synchrotron profiles of the cluster. It was de-bated (Pfrommer & Enßlin 2004) wether to use the steepersingle-dish profiles from Deiss et al. (1997) or the flatter in-terferometer profiles from Govoni et al. (2001). Howevernew single-dish GBT observations by Brown & Rudnick(2011) confirm the flat profile found previously by Govoniand present the deepest profile currently known.For the Coma cluster (Thierbach et al. 2003) and a smallnumber of other systems the spectrum of the diffuse emis-sion is observed to be a power-law with a spectral index of1.2 to 2.5. In figure 3, middle we show a few known exam-ples from Venturi (2011). Halos with a synchrotron spectralindex α ν > . (e.g. A521) are commonly named ultra-steep spectrum haloes. The Spectrum of the Coma clustershows a break/steepening of the emission, which has im-portant theoretical implications (see section 2.1). Howeverdifferences in instrumentation and analysis might contributeto the total shape of the spectrum. Individual points were es-timated with single-dish instruments as well as interferom-eters. While interferometers might lose diffuse flux becauseof limited UV-coverage, point source subtraction is compli-cated for single-dish data. Therefore the spectrum should betaken with a grain of salt. c (cid:13)(cid:13) J.M.F. Donnert: Giant Radio Haloes
Fig. 3
Left: Radio brightness at 1.4 GHz over bolometric X-ray luminosity for a sample of galaxy cluster. Upper limitsof non-detections are marked as blue arrows, tentative detection from stacking as red limits (see Brown et al. 2011, andreferences therein). Middle: Radio spectra of 4 clusters (Venturi 2011). Right: Radial profile of Compton-y parameter(pressure along the LoS) and radio synchrotron emission in the Coma cluster out to R (Planck Collaboration 2012).The PLANCK satellite observed the Sunyaev-Zeldovicheffect in a numebr clusters with unprecedented sensitivityand resolution. A correlation between the cluster Compton-y and radio luminosity was found from PLANCK databy Basu (2012), however without a clear indication for abimodal distribution. This suggests that selection effectsmight contribute to the bimodality found in the X-ray-radiocorrelation.In Coma, Planck Collaboration (2012) found that pres-sure and radio emission are tightly correlated up to r ≈ (figure 3, right). Previous studies had shown thatdensity and radio luminosity are correlated as well, howeverthe scaling differs between clusters (Govoni et al. 2001).Recent X-ray observations find a non-thermal pressureof ≤ in Coma (Churazov et al. 2012). The current observational status implies a number of con-strains on the available models for a cluster-wide radiobright population of CRe :1. The predicted radio emission has to be consistent withthe observed profile (e.g. Brown & Rudnick 2011).2. The non-thermal pressure by CRp, magnetic fields andturbulence in clusters may not exceed the non-thermalpressure constraint from the X-rays: .3. The observation of ultra steep spectrum halos like A512indicates very steep CRe spectra of > (Brunetti et al.2008). This implies either ”old” reaccelerated CRe or aCRp population with very steep spectral index in theseclusters.4. The non-detection of γ -rays constrains the amount ofCRp in clusters.5. The large number of non-detections in the radio regimesuggests that radio halos are transient phenomena.Statistics roughly indicates that given the observed pop-ulation a cluster may not stay on the correlation (i.e.be radio loud at 1.4 Ghz) for more than half a Gyr,with a transition period of roughly 200 Myr (Brunetti et al. 2009). The correlation reflects the structural self-similarity found in cluster properties, which is due to theself-similarity of the underlying DM halo and its gravi-tiational potential (Navarro et al. 1996). Relativistic protons ( p/m p c > ) might be injected inthe ICM alongside CRe. CRp have long life-times and aretrapped in the cluster volume (Berezinsky et al. 1997; V¨olk& Atoyan 1999). If their coupling to turbulence is ineffi-cient their spectrum may only be slightly modified duringtheir life-time so they are assumed to retain their power-lawinjection spectrum (Blasi & Colafrancesco 1999): n p ( p ) dp = K p p − α CRp dp, (2)with a spectral index α p . From accelerator data we knowthat CRp inject secondary CRe at GeV energies and γ -raysduring collisions via a hadronic cascade (Dermer 1986): p CR + p th ⇒ π + π + + π − + anything π ± ⇒ µ + ν µ µ ± ⇒ e ± + ν µ + ν e π ⇒ γ. This allows to calculate the CR electron spectrum n e ( p ) di-rectly from the thermal density and the injection Q ( r , p (cid:48) , t ) of CRe from the CRp spectrum 2. Under the assumptionof slow varying MHD conditions in the ICM, injection andlosses (eqs. 17+18) of CRe balance, the spectrum equilli-brates and becomes stationary. The CR transport equation16 can be simplified to: ∂∂p (cid:18) n e ( p ) d p d t (cid:12)(cid:12)(cid:12)(cid:12) loss (cid:19) = Q ( r , p, t ) (3) c (cid:13) stron. Nachr. / AN (2013) 5 The solution of this equation is then trivial (Dolag & Ensslin2000): n e ( p ) = (cid:12)(cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12)(cid:12) − ∞ (cid:90) p d p (cid:48) Q ( r , p (cid:48) , t ) . (4)Then the synchrotron emissivity follows analytically fromthe CRe for power-law spectra (Ginzburg & Syrovatskii1965; Longair 1994; Pacholczyk 1970; Rybicki & Light-man 1986). For a spectrum of CRp like equation 2 the syn-chrotron emissivity scales like (e.g. Dolag & Ensslin 2000): j ν ∝ n th K p ν − α ν B α ν +1 B + B , (5)with α ν = ( α e − / α p − / the spectral indexof the radio synchrotron emission and B CMB / (1 + z ) =3 . µ G the magnetic field equivalent of the CMB inducedIC effect (Longair 1994). More elaborate models can beconstructed which are based on a piecewise solution of thediffusion equation of CRp (Miniati et al. 2001) or adia-batic invariants (Ensslin et al. 2007). This has also beencombined with a prescription for primary CRe and CRp in-jection in shocks (i.e. models for radio relics) (Pfrommeret al. 2008). Most models share the simplification of sta-tionarity of the CRe spectrum obtained from the (sometimesnon-stationary) CRp spectrum. In that case CR spectra andsubsequent synchrotron spectra are limited to power-laws,sometimes in a number of energy bins. The coupling to tur-bulence is neglected in these works.In hadronic models the injection efficiency for CRp andCRe in shocks is a free parameter. The γ -ray emissivitycan be computed analytically as well (Pfrommer & Ensslin2004).The magnetic field in clusters is then often modelled rel-ative to the thermal density (Bonafede et al. 2010; Dolaget al. 2001): B ( r ) = B (0) (cid:18) ρ ( r ) ρ (0) (cid:19) η mag . (6)Free model parameters are then:1. K p ( r ) = X CRp ( r ) n th ( r ) : the CRp normalisation isoften given relative to the thermal density ( X CRp ∈ [10 − , . ) or simulated directly in a numerical ap-proach. It is constrained by the synchrotron profile andhas to be consistent with the observed non-thermal pres-sure and γ -ray limits.2. α p ∈ ]2 . , . : the CRp spectral index translates toCRe and synchrotron spectal index and is therefore con-strained by observed radio halo spectra. The spectral in-dex enters the γ -ray emissivity as well. The observedbreak in the radio spectrum of Coma is often (falsely)attributed to the SZ-decrement. This is then combined when considering the Thomson pp cross-section. The full energy de-pendent cross-section gives a slightly different scaling. Given the currentobservational data the difference can be ignored. with a flatter α p and a flat magnetic field to fit γ -rayconstrains (Veritas Collaboration et al. 2012). We willdiscuss this in section 3.2.1.3. B (0) ∈ [1 , µ G : the cluster magnetic field normal-isation is directly constrained from RM observations,equipartition arguments and simulations.4. η mag ∈ [0 , : the cluster magnetic field scaling rela-tive to the thermal density. While being constrained bythe same observations as the normalisation some modelsassume (nearly) flat magnetic fields. Exemplary we show here results from constrained cos-mological MHD simulations published in Donnert et al.(2010,?). These are consistent with analytic comparisonswith newest data (Brunetti et al. 2012).The constrained power-spectrum initial conditions pro-duces matter structures which on scales of a few Mpc (clus-ters) can be identified with their real counterparts (Mathiset al. 2002). The magnetic field was initialised from galac-tic outflows as shown in Donnert et al. (2009). The radioemission is then estimated with an analytic approach to theCR physics: we use the high energy approximation to thep-p cross-section in Brunetti & Blasi (2005) to compute thesynchrotron emissivity of a cluster. We estimate the γ -raybrightness from our models using the formalism by Pfrom-mer & Ensslin (2004).We model the radial profile of CRp normalisation X CRp ( r ) relative to the thermal density as: X CRp ( r ) = X CRp (0) (cid:18) ρ ( r ) ρ (0) (cid:19) − η CRp . (7)We will fix the normalisation X CRp (0) and the scaling η CRp in the next section to fit the observed radio profile. The pro-ton spectral index α CRp is another free parameter which wewill constrain from the observed spectrum. The magneticfield is taken from the simulation and is roughly consistentwith observations in Coma with η = 0 . and B ≈ µ G (Bonafede et al. 2010). The result is shown in figure 4,where we the magnetic field profile is plotted alongside thesquare root of the density. In this section we will focus on the Coma cluster first.We show how multi-frequency observations constrain thehadronic model. An overview of the parameters used can befound in table 1. Then we show the P14-LX correlation fora sample of simulated galaxy clusters and comment on theimplications of the PLANCK correlation between thermalpressure and radio emission in Coma.
Hadronic models predict a power-law synchrotron spectrum(equation 5), in contrast to the observed break in the Coma c (cid:13)(cid:13)
Hadronic models predict a power-law synchrotron spectrum(equation 5), in contrast to the observed break in the Coma c (cid:13)(cid:13) J.M.F. Donnert: Giant Radio Haloes
10 100 1000 10000 ν [MHz]0.010.101.0010.00100.00 F ν [ m J y ] Thierbach et al. 2003Model 1Model 2Model 3
100 1000r [kpc]10 -6 -5 -4 P . [ m J y a r c s ec - ] -3 -2 -1 P / P r = c
300 MHz, Brown & Rudnick 2011Model 1Model 2Model 3
100 1000r [kpc]10 -13 -12 -11 -10 P r e ss u r e [ g s - c m - ]
10% P thermal
Model 3Model 2Model 1 515 kpc400 kpc260 kpc -10 -9 -8 -7 F γ , - G e V [ γ c m - s - ] FERMI 1-3 GeV (Arlen et al. 2011)
Fig. 5
Top left: Observed radio spectrum in the Coma cluster, black diamonds (Thierbach et al. 2003). Spectra of threehadronic models with α CRp = 2 . , . , . and η CRp = 1 . , . , . . The SZ-decrement has been included pixel by pixelfrom the observed PLANCK map (Brunetti et al. in prep.) . Top right: Radial profile of radio synchrotron emission inComa at 300 MHz (top, normalised, Brown & Rudnick 2011) and 1.4 GHz (bottom, Deiss et al. 1997). In colors the threemodels with a CRp scaling to fit the conservative profile at 1.4GHz (red, blue) and the deep profile at 300 MHz (green).Bottom left: Non-thermal pressure constraint in Coma (black line, Churazov et al. 2012) and non-thermal pressure fromthe numerical models. Bottom right: Fermi upper limits at α CRp = 2 . , . from Veritas Collaboration et al. (2012) andexpected γ -ray flux from the three numerical models.
10 100 1000r [kpc]10 -7 -6 | B | [ G ] , ρ / ( a r b . un it s ) Magnetic fieldSquare root of thermal densityResolution scale
Fig. 4
Radial profile of magnetic field strength and squareroot of the density (dashed) of the simulated cluster. Thesmallest resolution scale h sml is marked as dotted line. spectrum. It has been argued that the break can be explainedby the Sunyaev-Zeldovich decrement in the unavoidableCMB radio signal measured alongside the actual halo emis-sion (Ensslin 2002; Pfrommer & Ensslin 2004). The SZ-decrement is the modification of the CMB thermal emissionby the clusters Sunyaev-Zeldovich effect at halo frequen-cies (Sunyaev & Zeldovich 1980). This is not connected tothe halo emission, but contaminates the total emission mea-sured from the halo. Naturally the influence is limited to theregion of the radio halo only. However Ensslin (2002) con-sidered the flux from a region more than twice the size ofthe radio halo (5 Mpc radius), therefore overestimating thedecrement.Due to the high resolution of the PLANCK measure-ments, the SZ-decrement can now be directly estimatedfrom observations. It is instructive to assume three spec-tral indices: α CRp = 2 . and α CRp = 2 . , as well as c (cid:13) stron. Nachr. / AN (2013) 7Parameter Model 1 Model 2 Model 3 α CRp η CRp X CRp (0) B [ µ G] η mag ν MHz
Table 1
Parameters and values for the three numericalmodels used in this comparison. α CRp = 2 . . In figure 5, top left we show the Coma ra-dio spectrum and three analytical hadronic models withthese spectral indexes, including the SZ-decrement from thePLANCK data (preliminary results from Brunetti in prep.).The spectra are normalised to 300 MHz, where the influenceof the decrement is negligible. Only the last model leads to amarginally consistent fit of the observed radio spectrum. Ahadronic model with spectral index of 2.6 is already morethe a factor of two away from the observations. Thereforeeven given the uncertainties in the observations (see section2) the SZ-decrement is not able to reproduce the break forindizes smaller than 2.6. The total radio flux and radial profile of the emission inComa can be used to constrain the amount of CRp in thecluster, i.e. X CRp ( r ) . We set the scaling according to equa-tion 7 to fit (table 1) the observed radio profile at 1.4 GHzfor model one and two (Deiss et al. 1997), and 300 MHz formodel three (Brown & Rudnick 2011). The fits are shownfor all three models in figure 5, top right graph. This im-plies η CRp = 1 . for the models with small spectral indexand η CRp = 2 . for the other one. The total luminosity ofthe models is fixed at 1.4 GHz to be ± / arcsec following Deiss et al. (1997).The first two models use the older radio profile at1.4GHz. The is very conservative estimates as at low fre-quencies the halo is observed to be much flatter and larger(Brown & Rudnick 2011; Govoni et al. 2001; Venturi et al.1990). Indeed the third model uses the state-of-the-art ra-dio observation of the COMA cluster and models the bestconstrains available on the radio halo problem today.CRp exert non-thermal pressure to the ICM, similar toturbulence and magnetic fields. Recent observational stud-ies on Coma set an upper limit to non-thermal pressureof 10 % in the cluster (Churazov et al. 2012). For com-parison we plot in figure 5 bottom left, the thermal pres-sure profile (dashed line) alongside the 10% limit (fullline) and the non-thermal pressure of the three models (red,green, blue). The pressure constrain is violated at radii of r ≈ , ,
260 kpc , while the observed radio halo at300 MHz extends to ≈ . Therefore observationsimply considerable non-thermal pressure at large radii forhadronic models. For the Coma cluster r could be in-creased by flatter magnetic field models for flat CRp spectra. Fig. 7
Radio brightness at 1.4GHz over X-ray brightnessbetween 0.2 and 2.4 keV for 16 clusters with X CRp ( r ) =1% and α CRp = 2 . . Observed halos as diamonds from(Cassano et al. 2007) and upper limits from (Venturi et al.2007).However this becomes less consistent with magnetic fieldsderived from observed rotation measures.The situation becomes increasingly problematic whenconsidering halos like A512. The steep spectrum of theseobjects require abundant CRp contents to reproduce theemission at lowest frequencies. Observed synchrotron spec-tra of α ν ≈ . imply α CRp = 3 . . This results in non-thermal pressures of more than 50 % within 3 core radii(Brunetti et al. 2008; Dallacasa et al. 2009). γ -ray brightness Observations of γ -ray emission from hadronic collisions inthe ICM are an independent way of constraining the CRpcontent in clusters. Newest upper limits in this frequencyband come from FERMI and VERITAS, while the for-mer ones are most constraining (Veritas Collaboration et al.2012).In figure 5 bottom right, we show the most constrain-ing upper limits from FERMI ( α CRp = 2 . , . ) as wellas our three numerical models. Only the first model withthe flattest CRp spectrum is marginally consistent with thenewest γ -limits given the simulated magnetic fields. Theother two steeper models violate the observed limits. Againthe situation could be eased by a stronger and flatter mag-netic field profile in the case of flat CRp spectra. The steep α CRp model however can be considered excluded by γ -rayobservations. It is the only one roughly consistent with theobserved radio spectrum though. Figure 7 shows 16 simulated clusters from the cosmolog-ical simulation in the P − L X − ray plane (triangles, X CRp = 1% , α CRp = 2 . ). Overplotted are the observedcorrelation (dotted line), observed radio halos (red & white c (cid:13)(cid:13)
Radio brightness at 1.4GHz over X-ray brightnessbetween 0.2 and 2.4 keV for 16 clusters with X CRp ( r ) =1% and α CRp = 2 . . Observed halos as diamonds from(Cassano et al. 2007) and upper limits from (Venturi et al.2007).However this becomes less consistent with magnetic fieldsderived from observed rotation measures.The situation becomes increasingly problematic whenconsidering halos like A512. The steep spectrum of theseobjects require abundant CRp contents to reproduce theemission at lowest frequencies. Observed synchrotron spec-tra of α ν ≈ . imply α CRp = 3 . . This results in non-thermal pressures of more than 50 % within 3 core radii(Brunetti et al. 2008; Dallacasa et al. 2009). γ -ray brightness Observations of γ -ray emission from hadronic collisions inthe ICM are an independent way of constraining the CRpcontent in clusters. Newest upper limits in this frequencyband come from FERMI and VERITAS, while the for-mer ones are most constraining (Veritas Collaboration et al.2012).In figure 5 bottom right, we show the most constrain-ing upper limits from FERMI ( α CRp = 2 . , . ) as wellas our three numerical models. Only the first model withthe flattest CRp spectrum is marginally consistent with thenewest γ -limits given the simulated magnetic fields. Theother two steeper models violate the observed limits. Againthe situation could be eased by a stronger and flatter mag-netic field profile in the case of flat CRp spectra. The steep α CRp model however can be considered excluded by γ -rayobservations. It is the only one roughly consistent with theobserved radio spectrum though. Figure 7 shows 16 simulated clusters from the cosmolog-ical simulation in the P − L X − ray plane (triangles, X CRp = 1% , α CRp = 2 . ). Overplotted are the observedcorrelation (dotted line), observed radio halos (red & white c (cid:13)(cid:13) J.M.F. Donnert: Giant Radio Haloes
Fig. 6
Full sky map of radio flux from a hadronic model with X CRp ( r ) = 1% and α CRp = 2 . in a cosmological MHDsimulation.diamonds, Venturi et al. (2007)) and upper limits (red lines).The observed correlation is well reproduced by the largestclusters in the simulated sample.However the brightness distribution from the simula-tion is not bimodal: all large clusters show significant radioemission. This is an intrinsic prediction of hadronic models.As CR protons are trapped in the cluster volume they accu-mulate in the ICM. In figure 6 we show a full sky projectionof the radio synchrotron emission from the simulation usingthe X CRp = const model and the simulated magnetic field.Every cluster shows diffuse radio emission. The radio skypredicted by this model is shown in figure 6. Given our simulated magnetic fields, hadronic models:1. reproduce the P − L X − ray correlation, but in thecase of classical models fail the bimodal distribution ofthe population.2. do not reproduce the deepest observed radio profile inComa within the non-thermal pressure constrains. Thiscould be eased using a flatter and stronger magneticfield distribution. However this tends to be not consis-tent with observed RM measures in the Coma cluster.3. are not consistent with present upper limits in the γ -rayregime when reproducing deepest Coma radio observa-tions and its radio spectrum. Models using α CRp = 2 . are indeed consistent with these limits if the 1.4GHzprofile is fitted. However they do not fit the Coma spec-trum (figure 5, top left).4. do not reproduce the break in the Coma spectrum when α CRp < .The observed constrains present serious challenges forhadronic models. The observed spectrum of the Coma clus-ter is impossible to fit without violating non-thermal pres-sure constrains and γ -ray limits. In the most recent attempts Veritas Collaboration et al. (2012); Zandanel et al. (2012)report a good fit to the observed constrains. However inthese papers Coma is modelled with α CRp = 2 . − . at 1.4 GHz only ! The PLANCK Compton-y - Radio Correlation:
Thesituation can be presented most elegantly from the observedCompton-y - 300 MHz radio profile correlation in Coma(Mazzotta & Planck Collaboration 2012). In the center ofthe cluster, the fit in figure 5 (and most other hadronicmodels) predict X CRp ≈ − − − (depending on theCRp spectral index) to reproduce the luminosity profile lo-cally. At the outer radii ( < r < ), where B << B
CMB , PLANCK found: y Compton ∝ j . ( r ) . (8)The Compton-y parameter measures pressure, so: y Compton ( r ) ∝ P ∝ n th T, (9) j . ∝ X CRp n B . (10)The temperature profile in clusters is roughly constant withradius, and we have X CRp ∝ n − B − , and (11) n th ∝ (1 + r /r ) , (12)for r ≤ r . It follows from the PLANCK correlation that X CRp ∝ (1+ r /r ) , even for a flat magnetic field. In Coma r c ≈ . r , so X CRp has to increase by a factor of 10 inthis case. If α CRp > . then X CRP ≈ . and the non-thermal pressure constraint of 10% is violated at 1Mpc evenfor flat magnetic field models. because of the over-estimation of the SZ -decrement c (cid:13) stron. Nachr. / AN (2013) 9 However observed rotation measures suggest B ∝ n − . (Bonafede et al. 2010). Here the increase is a fac-tor of 100 to 1000 and the non-thermal pressure constraintis violated in any case. Recently Enßlin et al. (2011) proposed a model based onCRp transport in the ICM in an attempt to explain the bi-modal distribution of halos in the hadronic framework. Inthis model the bimodality reflects two modes of CRp trans-port in clusters: – During mergers CRp transport is dominated by convec-tion through turbulent motions. CR protons are effec-tively stored between converging magnetic field lines and dragged along the turbulent motions. These motionsare injected at a scale of the core radius and induce anMHD cascade of cluster-wide turbulence. This way theCRp density in the center of clusters is increased. Theresulting flatter X CRp profile leads to the observed ra-dio bright clusters. – After turbulence has decayed CRp transport is supposedto be dominated by super-Alvenic streaming of CRpalong the magnetic field lines. If small scale turbulenceis not driven by external processes the streaming insta-bility is the only source of fluctuations at that scale.Scattering of CRs on these fluctuations limits the diffu-sion speed to the Alven speed (Wentzel 1974). Specif-ically the isotropisation of CRp for small pitch angles(i.e. reversal of direction along the field) is mediatedby scattering with large modes (i.e. mirror interactions)(Felice & Kulsrud 2001). However these modes aredamped efficiently by the ICM plasma through the ioncycloton resonance. Enßlin et al. (2011); Holman et al.(1979) argue that this way the streaming instablity is in-efficient to generate modes at small pitch angles. Thiswould allow streaming motions along the field linescomparable to the sound speed (highly super-Alvenic).Eventually this tends to remove CRp from the clustercenter and flatten the CRp profile of the cluster. Sub-sequently the radio brightness declines. This processmight as well be able to explain the steepening of theradio spectrum as a superposition of different CRp pop-ulations.However the assumption of streaming velocities higher thanthe Alven speed had been rejected earlier for CRe and onlylead to the cooling time dilemma. We therefore see threeproblems in the argumentation presented in (Enßlin et al.2011; Holman et al. 1979) and above: – Cosmological simulations show that even in relaxedclusters, structure formation causes a constant infallof small halos into the ICM, driving turbulence. TheICM itself is not a classical collisional plasma. Probably This is consistent with our fit (figure 5, left, cyan curve). Similar to magnetic bottles
Fig. 8
Diffusion coefficient over pitch angle in radians(Achterberg 1981). The quasi-linear result is shown dashed,the non-linear result as full curve. In clusters E > E M .collective effects mediate collisionality, which impliesvery high Reynolds numbers of the plasma (Brunetti& Lazarian 2011b). Therefore the infall of structuresalways causes turbulence on kpc scales which is dis-sipated at sub-pc scales. This is in contrast to thenon-classical hadronic approach which relies on smallmodes generated exclusively by the streaming instabil-ity. Even though damping processes are ill constrainedin the ICM it is likely that externally driven turbulenceis always present at some level on the damping scale ofCRp. – Holman et al. (1979) argue that at pitch angles of µ < turbulence is completely absent due to the strongdamping by the thermal ions. However even in the ab-sense of external driving, turbulence is constantly in-jected on a range of scales by CRp through the stream-ing instability. Due to mode coupling these motions willform a cascade and develop a break with finite steepness at the wavelength of interest (Schlickeiser 2002; Span-gler 1986). Therefore turbulence is unlikely to be com-pletely absent, even if ion cycloton damping is strongerthan the growth rate from the streaming instability. – Holman et al. (1979) calculate the diffusion coefficientin the quasi linear regime. In that formalism the res-onance function R j ( k , ω j ) is approximated by a deltafunction (e.g. Schlickeiser 2002): R j ( k , ω j ) = πδ ( vµk (cid:107) − ω R , j + n Ω) . (13)However, this approximation is only valid if Γ j → ∞ ,i.e. damping of plasma modes is negligible (Melrose1980). This is not the case in the situation consideredhere. Ion cycloton damping obviously has to be con-sidered, because it is supposed to dominate at smallpitch angles. The correct resonance function is of Breit-Wigner type (e.g. Schlickeiser 2002): R j ( k , ω j ) = Γ j ( k )Γ j ( k ) + ( vµk (cid:107) − ω R , j + n Ω) (14) c (cid:13)(cid:13)
Diffusion coefficient over pitch angle in radians(Achterberg 1981). The quasi-linear result is shown dashed,the non-linear result as full curve. In clusters E > E M .collective effects mediate collisionality, which impliesvery high Reynolds numbers of the plasma (Brunetti& Lazarian 2011b). Therefore the infall of structuresalways causes turbulence on kpc scales which is dis-sipated at sub-pc scales. This is in contrast to thenon-classical hadronic approach which relies on smallmodes generated exclusively by the streaming instabil-ity. Even though damping processes are ill constrainedin the ICM it is likely that externally driven turbulenceis always present at some level on the damping scale ofCRp. – Holman et al. (1979) argue that at pitch angles of µ < turbulence is completely absent due to the strongdamping by the thermal ions. However even in the ab-sense of external driving, turbulence is constantly in-jected on a range of scales by CRp through the stream-ing instability. Due to mode coupling these motions willform a cascade and develop a break with finite steepness at the wavelength of interest (Schlickeiser 2002; Span-gler 1986). Therefore turbulence is unlikely to be com-pletely absent, even if ion cycloton damping is strongerthan the growth rate from the streaming instability. – Holman et al. (1979) calculate the diffusion coefficientin the quasi linear regime. In that formalism the res-onance function R j ( k , ω j ) is approximated by a deltafunction (e.g. Schlickeiser 2002): R j ( k , ω j ) = πδ ( vµk (cid:107) − ω R , j + n Ω) . (13)However, this approximation is only valid if Γ j → ∞ ,i.e. damping of plasma modes is negligible (Melrose1980). This is not the case in the situation consideredhere. Ion cycloton damping obviously has to be con-sidered, because it is supposed to dominate at smallpitch angles. The correct resonance function is of Breit-Wigner type (e.g. Schlickeiser 2002): R j ( k , ω j ) = Γ j ( k )Γ j ( k ) + ( vµk (cid:107) − ω R , j + n Ω) (14) c (cid:13)(cid:13) Considering this resonance function the diffusion co-efficient has been investigated in non-linear theoreticalapproaches (Achterberg 1981; Ben-Israel et al. 1975;Dupree 1966; Goldstein 1976; Jones et al. 1978; V¨olk1973; Weinstock 1969, 1970; Yan & Lazarian 2008).The resulting diffusion coefficient ( D µ ) is consistentbetween authors. We show in figure 8 the result fromAchterberg (1981) which does not vanish for µ → ,in the case of fully relativistic particles. CR scatteringthrough µ = 0 appears not to be a problem in the cor-rect non-linear approach.We conclude that CR streaming is not important in galaxyclusters. The ICM thermal plasma might be dominated by field-particle interactions to establish the observed collisionalityof the plasma. This can be seen when comparing particle-particle and particle-field interactions: In a cluster, parti-cle collision lengths are > and the speed of a par-ticle in the collisionless regime is the sound speed, ≈ cm / s . However the debye length is of the order of
20 km (Schlickeiser 2002). That means every charged par-ticle is coupled to ≈ neighbouring particles throughits em-field, with quasi instantaneous interaction speed (thespeed of light). In contrast the time between collisions is cm / × cm / s ≈ .One may consider, that major mergers drag erg ofkinetic energy into the ICM. The most part is dissipatedthrough shocks into heat, which is prominently seen in ra-dio relics (e.g. Vazza et al. 2012, 2009). Additionally, vor-tical motions, shear flows and instabilities efficiently driveturbulence in the ICM (Subramanian et al. 2006), becauseof its high Reynolds number. This amplifies magnetic fieldsto µG values (Beck et al. 2012; Dolag et al. 2001; Don-nert et al. 2009). Furthermore, turbulent modes (local fieldfluctuations) resonantly couple to the cosmic-ray electronsin the ICM on pc scales (Schlickeiser 2002). This way asynchrotron dark, long-lived population of CRe can dif-fuse throughout a cluster, and is accelerated to synchrotronbright energies by turbulence, on a time-scale shorter thanthe life-time of radio halos (Brunetti et al. 2001; Petrosian2001; Schlickeiser et al. 1987). Models involving reacceleration do not assume stationar-ity of the CR population. Therefore the analytical formal-ism based on the stationarity condition, eq. 3, can not beused in this framework. Again the population of cosmic-rays is described by its isotropic spectral density n ( p, t ) inmomentum space. The dynamics of this spectrum is deter-mined by the interplay between cooling, due to radiative and the distance it takes to shield the field of an extra charge Coulomb losses, and acceleration, due to turbulence (figure1,left). The momentum transport is governed by a Fokker-Planck equation (16), in which these processes are realisedas coefficients. The loss terms (eqs. 17, 18) define a cool-ing time scale which is given by equation 1, and is of theorder of yrs for synchrotron bright CRe in the ICM (seesection 1).The coupling to turbulence through resonant scatteringon plasma waves is realised through the D pp coefficient (eq.23), which defines an acceleration time scale (e.g. Cassano& Brunetti 2005) : t − = 4 D pp p . (15)Stochastic acceleration is efficient only , if this time scaleis comparable or smaller than the cooling time scale in themedium. For the ICM this is true, when the local turbulentvelocity is of the order of ≈
300 km / s on scales of
50 kpc . The change of the spectrum n ( p ) in momentum space is de-scribed by a Fokker-Planck equation. It follows from the rel-ativistic Maxwell-Vlasov system of equations (see Schlick-eiser 2002, and references therein). Here one neglects thespatial diffusion of cosmic-rays in the ICM for reasonslayed out in section 3.4 (Wentzel 1974). ∂n ( p, t ) ∂t = ∂∂p (cid:20) n ( p, t ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12)(cid:12) rad + (cid:12)(cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12)(cid:12) ion − p D pp ( p ) (cid:19)(cid:21) + ∂∂p (cid:20) D pp ( p ) ∂n ( p, t ) ∂p (cid:21) + Q e ( p, t ) . (16)Conceptually this is a non-linear momentum diffusion equa-tion. Setting D pp = 0 , one recovers the usual diffusionequation with the loss terms as positive definite diffusioncoefficient. This leads to diffusion to lower momenta, i.e.cooling. For D pp (cid:54) = 0 this diffusion coefficient is modi-fied and can become negative, leading to diffusion to highermomenta, i.e. heating. Additionally, the second term on theright hand side describes a non-linear broadening of the dis-tribution. The evolution of a spectrum is shown in figure 11,left for constant injection of power-law CRe at momenta p ∈ [50 , ] m e c . The injection of CRe in the Fokker-Planck equation isdescribed by an injection function Q ( p, t ) . Possible injec-tion sources include shock acceleration, hadronic processes,reconnection, AGN and galactic outflows (Blasi et al. 2007;Lazarian & Brunetti 2011). In general these processes arefound to inject power-law distributions of CRe into the ICM(see e.g. Blasi 2010, for details). They are missing a factor of two in the corresponding formula c (cid:13) stron. Nachr. / AN (2013) 11 The loss terms due to inverse Compton scattering withCMB photons and synchrotron radiation and Coulomblosses at z = 0 take the form (Cassano & Brunetti 2005): d p d t (cid:12)(cid:12)(cid:12)(cid:12) rad = − . × − p (cid:34)(cid:18) B µG . (cid:19) + 1 (cid:35) , (17) d p d t (cid:12)(cid:12)(cid:12)(cid:12) ion = − . × − n th (cid:20) (cid:18) γn th (cid:19) / (cid:21) , (18)where B µ G is the magnetic field in micro Gauss, n th thenumber density of thermal particles and γ the Lorentz fac-tor. One may observe that synchrotron emission can be de-scribed with an effective magnetic field in the IC formula.This stems from the fact that IC and synchrotron mech-anisms use the same quantum-mechanical scattering pro-cess (Longair 1994). They are dominant at higher momenta( p > m e c ). Ionisation losses and Coulomb scatter-ing depend on the thermal number density of particles inthe ICM n th and are dominant in the low energy regime( p < m e c ). The momentum diffusion coefficient follows from apertubative approach to the relativistic Maxwell-Vlasovsystem of equations. Here it is assumed that the change ofthe field perturbation by the particle can be neglected . Thisis referred to as quasi-linear-theory . The rigorous derivationis lengthy, see Schlickeiser (2002) for details. We followhere a more physical the approach by Brunetti & Lazarian(2007).The D pp coefficient describes the interaction of cosmic-rays with turbulence in a plasma. Turbulent fluctuationsmanifest in a plasma, amongst others, in fluctuations of theelectric and magnetic field. In the ICM these fluctuationscan be damped by cosmic-rays, similar to how a dielectricdamps an infalling light-wave.The dielectric tensor describes the action of themedium/CRs on the fluctuating e/m-fields. In the core ofthis process is the resonance condition (eq. 19), which de-scribes the coupling of the particle population to the e/m-fields. However turbulence is a statistical process and theturbulent fluctuations are better described by a spectrum in k -space. This spectrum evolves according to a damped dif-fusion equation (eq. 21). To describe the action of the di-electric on the damping of the waves the dielectric tensorhas to be translated to a damping coefficient (eq. 20). Usingdetailed balancing and a Kraichnan spectrum, the diffusioncoefficient (eq. 23) can then be derived from an energy ar-gument (eq. 22).Brunetti & Lazarian (2007) consider only the scatteringby fast compressive turbulent modes . This is the least effi-cient coupling, which makes this a conservative approach .Alven waves have been considered in e.g Brunetti et al. I.e. damping can be neglected; compare with CR streaming, section3.4, where this is not the case. An MHD version of sound waves (2004) and Brunetti & Blasi (2005), but require CR protonsand their back-reaction on the spectrum.In the presence of compressible low-frequency MHD-waves with an energy spectrum W ( k ) , relativistic particleswith velocity v in a magnetic field are accelerated by elec-tric field fluctuations. In quasi-linear theory this gyroreso-nant interaction applies for the resonance condition (Mel-rose 1968): ω − k (cid:107) v (cid:107) − n Ω γ = 0 , (19)with ω the frequency of the wave, k (cid:107) and v (cid:107) the wave-vectorand particle velocity parallel to the magnetic field, Ω theLarmor frequency, γ the Lorentz factor and n the resonanceorder. Here only n = 0 case is considered (Transit timedamping), which requires effective pitch-angle isotropisa-tion of the particles by other processes.The resonance condition (19) can be used to derive thedamping rate Γ of turbulence from the dielectric tensor K a ij in the limit of long wavelengths. Note that here indeed oneis allowed to use the quasi-linear approximation equation13, because the damping is very small. The damping ratethen follows from the dielectric tensor via (Melrose 1968): Γ = − i (cid:18) E ∗ i K a ij E j πW ( k ) (cid:19) ω i =0 ω r , (20)where E i are the electric fields, ω r is the real part of thefrequency of the fluctuating field, and ω i its imaginary part.The turbulence spectrum follows the damped diffusionequation: ∂W k ( t ) ∂t = ∂∂k (cid:18) D kk ∂W k ( t ) ∂k (cid:19) − N (cid:88) i =1 Γ i ( k ) W k ( t ) , (21)with the diffusion coefficient due to mode coupling D kk = k /τ s , τ s the spectral energy transfer time. Here i sums overall damping processes of turbulence in the plasma. For thehigh- β ICM thermal particles and CRp argueably do notdamp fast modes efficiently. Therefore we only considerCRe damping here ( N = 1 ).The argument of detailed balancing then states that theenergy lost by the turbulent waves below a scale l due toCRe damping equals the energy change in the relativisticparticles. It relates the total energy in a mode W ( k ) d k tothe particle spectrum (Achterberg 1981; Eilek 1979): (cid:90) d p E CRe (cid:18) ∂f CRe ( p ) ∂t (cid:19) = (cid:90) d k Γ CRe ( k, θ ) W ( k ) . (22)From this argument D pp can be found: D pp ≈ × − p v c l η (23)for Kraichnan turbulence P ( k ) ∝ k − / and an energy frac-tion in magneto-sonic waves η turb . c (cid:13)(cid:13)
50 kpc . The change of the spectrum n ( p ) in momentum space is de-scribed by a Fokker-Planck equation. It follows from the rel-ativistic Maxwell-Vlasov system of equations (see Schlick-eiser 2002, and references therein). Here one neglects thespatial diffusion of cosmic-rays in the ICM for reasonslayed out in section 3.4 (Wentzel 1974). ∂n ( p, t ) ∂t = ∂∂p (cid:20) n ( p, t ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12)(cid:12) rad + (cid:12)(cid:12)(cid:12)(cid:12) d p d t (cid:12)(cid:12)(cid:12)(cid:12) ion − p D pp ( p ) (cid:19)(cid:21) + ∂∂p (cid:20) D pp ( p ) ∂n ( p, t ) ∂p (cid:21) + Q e ( p, t ) . (16)Conceptually this is a non-linear momentum diffusion equa-tion. Setting D pp = 0 , one recovers the usual diffusionequation with the loss terms as positive definite diffusioncoefficient. This leads to diffusion to lower momenta, i.e.cooling. For D pp (cid:54) = 0 this diffusion coefficient is modi-fied and can become negative, leading to diffusion to highermomenta, i.e. heating. Additionally, the second term on theright hand side describes a non-linear broadening of the dis-tribution. The evolution of a spectrum is shown in figure 11,left for constant injection of power-law CRe at momenta p ∈ [50 , ] m e c . The injection of CRe in the Fokker-Planck equation isdescribed by an injection function Q ( p, t ) . Possible injec-tion sources include shock acceleration, hadronic processes,reconnection, AGN and galactic outflows (Blasi et al. 2007;Lazarian & Brunetti 2011). In general these processes arefound to inject power-law distributions of CRe into the ICM(see e.g. Blasi 2010, for details). They are missing a factor of two in the corresponding formula c (cid:13) stron. Nachr. / AN (2013) 11 The loss terms due to inverse Compton scattering withCMB photons and synchrotron radiation and Coulomblosses at z = 0 take the form (Cassano & Brunetti 2005): d p d t (cid:12)(cid:12)(cid:12)(cid:12) rad = − . × − p (cid:34)(cid:18) B µG . (cid:19) + 1 (cid:35) , (17) d p d t (cid:12)(cid:12)(cid:12)(cid:12) ion = − . × − n th (cid:20) (cid:18) γn th (cid:19) / (cid:21) , (18)where B µ G is the magnetic field in micro Gauss, n th thenumber density of thermal particles and γ the Lorentz fac-tor. One may observe that synchrotron emission can be de-scribed with an effective magnetic field in the IC formula.This stems from the fact that IC and synchrotron mech-anisms use the same quantum-mechanical scattering pro-cess (Longair 1994). They are dominant at higher momenta( p > m e c ). Ionisation losses and Coulomb scatter-ing depend on the thermal number density of particles inthe ICM n th and are dominant in the low energy regime( p < m e c ). The momentum diffusion coefficient follows from apertubative approach to the relativistic Maxwell-Vlasovsystem of equations. Here it is assumed that the change ofthe field perturbation by the particle can be neglected . Thisis referred to as quasi-linear-theory . The rigorous derivationis lengthy, see Schlickeiser (2002) for details. We followhere a more physical the approach by Brunetti & Lazarian(2007).The D pp coefficient describes the interaction of cosmic-rays with turbulence in a plasma. Turbulent fluctuationsmanifest in a plasma, amongst others, in fluctuations of theelectric and magnetic field. In the ICM these fluctuationscan be damped by cosmic-rays, similar to how a dielectricdamps an infalling light-wave.The dielectric tensor describes the action of themedium/CRs on the fluctuating e/m-fields. In the core ofthis process is the resonance condition (eq. 19), which de-scribes the coupling of the particle population to the e/m-fields. However turbulence is a statistical process and theturbulent fluctuations are better described by a spectrum in k -space. This spectrum evolves according to a damped dif-fusion equation (eq. 21). To describe the action of the di-electric on the damping of the waves the dielectric tensorhas to be translated to a damping coefficient (eq. 20). Usingdetailed balancing and a Kraichnan spectrum, the diffusioncoefficient (eq. 23) can then be derived from an energy ar-gument (eq. 22).Brunetti & Lazarian (2007) consider only the scatteringby fast compressive turbulent modes . This is the least effi-cient coupling, which makes this a conservative approach .Alven waves have been considered in e.g Brunetti et al. I.e. damping can be neglected; compare with CR streaming, section3.4, where this is not the case. An MHD version of sound waves (2004) and Brunetti & Blasi (2005), but require CR protonsand their back-reaction on the spectrum.In the presence of compressible low-frequency MHD-waves with an energy spectrum W ( k ) , relativistic particleswith velocity v in a magnetic field are accelerated by elec-tric field fluctuations. In quasi-linear theory this gyroreso-nant interaction applies for the resonance condition (Mel-rose 1968): ω − k (cid:107) v (cid:107) − n Ω γ = 0 , (19)with ω the frequency of the wave, k (cid:107) and v (cid:107) the wave-vectorand particle velocity parallel to the magnetic field, Ω theLarmor frequency, γ the Lorentz factor and n the resonanceorder. Here only n = 0 case is considered (Transit timedamping), which requires effective pitch-angle isotropisa-tion of the particles by other processes.The resonance condition (19) can be used to derive thedamping rate Γ of turbulence from the dielectric tensor K a ij in the limit of long wavelengths. Note that here indeed oneis allowed to use the quasi-linear approximation equation13, because the damping is very small. The damping ratethen follows from the dielectric tensor via (Melrose 1968): Γ = − i (cid:18) E ∗ i K a ij E j πW ( k ) (cid:19) ω i =0 ω r , (20)where E i are the electric fields, ω r is the real part of thefrequency of the fluctuating field, and ω i its imaginary part.The turbulence spectrum follows the damped diffusionequation: ∂W k ( t ) ∂t = ∂∂k (cid:18) D kk ∂W k ( t ) ∂k (cid:19) − N (cid:88) i =1 Γ i ( k ) W k ( t ) , (21)with the diffusion coefficient due to mode coupling D kk = k /τ s , τ s the spectral energy transfer time. Here i sums overall damping processes of turbulence in the plasma. For thehigh- β ICM thermal particles and CRp argueably do notdamp fast modes efficiently. Therefore we only considerCRe damping here ( N = 1 ).The argument of detailed balancing then states that theenergy lost by the turbulent waves below a scale l due toCRe damping equals the energy change in the relativisticparticles. It relates the total energy in a mode W ( k ) d k tothe particle spectrum (Achterberg 1981; Eilek 1979): (cid:90) d p E CRe (cid:18) ∂f CRe ( p ) ∂t (cid:19) = (cid:90) d k Γ CRe ( k, θ ) W ( k ) . (22)From this argument D pp can be found: D pp ≈ × − p v c l η (23)for Kraichnan turbulence P ( k ) ∝ k − / and an energy frac-tion in magneto-sonic waves η turb . c (cid:13)(cid:13) As reacceleration models are non-stationary, all parametershave to be estimated as a function of time. One may modelthe thermal quantities with the usual isothermal beta model.This leaves the following model parameters: – Magnetic field distribution, i.e. B ( t ) and η B ( t ) . – Spectral distribution of turbulence W ( t ) : e.g. a power-law with index α turb , cut-off k min , k max and normalisa-tion W ( t ) . For Kraichnan turbulence this translates toeq. 23 with the turbulent velocity v turb at scale l . – The fraction of turbulence in magneto-sonic waves η turb ∈ [0 . , . . This parameter is not well con-strained, but given the theoretical uncertainties in thecoupling (see the motivation of D pp above) values be-tween 0.01 and 0.5 seem acceptable. – A source of CRe, i.e. Q ( t ) .The time evolution of the particle spectrum as well as theresulting synchrotron emission have to be estimated numer-ically.While semi-analytical models have been applied suc-cessfully in the past (e.g. Fujita et al. 2003) the complexityof the problem really requires a fully numerical approach. We show results from a first numerical study of reacceler-ation in cluster mergers (Donnert et al. 2012). This workmakes use of the MHDSPH code
GADGET3 (Dolag & Sta-syszyn 2009; Springel 2005) and a model for isolated clus-ter collisions (Donnert in prep., section 4.3.2). We self-consistently follow the collisionless dynamics of dark mat-ter (DM), and the gas in the MHD-approximation. This sim-ulates the temperature, density and magnetic field distribu-tion of the merger over time. In addition the following pa-rameters are used: – the magnetic field is setup with η mag = 1 – Turbulence follows a Kraichnan spectrum below the nu-merical resolution, so the diffusion coefficient equation23 is used. Above the resolution scale turbulence is com-puted explicitely by the code. – A numerical model for smoothed particle hydrodynam-ics (SPH) turbulence gives v turb on the kernel scale l = 2 h sml . Here h sml is the compact support of the ker-nel. – η turb = 0 . – CRe are assumed to be constantly injected into the ICM.The injection coefficient is modelled as: Q e ( γ, t ) = K e ε γ − (cid:114) − γ (1 − γ ) , (24)which is a power-law with smooth cut-offs at γ = p/ m e c = 50 , . As the injection scales as the ther-mal energy density ε th it is equivalent to an hadronicinjection. K e ≈ − p [m e c]10 n ( p ) ( a r b . un it s ) Fig. 9
Comparison of CRe spectra from our code(colours) using 100 gridcells with another code (black) usedin Cassano & Brunetti (2005) with 10000 cells.
In post-processing to the simulation we follow dynamicsof the CRe spectrum by solving the transport equation 16(Donnert in prep.) in parallel. We linearly interpolate be-tween simulation outputs every 10 Myr and use a conser-vative timestep of 1 Myr in the code. The Fokker-Planckcoefficients are computed from equations 17, 18, 23 and 24using the thermal and turbulence properties from the simu-lation.We sample the CRe spectrum in 100 logarithmic binsbetween p/m e c = 1 and . We realise open boundaryconditions with the method from Borovsky & Eilek (1986).We use the method of Chang & Cooper (1970) to computethe time evolution of the spectrum. They derive a first-orderaccurate adaptive upwind scheme which can be analyticallyshown to: – be unconditionally stable – guarantee positivity – converge to the steady-state solution. – conserve particle number.This means it is ideally suited for our purpose to com-pute millions of spectra on particles of an SPH simulation.Specifically it allows us to use a logarithmic grid with onlya small number of cells. The convergence and accuracy istested against a run using the Fokker-Planck code from Cas-sano & Brunetti (2005). In figure 9 we compare results fromtheir code using 10000 gridcells (black) with ours using 100gridcells (colours). The input data are taken from Cassano& Brunetti (2005). Deviations at low momenta occur duedifferent implementations of the open boundary conditions. We use an idealised model for major cluster merger basedon the Hernquist profile (Hernquist 1990) for the collision-less matter. We identify a Hernquist profile with an NFW-profile (Navarro et al. 1996) with concentration parameter c c (cid:13) stron. Nachr. / AN (2013) 13Value Unit Cluster 0 Cluster 1 a Hernq kpc 811 473 r kpc 2192 1380 r core kpc 237 130 M M (cid:12) ρ , gas − g / cm T( r core ) 10 K Table 2
Model parameters of the initial conditions basedon Donnert in prep.according to Springel et al. (2005) The ICM is modelled asa β -model (Cavaliere & Fusco-Femiano 1978; King 1966)with β = 2 / and r core = c/ . The hydrostatic equationcan then be solved analytically to give a roughly constanttemperature profile (Donnert in prep.).We simulate a merging system with a mass ratio of1:8, total mass of . × M (cid:12) and a baryon fraction b f = 0 . . The two clusters are relaxed separately and thenjoined in a periodic box of 10 Mpc size. They are set ona zero energy orbit with an impact parameter of 300 Mpc.The magnetic field is set-up divergence free in k-space froma Gaussian random vector potential. This is transformed to agrid of 150 kpc size and the particles are initialised via NGPsampling. The field is attenuated according to η mag = 1 ,which introduces divergence. We rely on the divergencecleaning of the code to deal with this in the beginning ofthe simulation. To catch the bulk of turbulence generated in our simulationwe use an SPH algorthim tuned towards minimising vis-cosity in the flow. We use a time-dependent viscosity ap-proach (Dolag et al. 2005) and a description for artificialthermal conduction and magnetic diffusion to resolve insta-bilities (Price 2007). We use the high-order C4 Wendlandkernel with 210 kernel-weighted neighbours (Dehnen & Aly2012).The local turbulent energy is estimated from the RMSvelocity dispersion of neighbours within the SPH kernel. Byusing a kernel with large compact support we make sure thatviscous damping happens within the kernel scale. We set thelow viscosity scheme to retain a minimum viscosity, corre-sponding to α = 0 . in the viscosity formulation of Dolaget al. (2005). This way our approach remains conservative. We use our parallel projection routine
SMAC2 to extractthermal and non-thermal synthetic observations from thesystem. In figure 10 we show projections of the X-ray emis-sion at three different radio bright stages of the system.
The thermal evolution of the system can be sum-marised as follows: – Upon infall a large shock develops in front of the smallcluster. This boosts the X-ray emission of the system,which peaks at the core passage (infall phase). – As the small DM core leaves the host cluster, the X-rayluminosity declines rapidly. The small core drags a partof the ICM along (effectively displacing the gas afterthe first encounter), causing turbulence (reaccelerationphase). – When the DM core reaches its turn-around point the tur-bulence has decayed, the ICM relaxed and the cluster isradio dark (decay phase).These merging phases are repeated two times as the DMcore oscillates in the host potential. The mass of the sub-cluster successively declines and the host system is pre-disturbed. The core oscillations decay with time and the re-sulting emission peaks in the X-rays decline.Due to the impact parameter, the ICM of the host clus-ter recieves angular momentum and an infalling stream ofhot gas develops after the second core passage. This hasbeen seen in other simulations of this kind as well (Ricker& Sarazin 2001). The system relaxes after roughly 6 Gyr.
The non-thermal emission follows the spatial and tem-poral evolution of turbulence in the simulation. Shortly af-ter the first passage of the smaller DM core a large vol-ume filling off-center halo develops (fig. 10, left), whichdecays within 1 Gyr. After the second passage an elongatedsmaller halo centered on the peak X-ray emission of the pre-disturbed cluster can be seen (fig. 10, middle). After relax-ation of the DM cores, large off-center emission develops ontop of the turbulent infall stream (fig. 10, right). In betweenthese phases the system becomes radio quiet as expectedfrom analytical models.
For a more quantative analysis we will focus on thefirst passage only. In figure 11 (left) we show CRe elec-tron spectra from a single particle in the simulation at seventimes between 1 Gyr and 3.5 Gyrs. At 1 Gyr the CRe spec-trum is in the equillibrium state between injection and cool-ing (blue). At 1.5 Gyr (maximum halo brightness) the spec-trum shows the typical bending from turbulent reaccelera-tion (light red). In the following times the population cools,i.e. reacceleration is not efficient anymore and CRe diffuseto synchrotron dark momenta.In figure 11, middle panel, we show the evolution of thesystem in the P14-LX plane (black line). We mark every0.2 Myr with a small dot on the line and the times shownin the other two plots with corresponding colours. The in-jection only model is shown as a dashed line. The observedcorrelation (straight black line), observed halos (diamonds)and ultra-steep spectrum halos (asterisks) and upper-limits(arrows) are added as well. In this model, the system is be-low the upper-limits in its initial equillibrium state (bluedot). Upon infall the shock rapidly increases X-ray and syn-chrotron luminosity. In the reacceleration phase the X-rayluminosity declines, because the DM core drags ICM gas c (cid:13) Fig. 10
Bolometric X-ray emission from the system with radio synchrotron emission at 1.4 GHz overlayed as contours(Donnert et al. 2012). We show three different times: halo phase after the first (left) and second (middle) merger andoff-center emission from an infalling stream (right). p [m e c]10 -14 -13 -12 -11 -10 -9 -8 -7 n ( p ) ( a r b . un it s ) L X-ray [0.2,2.4] keV [erg/s]10 P . GH z [ e r g / s / H z ] non-detectionsnon-detectionsdetectionsdetectionsComaComasteep spectrumsteep spectrum ν [Hz]10 ν P ν / P / M H z Fig. 11
Left: Exemplary CRe spectrum of a particle in the simulation at different times. Middle: Observed radio-X-raycorrelation (black line) and observed halos (diamonds, Coma: cross, ultra-steep: asterisks), upper limits (arrows). Simulatedsystem (black curve, 10 Myr intervalls marked), simulated injection only (dashed curve). Colored dots correspond to thecurves shown in the other panels. Right: Observed radio spectra of Coma (dots) and A521 (diamonds) and of the system atdifferent times.out of the host system. This causes turbulence and reac-celeration, which boosts the radio luminosity to maximumbrightness and on the correlation (light red point). As tur-bulence decays the radio emission declines and the clusterleaves the correlation. Within one Gyr the radio luminosityfalls below the upper limits again.In figure 11, right panel we show synchrotron spectra ofthe system. The colours correspond to the times discussedbefore. The synchrotron spectrum at 1Gyr (dark blue) isconsistent with the analytical expectation for the equillib-rium of cooling and injection (eq. 4). At its maximumbrightness of the system, its radio spectrum is flatter thanthe observed Coma spectrum. The simulated spectrum thensteepens with time during the cooling phase. At 1.7 Gyr itfits the Coma spectrum (filled dots) and at 1.8 Gyr the ultra-steep spectrum of A521 (diamonds). With time the spectrumflattens again and approaches the equillibrium spectrum at3.4 Gyrs.
We reported on the status of models for giant radio halos.Reviewing the current picture on radio halos from observa-tions, we presented a short introduction to hadronic models.Using a pure hadronic model in a cosmological MHD sim-ulation, we argued that hadronic models fail to simultane-ously reproduce key observables: – the Coma radio profile within the non-thermal pressureconstrains, – the Coma Compton-y - radio correlation – the Coma spectrum and radio brightness within the cur-rent γ -ray upper limits, – the bimodal distribution in cluster brightness.We put forward three theoretical arguments against non-classical hadronic (streaming) models: – Galaxy clusters are unlikely to be free of turbulence onsmall scales due to high Reynolds numbers and constantinfall of small DM haloes. c (cid:13) stron. Nachr. / AN (2013) 15 – Efficient damping does not imply complete absense ofturbulence if a cascade is present. – Linear theory is not applicable for non-negligable damp-ing. In turn non-linear theory predicts efficient CR scat-tering also for small pitch angles.Considering this evidence we presented as short introduc-tion to reacceleration models. We motivated the transportequation and its terms governing cooling, injection andreacceleration of the CRe population. We presented firstresults from a numerical approach to the problem in theframework of idealised cluster collisions. Using a constantinjection of CRe, the idealised system reproduces key ob-servables, specifically: – the variety of observed spectral indices, from flat toultra-steep – the transient nature of radio halosThis reaffirms prior expectations from theoretical ap-proaches. Naturally this idealised model is only a first steptowards a more detailed modelling of the non-thermal com-ponents of the ICM. Upcoming radio telescopes like LO-FAR will test our current understanding of CR dynamicsand detailed numerical modelling in a cosmological frame-work seems highly appropriate.We would like to emphasize that future approaches mustinclude CR protons as well, and the absense of any clear ob-servational signature of these particles remains an excitingproblem in the next decades. One may also consider thatstate-of-the-art models for radio haloes require a plasma, not a fluid model of the ICM on small scales. This comeswith significant complications in the physics, but bares thechance of a more complete understanding of the micro-physics of the ICM, a truly unique plasma. Acknowledgements.
The author thanks the German AstronomicalSociety for the PhD award 2012. This work has been done withthe support of many people, special thanks go to K.Dolag andG.Brunetti. J.D. acknowledges support by PRIN-INAF2009 andthe FP7 Marie Curie programme ’People’ of the European Union.The computations where performed at the “Rechenzentrum derMax-Planck-Gesellschaft”, with resources assigned to the “Max-Planck-Institut f¨ur Astrophysik”.
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