# Curved Radio Spectra of Weak Cluster Shocks

Hyesung Kang and Dongsu Ryu Department of Earth Sciences, Pusan National University, Pusan 609-735, Republic of Korea: hskang@pusan.ac.kr
Department of Physics, UNIST, Ulsan 689-798, Korea: ryu@sirius.unist.ac.kr
###### Abstract

In order to understand certain observed features of arc-like giant radio relics such as the rareness, uniform surface brightness, and curved integrated spectra, we explore a diffusive shock acceleration (DSA) model for radio relics in which a spherical shock impinges on a magnetized cloud containing fossil relativistic electrons. Toward this end, we perform DSA simulations of spherical shocks with the parameters relevant for the Sausage radio relic in cluster CIZA J2242.8+5301, and calculate the ensuing radio synchrotron emission from re-accelerated electrons. Three types of fossil electron populations are considered: a delta-function like population with the shock injection momentum, a power-law distribution, and a power-law with an exponential cutoff. The surface brightness profile of radio-emitting postshock region and the volume-integrated radio spectrum are calculated and compared with observations. We find that the observed width of the Sausage relic can be explained reasonably well by shocks with speed and sonic Mach number . These shocks produce curved radio spectra that steepen gradually over with break frequency GHz, if the duration of electron acceleration is Myr. However, the abrupt increase of spectral index above GHz observed in the Sausage relic seems to indicate that additional physical processes, other than radiative losses, operate for electrons with .

acceleration of particles — cosmic rays — galaxies: clusters: general — shock waves
slugcomment: draft of July 13, 2019

## 1 Introduction

Radio relics are diffuse radio sources found in the outskirts of galaxy clusters and they are thought to trace synchrotron-emitting cosmic-ray (CR) electrons accelerated via diffusive shock acceleration (DSA) at cluster shocks (e.g. Ensslin et al., 1998; Bagchi et al., 2006; van Weeren et al., 2010; Brüggen et al., 2012). So far several dozens of clusters have been observed to have radio relics with a variety of morphologies and most of them are considered to be associated with cluster merger activities (see for reviews, e.g., Feretti et al., 2012; Brunetti & Jones, 2014). For instance, double radio relics, such as the ones in ZwCl0008.8+5215, are thought to reveal the bow shocks induced by a binary major merger (van Weeren et al., 2011b; de Gasperin et al., 2014). On the other hand, recently it was shown that shocks induced by the infall of the warm-hot intergalactic medium (WHIM) along adjacent filaments into the hot intracluster medium (ICM) can efficiently accelerate CR electrons, and so they could be responsible for some radio relics in the cluster outskirts (see, e.g., Hong et al., 2014). The radio relic 1253+275 in Coma cluster observed in both radio (Brown & Rudnick, 2011) and X-ray (Ogrean & Brüggen, 2013) provides an example of such infall shocks.

The so-called Sausage relic in CIZA J2242.8+5301 () contains a thin arc-like structure of kpc width and  Mpc length, which could be represented by a portion of spherical shell with radius Mpc (van Weeren et al., 2010). Unique features of this giant radio relic include the nearly uniform surface brightness along the length of the relic and the strong polarization of up to with magnetic field vectors aligned with the relic (van Weeren et al., 2010). A temperature jump across the relic that corresponds to a shock has been detected in X-ray observations (Akamatsu & Kawahara, 2013; Ogrean et al., 2014). This was smaller than estimated from the above radio observation. Several examples of Mpc-scale radio relics include the Toothbrush relic in 1RXS J0603.3 with a peculiar linear morphology (van Weeren et al., 2012) and the relics in A3667 (Rottgering et al., 1997) and A3376 (Bagchi et al., 2006). The shock Mach numbers of radio relics estimated based on X-ray observation are often lower than those inferred from the radio spectral index using the DSA model, for instance, in the Toothbrush relic (Ogrean et al., 2013) and in the radio relic in A2256 (Trasatti et al., 2015). Although such giant radio relics are quite rare, the fraction of X-ray luminous clusters hosting some radio relics is estimated to be % or so (Feretti et al., 2012).

Through a number of studies using cosmological hydrodynamical simulations, it has been demonstrated that during the process of hierarchical structure formation, abundant shocks are produced in the large-scale structure of the universe, especially in clusters (e.g., Ryu et al., 2003; Pfrommer et al., 2006; Skillman et al., 2008; Hoeft et al., 2008; Vazza et al., 2009; Vazza et al, 2011; Hong et al., 2014). Considering that the characteristic time-scale of cluster dynamics including mergers is Gyr, typical cluster shocks are expected to last for about the same period. Yet, the number of observed radio relics, which is thought to trace such shocks, is still limited. So it is plausible to conjecture that cluster shocks may ‘turn on’ to emit synchrotron radiation only for a fraction of their lifetime. One feasible scenario is that a cluster shock lights up in radio when it sweeps up a fossil cloud, i.e., a magnetized ICM gas with fossil relativistic electrons left over from either a radio jet from AGN or a previous episode of shock/turbulence acceleration (see the discussions in Section 2.5 and Figure 1).

Pre-exiting seed electrons and/or enhanced magnetic fields are the requisite conditions for possible lighting-up of relic shocks. In particular, the elongated shape with uniform surface brightness and high polarization fraction of radio emission in the Sausage relic, may be explained, if a Mpc-scale thermal gas cloud, containing fossil relativistic electrons and permeated with regular magnetic field of a few to several G, is adopted. A more detailed description will be given later in Section 2.5. In this picture, fossil electrons are expected to be re-accelerated for less than cloud-crossing time (), which is much shorter than the cluster dynamical time-scale. In addition, only occasional encounters with fossil clouds combined with the short acceleration duration could alleviate the strong constraints on the DSA theory based on non-detection of -ray emission from clusters by Fermi-LAT (Ackermann et al., 2014; Vazza et al., 2015). A similar idea has been brought up by Shimwell et al. (2015), who reported the discovery of a Mpc-scale, elongated relic in the Bullet cluster 1E 0657-55.8. They also proposed that the arc-like shape of uniform surface brightness in some radio relics may trace the underlying regions of pre-existing, seed electrons remaining from old radio lobes. On the other hand, Ensslin & Gopal-Krishna (2001) suggested that radio relics could be explained by revival of fossil radio plasma by compression due to a passage of a shock, rather than DSA. In a follow-up study, Ensslin & Brüggen (2002) showed using MHD simulations that a cocoon of hot radio plasma swept by a shock turns into a filamentary or toroidal structure. Although this scenario remains to be a viable explanation for some radio relics, it may not account for the uniform arc-like morphology of the Sausage relic.

It is now well established, through observations of radio halos/relics and Faraday rotation measures of background radio sources, that the ICM is permeated with G-level magnetic fields (e.g. Bonafede et al., 2011; Feretti et al., 2012). The observed radial profile of magnetic field strength tends to peak at the center with a few G and decrease outward to G in the cluster outskirts (Bonafede et al., 2010). A variety of physical processes that could generate and amplify magnetic fields in the ICM have been suggested: primordial processes, plasma processes at the recombination epoch, and Biermann battery mechanism, combined with turbulence dynamo, in addition to galactic winds and AGN jets (e.g. Ryu et al., 2008; Dolag et al., 2008; Brüggen et al., 2012; Ryu et al., 2012; Cho, 2014). Given the fact that G fields are required to explain the amplitude and width of the observed radio flux profile of the Sausage relic (e.g. van Weeren et al., 2010; Kang et al., 2012), the presence of a cloud with enhanced magnetic fields of several G might be preferred to the background fields of G in the cluster periphery. Alternatively, Iapichino & Brüggen (2012) showed that the postshock magnetic fields can be amplified to G level leading to high degrees of polarization, if there exists dynamically significant turbulence in the upstream region of a curved shock. Although it is well accepted that magnetic fields can be amplified via various plasma instabilities at collisionless shocks, the dependence on the shock parameters such as the shock sonic and Alfvénic Mach numbers, and the obliquity of background magnetic fields remains to be further investigated (see Schure et al., 2012). For example, the acceleration of protons and ensuing magnetic field amplification via resonant and non-resonant streaming instabilities are found to be ineffective at perpendicular shocks (Bell, 1978, 2004; Caprioli & Spitkovsky, 2014).

According to the DSA theory, in the case of a steady planar shock with constant postshock magnetic field, the electron distribution function at the shock location becomes a power-law of , and so the synchrotron emissivity from those electrons becomes a power-law of . The power-low slopes depend only on the shock sonic Mach number, , and are given as and for the gasdynamic shock with the adiabatic index (Drury, 1983; Blandford & Eichler, 1987; Ensslin et al., 1998). Here we refer as the injection spectral index for a steady planar shock with constant postshock magnetic field. Then, the volume-integrated synchrotron spectrum downstream of the shock also becomes a simple power-law of with the spectral index above the break frequency, , since electrons cool via synchrotron and inverse-Compton (IC) losses behind the shock (e.g., Ensslin et al., 1998; Kang, 2011).111Note that radio observers commonly use ‘’ as the spectral index of the flux density, for unresolved sources, so in that case is the same as . Here is defined as the spectral index of the local emissivity, . See Equations (9)-(10). Such predictions of the DSA theory have been applied to explain the observed properties of radio relics, e.g., the relation between the injection spectral index and the volume-integrated spectral index, and the gradual steepening of spatially resolved spectrum downstream of the shock.

Kang et al. (2012) performed time-dependent, DSA simulations of CR electrons for steady planar shocks with and constant postshock magnetic fields. Several models with thermal leakage injection or pre-existing electrons were considered in order to reproduce the surface brightness and spectral aging profiles of radio relics in CIZA J2242.8+5301 and ZwCl0008.8+5215. Adopting the same geometrical structure of radio-emitting volume as described in Section 2.5, they showed that the synchrotron emission from shock accelerated electrons could explain the observed profiles of the radio flux, , of the Sausage relic, and the observed profiles of both and of the relic in ZwCl0008.8+5215. Here is the distance behind the projected shock edge in the plane of the sky.

In the case of spherically expanding shocks with varying speeds and/or nonuniform magnetic field profiles, on the other hand, the electron spectrum and the ensuing radio spectrum could deviate from those simple power-law forms, as shown in Kang (2015a, b). Then even the injection slope should vary with the frequency, i.e., . Here we follow the evolution of a spherical shock expanding outward in the cluster outskirts with a decreasing density profile, which may lead to a curvature in both the injected spectrum and the volume-integrated spectrum. Moreover, if the shock is relatively young or the electron acceleration duration is short ( Myr), then the break frequency falls in GHz and the volume-integrated spectrum of a radio relic would steepen gradually with the spectral index from to over (e.g. Kang, 2015b).

In the case of the Sausage relic, van Weeren et al. (2010) and Stroe et al. (2013) originally reported observations of and , which imply a shock of . Stroe et al. (2014b), however, found a spectral steepening of the volume-integrated spectrum at 16 GHz, which would be inconsistent with the DSA model for a steady planar shock. Moreover, Stroe et al. (2014a), by performing a spatially-resolved spectral fitting, revised the injection index to a steeper value, . Then, the corresponding shock Mach number is reduced to . They also suggested that the spectral age, calculated under the assumption of freely-aging electrons downstream of a steady planar shock, might not be compatible with the shock speed estimated from X-ray and radio observations. Also Trasatti et al. (2015) reported that for the relic in A2256, the volume-integrated index steepens from for MHz to for GHz, which was interpreted as a broken power-law.

Discoveries of radio relic shocks with in recent years have brought up the need for more accurate understanding of injections of protons and electrons at weak collisionless shocks, especially at high plasma beta () ICM plasmas (e.g. Kang et al., 2014). Here is the ratio of the gas to magnetic field pressure. Injection of electrons into the Fermi 1st-order process has been one of long-standing problems in the DSA theory for astrophysical shocks, because it involves complex plasma kinetic processes that can be studied only through full Particle-in-Cell (PIC) simulations (e.g. Amano & Hoshino, 2009; Riquelme & Spitkovsky, 2011). It is thought that electrons must be pre-accelerated from their thermal momentum to several times the postshock thermal proton momentum to take part in the DSA process, and electron injection is much less efficient than proton injection due to smaller rigidity of electrons. Several recent studies using PIC simulations have shown that some of incoming protons and electrons gain energies via shock drift acceleration (SDA) while drifting along the shock surface, and then the particles are reflected toward the upstream region. Those reflected particles can be scattered back to the shock by plasma waves excited in the foreshock region, and then undergo multiple cycles of SDA, resulting in power-law suprathermal populations (e.g., Guo et al., 2014a, b; Park et al., 2015). Such ‘self pre-acceleration’ of thermal electrons in the foreshock region could be sufficient enough even at weak shocks in high beta ICM plasmas to explain the observed flux level of radio relics. In these PIC simulations, however, subsequent acceleration of suprathermal electrons into full DSA regime has not been explored yet, because extreme computational resources are required to follow the simulations for a large dynamic range of particle energy.

The main reasons that we implement the fossil electron distribution, instead of the shock injection only case, are (1) the relative scarcity of radio relics compared to the abundance of shocks expected to form in the ICM, (2) the peculiar uniformity of the surface brightness of the Sausage relic, and (3) curved integrated spectra often found in some radio relics, implying the acceleration duration  Myr, much shorter than the cluster dynamical time.

In this paper, we consider a DSA model for radio relics; a spherical shock moves into a magnetized gas cloud containing fossil relativistic electrons, while propagating through a density gradient in the cluster outskirts. Specifically, we perform time-dependent DSA simulations for several spherical shock models with the parameters relevant for the Sausage relic. We then calculate the surface brightness profile, , and the volume-integrated radio spectrum, , by adopting a specific geometrical structure of shock surface, and compare them with the observational data of the Sausage relic.

In Section 2, the DSA simulations and the model parameters are described. The comparison of our results with observations is discussed in Section 3. A brief summary is given in Section 4.

## 2 Dsa Simulations of Cr Electrons

### 2.1 1D Spherical CRASH Code

We follow the evolution of the CR electron population by solving the following diffusion-convection equation in the one-dimensional (1D) spherical geometry:

 ∂ge∂t+u∂ge∂r=13r2∂(r2u)∂r(∂ge∂y−4ge)+1r2∂∂r[r2D(r,p)∂ge∂r]+p∂∂y(bp2ge), (1)

where is the pitch-angle-averaged phase space distribution function of electrons, is the flow velocity and with the electron mass and the speed of light (Skilling, 1975). The spatial diffusion coefficient, , is assumed to have a Bohm-like dependence on the rigidity,

 D(r,p)=1.7×1019cm2s−1(B(r)1 μG)−1(pmec). (2)

The cooling coefficient accounts for radiative cooling, and the cooling time scale is defined as

where is the Lorentz factor of electrons. Here the ‘effective’ magnetic field strength, with , takes account for the IC loss due to the cosmic background radiation as well as synchrotron loss. The redshift of the cluster CIZA J2242.8+5301 is .

Assuming that the test-particle limit is applied at weak cluster shocks with several (see Table 1), the usual gasdynamic conservation equations are solved to follow the background flow speed, , using the 1D spherical version of the CRASH (Cosmic-Ray Amr SHock) code (Kang & Jones, 2006). The structure and evolution of are fed into Equation (1), while the gas pressure, , is used in modeling the postshock magnetic field profile (see Section 2.3). We do not consider the acceleration of CR protons in this study, since the synchrotron emission from CR electrons is our main interest and the dynamical feedback from the CR proton pressure can be ignored at weak shocks () in the test-particle regime (Kang & Ryu, 2013). In order to optimize the shock tracking of the CRASH code, a comoving frame that expands with the instantaneous shock speed is adopted. Further details of DSA simulations can be found in Kang (2015a).

### 2.2 Shock Parameters

To set up the initial shock structure, we adopt a Sedov self-similar blast wave propagating into a uniform static medium, which can be specified by two parameters, typically, the explosion energy, , and the background density, (Ostriker & McKee, 1988; Ryu & Vishniac, 1991). For our DSA simulations, we choose the initial shock radius and speed, and , respectively, and adopt the self-similar profiles of the gas density , the gas pressure , and the flow speed behind the shock in the upstream rest-frame. For the fiducial case (SA1 model in Table 1), for example, the initial shock parameters are , and . For the model parameters for the shock and upstream conditions, refer to Table 1 and Section 3.1.

We suppose that at the onset of the DSA simulations, this initial shock propagates through the ICM with the gas density gradient described by a power law of . Typical X-ray brightness profiles of observed clusters can be represented approximately by the so-call beta model for isothermal ICMs, with (Sarazin, 1986). In the outskirts of clusters, well outside of the core radius (), it asymptotes as . We take the upstream gas density of as the fiducial case (SA1), but also consider (SA3) and constant (SA4) for comparison. The shock speed and Mach number decrease in time as the spherical shock expands, depending on the upstream density profile. Kang (2015b) demonstrated that the shock decelerates approximately as for constant and as for , while the shock speed is almost constant in the case of . As a result, nonlinear deviations from the DSA predictions for steady planar shocks are expected to become the strongest in SA4 model, while the weakest in SA3 model. In the fiducial SA1 model, which is the most realistic among the three models, the effects of the evolving spherical shock are expected to be moderate.

The ICM temperature is set as keV, adopted from Ogrean et al. (2014), in most of the models. Hereafter, the subscripts “1” and “2” are used to indicate the quantities immediately upstream and downstream of the shock, respectively. Although the ICM temperature is known to decrease slightly in the cluster outskirt, it is assumed to be isothermal, since the shock typically travels only  Mpc for the duration of our simulations  Myr.

### 2.3 Models for Magnetic Fields

Magnetic fields in the downstream region of the shock are the key ingredient that governs the synchrotron cooling and emission of CR electrons in our models. We assume that the fossil cloud is magnetized to G level. As discussed in the Introduction, observations indicate that the magnetic field strength decreases from in the core region to in the periphery of clusters (e.g., Bonafede et al., 2010; Feretti et al., 2012). This corresponds to the plasma beta of in typical ICMs (e.g., Ryu et al., 2008, 2012). On the other hand, it is well established that magnetic fields can be amplified via resonant and non-resonant instabilities induced by CR protons streaming upstream of strong shocks (Bell, 1978, 2004). In addition, magnetic fields can be amplified by turbulent motions behind shocks (Giacalone & Jokipii, 2007). Recent hybrid plasma simulations have shown that the magnetic field amplification factor due to streaming CR protons scales with the Alfvénic Mach number, , and the CR proton acceleration efficiency as (Caprioli & Spitkovsky, 2014). Here, is the turbulent magnetic field perpendicular to the mean background magnetic field, is the downstream CR pressure, and is the upstream ram pressure. For typical radio relic shocks, the sonic and Alfvénic Mach numbers are expected to range and , respectively (e.g., Hong et al., 2014).

The magnetic field amplification in both the upstream and downstream of weak shocks is not yet fully understood, especially in high beta ICM plasmas. So we consider simple models for the postshock magnetic fields. For the fiducial case, we assume that the magnetic field strength across the shock transition is increased by compression of the two perpendicular components:

 B2(t)=B1√1/3+2σ(t)2/3, (4)

where and are the magnetic field strengths immediately upstream and downstream of the shock, respectively, and is the time-varying compression ratio across the shock. For the downstream region (), the magnetic field strength is assumed to scale with the gas pressure:

 Bdn(r,t)=B2(t)⋅[Pg(r,t)/Pg,2(t)]1/2, (5)

where is the gas pressure immediately behind the shock. This assumes that the ratio of the magnetic to thermal energy density is constant downstream of the shock. Since decreases behind the spherical blast wave, also decreases downstream as illustrated in Figure 2. This fiducial magnetic field model is adopted in most of the models described in Table 1, except SA2 and SA4 models. The range of is shown for the acceleration duration of Myr in Table 1, reflecting the decrease of shock compression ratio during the period, but the change is small.

In the second, more simplified (but somewhat unrealistic) model, it is assumed that and , and the downstream magnetic field strength is constant, i.e., , for . This model was adopted in Kang et al. (2012) and also for SA2 and SA4 models for comparison in this study.

Kang (2015b) showed that the postshock synchrotron emission increases downstream away from the shock in the case of a decelerating shock, because the shock is stronger at earlier time. But such nonlinear signatures become less distinct in the model with decreasing downstream magnetic fields, compared with the model with constant downstream magnetic fields, because the contribution of synchrotron emission from further downstream region becomes weaker.

### 2.4 Injection Momentum

As described in the Introduction, the new picture of particle injection emerged from recent PIC simulations is quite different from the ‘classical’ thermal leakage injection model commonly employed in previous studies of DSA (e.g., Kang et al., 2002). However, the requirement of for particles to take part in the full Fermi 1st-order process, scattering back and forth diffusively across the shock transition zone with thickness, , seems to remain valid (Caprioli et al., 2015; Park et al., 2015). Here, is the most probable momentum of thermal protons with postshock temperature and is the gyroradius of particles. In other words, only suprathermal particles with the gyro-radius greater than the shock thickness are expected to cross the shock transition layer. Hence, we adopt the traditional phenomenological model in which only particles above the injection momentum, , are allowed to get injected into the CR populations at the lowest momentum boundary (Kang et al., 2002). This can be translated to the electron Lorentz factor, . In the case of expanding shocks considered in this study, decreases as the shock slows down in time.

### 2.5 Fossil Electrons

As mentioned in the Introduction, one peculiar feature of the Sausage relic is the uniform surface brightness along the Mpc-scale arc-like shape, which requires a special geometrical distribution of shock-accelerated electrons (van Weeren et al., 2011a). Some of previous studies adopted the ribbon-like curved shock surface and the downstream swept-up volume, viewed edge-on with the viewing extension angle (e.g., van Weeren et al., 2010; Kang et al., 2012; Kang, 2015b). We suggest a picture where a spherical shock of radius Mpc passes through an elongated cloud with width  kpc and length  Mpc, filled with fossil electrons. Then the shock surface penetrated into the cloud becomes a ribbon-like patch, distributed on a sphere with radius Mpc with the angle . The downstream volume of radio-emitting, reaccelerated electrons has the width , as shown in Figure 1 of Kang (2015b). Hereafter the ‘acceleration age’, , is defined as the duration of electron acceleration since the shock encounters the cloud. This model is expected to produce a uniform surface brightness along the relic length. Moreover, if the acceleration age is , the volume-integrated radio spectrum is expected to steepen gradually over 0.1-10 GHz (Kang, 2015b).

There are several possible origins for such clouds of relativistic electrons in the ICMs: (1) old remnants of radio jets from AGNs, (2) electron populations that were accelerated by previous shocks and have cooled down below , and (3) electron populations that were accelerated by turbulence during merger activities. Although an AGN jet would have a hollow, cocoon-like shape initially, it may turn into a filled, cylindrical shape through diffusion, turbulent mixing, or contraction of relativistic plasmas, as the electrons cool radiatively. During such evolution relativistic electrons could be mixed with the ICM gas within the cloud. We assume that the cloud is composed of the thermal gas of , whose properties (density and temperature) are the same as the surrounding ICM gas, and an additional population of fossil electrons with dynamically insignificant pressure. In that regard, our fossil electron cloud is different from hot bubbles considered in previous studies of the interaction of a shock with a hot rarefied bubble (e.g., Ensslin & Brüggen, 2002). In the other two cases where electrons were accelerated either by shocks or by turbulence (see, e.g., Farnsworth et al., 2013), it is natural to assume that the cloud medium should contain both thermal gas and fossil electrons.

Three different spectra for fossil electron populations are considered. In the first fiducial case (e.g., SA1 model), nonthermal electrons have the momentum around , which corresponds to for the model shock parameters considered here. So in this model, seed electrons with are injected from the fossil population and re-accelerated into radio-emitting CR electrons with . Since we compare the surface brightness profiles in arbitrary units here, we do not concern about the normalization of the fossil population.

In the second model, the fossil electrons are assumed to have a power-law distribution extending up to ,

 fe,up(p)=f0⋅(ppinj)−s. (6)

For the modeling of the Sausage relic, the value of is chosen as with (SA1p model).

As mentioned in the Introduction, the volume-integrated radio spectrum of the Sausage relic seems to steepen at high frequencies, perhaps more strongly than expected from radiative cooling alone (Stroe et al., 2014b). So in the third model, we consider a power-law population with exponential cutoff as follows:

 (7)

where is the cutoff Lorentz factor. This may represent fossil electrons that have cooled down to from the power-law distribution in Equation (6). The integrated spectrum of the Sausage relic shows a strong curvature above GHz (Stroe et al., 2014b), which corresponds to the characteristic frequency of synchrotron emission from electrons with when the magnetic field strength ranges (see Equation (12) below). So and are chosen for SC1pex1 and SC1pex2 models, respectively.

Figure 3 shows the electron distributions in the cloud medium: the thermal distribution for the ICM gas with keV and the fossil population. The characteristics of the different models, SA1 (fiducial model), SA1p (power-law), and SC1pex1 (power-law with an exponential cutoff) will be described in Section 3.1. Note that there could be ‘suprathermal’ distribution between the thermal and CR populations. But it does not need to be specified here, since only around controls the spectrum of re-accelerated electrons.

## 3 Results of Dsa Simulations

### 3.1 Models

We consider several models whose characteristics are summarized in Table 1. For the fiducial model, SA1, the upstream density is assumed to decrease as , while the upstream temperature is taken to be keV. Since we do not concern about the absolute radio flux level in this study, needs not to be specified. The preshock magnetic field strength is and the immediate postshock magnetic field strength is during 60 Myr, while the downstream field, , is given as in equation (5). The initial shock speed is , corresponding to the sonic Mach number of at the onset of simulation. In all models with , the shock slows down as , and at the electron acceleration age of 60 Myr, with and . The fossil seeds electrons are assumed to have a delta-function-like distribution around .

In SA1b model, the upstream magnetic field strength is weaker with and during 60 Myr. Otherwise, it is the same as the fiducial model. So the character ‘b’ in the model name denotes ‘weaker magnetic field’, compared to the fiducial model. Comparison of SA1 and SA1b models will be discussed in Section 3.4.

In SA1p model, the fossil electrons have a power-law population of , while the rest of the parameters are the same as those of SA1 model. The character ‘p’ in the model name denotes a ‘power-law’ fossil population.

In SA2 model, both the preshock and postshock magnetic field strengths are constant, i.e.,  and , otherwise it is the same as SA1 model.

In SA3 model, the upstream gas density decreases as , so the shock decelerates more slowly, compared to SA1 model. Considering this, the initial shock speed is set to with . At the acceleration age of 60 Myr, the shock speed decreases to corresponding to .

In SA4 model, the upstream density is constant, so the shock decelerates approximately as , more quickly, compared to SA1 model. The upstream and downstream magnetic field strengths are also constant as in SA2 model. Figure 2 compares the profiles of the flow speed, , and the magnetic field strength, , in SA1 and SA4 models. Note that although the shocks in SA1 and SA4 models are not very strong with the initial Mach number , they decelerate approximately as and , respectively, as in the self-similar solutions of blast waves (Ostriker & McKee, 1988; Ryu & Vishniac, 1991). The shock speed decreases by during the electron acceleration age of 60 Myr in the two models.

In SB1 model, the preshock temperature, , is lower by a factor of , and so the initial shock speed corresponds to . The shock speed at the age of 60 Myr corresponds to , so the injection spectral index . The ‘SB’ shock is different from the ‘SA’ shock in terms of only the sonic Mach number.

In SC1pex1 and SC1pex2 models, the fossil electrons have with the cutoff at and , respectively. The character ‘pex’ in the model name denotes a ‘power-law with an exponential cutoff’. A slower initial shock with and is chosen, so at the acceleration age of 80 Myr the shock slows down to with . The ‘SC’ shock differs from the ‘SA’ shock in terms of the shock speed and the sonic Mach number. The integrated spectral index at high frequencies would be steep with , while at low frequencies due to the flat fossil electron population. They are intended to be toy models that could reproduce the integrated spectral indices, for GHz and for GHz, compatible with the observed curved spectrum of the Sausage relic (Stroe et al., 2014b).

### 3.2 Radio Spectra and Indices

The local synchrotron emissivity, , is calculated, using the electron distribution function, , and the magnetic field profile, . Then, the radio intensity or surface brightness, , is calculated by integrating along lines-of-sight (LoSs).

 Iν(R)=2∫hmax0jν(r)dh. (8)

is the distance behind the projected shock edge in the plane of the sky, as defined in the Introduction, and is the path length along LOSs; , , and are related as . The extension angle is (see Section 2.5). Note that the radio flux density, , can be obtained by convolving with a telescope beam as , if the brightness distribution is broad compared to the beam size of .

The volume-integrated synchrotron spectrum, , is calculated by integrating over the entire downstream region with the assumed geometric structure described in Section 2.5. The spectral indices of the local emissivity, , and the integrated spectrum, , are defined as follows:

 αν(r)=−dlnjν(r)dlnν, (9)
 Aν=−dlnJνdlnν. (10)

As noted in the Introduction, unresolved radio observations usually report the spectral index, ‘’ at various frequencies, which is equivalent to here.

In the postshock region, the cutoff of the electron spectrum, , decreases downstream from the shock due to radiative losses, so the volume-integrated electron spectrum steepens from to above the break Lorentz factor (see Figure 5). At the electron acceleration age , the break Lorentz factor can be estimated from the condition (Kang et al., 2012):

Hereafter, the postshock magnetic field strength, , and are expressed in units of G. Since the synchrotron emission from mono-energetic electrons with peaks around

 νpeak≈0.3(3eB24πmec)γ2e≈0.63 GHz⋅(B25)(γe104)2, (12)

the break frequency that corresponds to becomes

So the volume-integrated synchrotron spectrum, , has a spectral break, or more precisely a gradual increase of the spectral index approximately from to around .

### 3.3 Width of Radio Shocks

For electrons with , the radiative cooling time in Equation (3) becomes shorter than the acceleration age if Myr. Then, the width of the spatial distribution of those high-energy electrons downstream of the shock becomes

where is the downstream flow speed.

With the characteristic frequency of electrons with in Equation (12), the width of the synchrotron emitting region behind the shock at the observation frequency of would be similar to :

Here, is a numerical factor of . This factor takes account for the fact that the spatial distribution of synchrotron emission at is somewhat broader than that of electrons with the corresponding , because more abundant, lower energy electrons also make contributions (Kang, 2015a). One can estimate two possible values of the postshock magnetic field strength, , from this relation, if can be determined from the observed profile of surface brightness and is known. For example, van Weeren et al. (2010) inferred two possible values of the postshock magnetic field strength of the Sausage relic, and , by assuming that the FWHM of surface brightness, , is the same as kpc (i.e., ).

On the other hand, Kang et al. (2012) showed that the profile of surface brightness depends strongly on in the case of the assumed geometrical structure. Figure 4 illustrates the results of a planar shock with different ’s. The shock has and . The magnetic field strengths are assumed to be and , upstream and downstream of the shock, respectively. The results shown are at the acceleration age of 80 Myr. Note that the quantities, , , and , are multiplied by arbitrary powers of and , so they can be shown for different values of and with single linear scales in Figure 4. Here, the FWHM of is, for instance, kpc for (red dotted line in the top panel), while the FWHM of is kpc for GHz (red dotted line in the middle panel). Note that the values of and are chosen from the sets of discrete simulation bins, so they satisfy only approximately the Equation (12). The bottom panel demonstrates that the FWHM of , , strongly depends on . For instance, kpc for GHz, if (red dotted line). This implies that due to the projection effect, could be substantially different from . So of the observed radio flux profile may not be used to infer possible values of in radio relics, unless the geometrical structure of the shock is known and so the projection effect can be modeled accordingly.

Note that the quantity in Equation (15) also appears as in Equation (13). So the break frequency becomes identical for the two values of , and , that give the same value of . For example, SA1 model with and SA1b model with (see Table 1) produce not only the similar but also the similar spectral break in . But the corresponding values of in Equation (11) are different for the two models with and . Moreover, the amplitude of the integrated spectrum would scale as , if the electron spectrum for is similar for the two models . We compare these two models in detail in the next section.

### 3.4 Comparison of the Two Models with Bhigh and Blow

In Figure 5, we first compare the electron spectra and the synchrotron emission spectra in SA1 and SA1b models. Here, the plotted quantities, , , and , are in arbitrary units. The postshock magnetic field strength decreases from to in 110 Myr in SA1 model, while to in SA1b model. In these two models, the values of are similar, so the spectral break in the integrated spectra should be similar as well.

The left panels of Figure 5 show that the electron spectrum at the shock, , steepens in time as the shock weakens. As expected, the volume-integrated electron spectrum, , steepens by one power of above the break momentum, , which decreases in time due to radiative losses. In both models, however, the slopes, and , deviate from the simple steepening of to above , because of the time-dependence of the shock properties.

The right panels of Figure 5 show the synchrotron spectrum at the shock, , the volume-integrated synchrotron spectrum, , and their spectral indices, and . Note that in both models, the transition of from to is broader than the transition of from to . This is because the synchrotron emission at a given frequency comes from electrons with a somewhat broad range of ’s. As in the case of , does not follow the simple steepening, but rather shows nonlinear features due to the evolving shock properties. This implies that the simple relation, , which is commonly adopted in order to confirm the DSA origin of synchrotron spectrum, should be applied with some caution in the case of evolving shocks.

The highest momentum for , , is higher in SA1 model with stronger magnetic field, but the amplitude of near , say at , is greater in SA1b model with weaker magnetic field. As a consequence, the ratio of the synchrotron emission of the two models is somewhat less than the ratio of the magnetic energy density, . In SA1b model, for example, the amplitudes of and at GHz are reduced by a factor of , compared to the respective values in SA1 model. Also the cutoff frequencies for both and are lower in SA1b model, compared to those in SA1 model. As pointed above, is almost the same in the two models, although is different. This confirms that we would get two possible solutions for , if we attempt to estimate the magnetic field strength from the value of in the integrated spectrum.

### 3.5 Surface Brightness Profile

As noted in Section 3.3, the width of the downstream electron distribution behind the shock is determined by the advection length, , for low-energy electrons or by the cooling length, , for high-energy electrons. As a result, at high frequencies the width of the synchrotron emission region, , varies with the downstream magnetic field strength for given and as in Equation (15). In addition, the surface brightness profile, , and its FWHM, , also depend on the extension angle, , as demonstrated in Figure 4.

In Figures 6 and 7, the spatial profiles of are shown for eight models listed in Table 1. Here, we choose the observation frequencies MHz; the source frequencies are given as with . While the downstream flow speed ranges in most of the models, in SC1pex1 model. So the results are shown at and , except in SC1pex1 model for which the results are shown at and . Note that the quantity, , is shown in order to be plotted with one linear scale, where is the numerical scale factor specified in each panel. In SB1 model with a stronger shock, for example, the electron acceleration is more efficient by a factor of about , compared to other models, so the profiles shown are reduced by similar factors.

For a fixed value of , the surface brightness profile is determined by the distribution of behind the shock along the path length , as given in Equation (8). The intensity increases gradually from the shock position () to the first inflection point, kpc with , mainly due to the increase of the path length along LoSs. Since the path length starts to decrease beyond , if the emissivity, , is constant or decreases downstream of the shock, then should decrease for . In all the models considered here, however, at low frequencies, increases downstream of the shock, because the model spherical shock is faster and stronger at earlier times. As a result, at 150 MHz is almost constant or decrease very slowly beyond , or even increases downstream away from the shock in some cases (e.g., SA2 and SA4 models). So the downstream profile of the synchrotron radiation at low frequencies emitted by uncooled, low-energy electrons could reveal some information about the shock dynamics, providing that the downstream magnetic field strength is known.

The second inflection in occurs roughly at for low frequencies, and at for high frequencies where , with . Here, , and is given in Equation (14). Figures 6 and 7 exhibit that at 30 Myr (), the second inflection appears at the same position ( kpc) for the three frequencies shown. At later times, the position of the second inflection depends on at low frequencies, while it varies with and in addition to . Thus, only if , at high frequencies can be used to infer , providing that and are known.

The width of the Sausage relic, defined as the FWHM of the surface brightness, was estimated to be kpc at 600 MHz (van Weeren et al., 2010). As shown in Figures 6 and 7, in all the models except SA4, the calculated FWHM of at 600 MHz (red dotted lines) at the acceleration age of Myr would be compatible with the observed value. In SA4 model, the intensity profile at 600 MHz seems a bit too broad to fit the observed profile.

### 3.6 Volume-Integrated and Postshock Spectra

Figures 8 and 9 show the volume-integrated synchrotron spectrum, , and its slope, , for the same models at the same ages as in Figures 6 and 7. Here is integrated over the curved, ribbon-like downstream volume with , as described in Section 3.2. In the figures, is plotted in arbitrary units. The filled circles represent the data points for the integrated flux of the Sausage relic, which were taken from Table 1 of Stroe et al. (2014b) and re-scaled roughly to fit by eye the spectrum of SA1 model at (the red dotted line in the top-left panel of Figure 8). Observational errors given by Stroe et al. (2014b) are about 10 %, so the error bars are in fact smaller than the size of the filled circles in the figures, except the one at 16 GHz with 25 %.

At first it looks that in most of the models, the calculated reasonably fits the observed data including the point at 16 GHz. But careful inspections indicate that our models fail to reproduce the rather abrupt increase in the spectral curvature at GHz. The calculated is either too steep at low frequencies GHz, or too flat at high frequencies GHz. For instance, of the fiducial model, SA1, is too steep at GHz. On the other hand, SA1p model with a flatter fossil population and SB1 model with a flatter injection spectrum (due to higher ) seem to have the downstream electron spectra too flat to explain the observed data for GHz.

As mentioned before, the transition of the integrated spectral index from to occurs gradually over a broad range of frequency, . If we estimate as the frequency at which the gradient, , has the largest value, it is , 1, 0.3 GHz at , respectively, for all the models except SC1pex1 model (for which is shown at different epochs). Since is determined mainly by the magnetic field strength and the acceleration age, it does not sensitively depend on other details of the models. In SC1pex1 model, the power-law portion () gives a flatter spectrum with at low frequencies, while the newly injected population at the shock with results in a steeper spectrum with at high frequencies. Note that the spectral index estimated with the observed flux in Stroe et al. (2014b) between 2.3 and 16 GHz is , implying .

In Figure 10, ’s from all the models considered here are compared at Myr (SA and SB shocks) or Myr (SC shocks). The observed data points of the Sausage relic are also shown. In most of the models (SA models), the shocks have at 60 Myr, so the predicted between 2.3 and 16 GHz is a bit flatter with than the observed spectrum with at the same frequency range. As mentioned above, SA1p and SB1 models produce , which is significantly flatter at high frequencies than the observed spectrum. The toy model, SC1pex1, seems to produce the best fit to the observed spectrum, as noted before.

In short, the volume-integrated synchrotron spectra calculated here steepen gradually over with the break frequency GHz, if Myr. However, all the models considered here seem to have some difficulties fitting the sharp curvature around GHz in the observed integrated spectrum. This implies that the shock dynamics and/or the downstream magnetic field profile could be different from what we consider here. Perhaps some additional physics that can introduce a feature in the electron energy spectrum, stronger than the ‘’ steepening due to radiative cooling, might be necessary.

In Figure 11, we present for the three models, SA1, SA1b, and SC1pex1, the mean intensity spectrum in the downstream regions of , , where and runs from 1 to 6. This is designed to be compared with the ‘spatially resolved’ spectrum behind the shock in radio observations (e.g., Stroe et al., 2013). The figure shows how the downstream spectrum cuts off at progressively lower frequencies due to radiative cooling as the observed position moves further away from the shock.

## 4 Summary

We propose a model that may explain some characteristics of giant radio relics: the relative rareness, uniform surface brightness along the length of thin arc-like radio structure, and spectral curvature in the integrated radio spectrum over GHz. In the model, a spherical shock encounters an elongated cloud of the ICM thermal gas that is permeated by enhanced magnetic fields and an additional population of fossil relativistic electrons. As a result of the shock passage, the fossil electrons are re-accelerated to radio-emitting energies (), resulting in a birth of a giant radio relic.

In order to explore this scenario, we have performed time-dependent, DSA simulations of spherical shocks with the parameters relevant for the Sausage radio relic in cluster CIZA J2242.8+5301. In the fiducial model, the shock decelerates from () to () at the acceleration age of 60 Myr. The seed, fossil electrons with are assumed to be injected into the CR population, which is subsequently re-accelerated to higher energies. Such shocks are expected to produce the electron energy spectrum, , resulting in the synchrotron radiation spectrum with the injection index, , and the integrated index, , at high frequencies ( GHz). We consider various models with a range of shock parameters, different upstream gas density profiles, different downstream magnetic field profiles, and three types of fossil electron populations, as summarized in Table 1. Adopting a ribbon-like curved shock surface and the associated downstream volume, which are constrained by the extension angle (or viewing depth) of as detailed in Section 2.5 (e.g van Weeren et al., 2010; Kang et al., 2012), the radio surface brightness profile, , and the volume-integrated spectrum, , are calculated.

The main results are summarized as follows.

1) Two observables, the break frequency in the integrated synchrotron spectrum, , and the width of the synchrotron emission region behind the shock, , can have identical values for two values of postshock magnetic field strength (see Equations [13] and [15]).

2) The observed width of the surface brightness projected onto the sky plane, , strongly depends on the assumed value of (see Figure 4). So may not be used to estimate the postshock magnetic field strength, unless the projection effects can be modeled properly.

3) The integrated synchrotron spectrum is expected to have a spectral curvature that runs over a broad range of frequency, typically for . For a shock of with the postshock magnetic field strength, or , the integrated spectral index increases gradually from to over GHz, if the duration of the shock acceleration is Myr.

4) Assuming that the upstream sound speed is ( keV) as inferred from X-ray observation, a shock of and (e.g., SA1 model) can reasonably explain the observed width,  kpc (van Weeren et al., 2010), and the curved integrated spectrum of the Sausage relic (Stroe et al., 2014b). SB1 model with a shock of , however, produces the integrated spectrum that seems too flat to explain the observed spectrum above  GHz.

5) We also consider two toy models with power-law electron populations with exponential cutoffs at , (SC1pex1 and SC1pex2 models). They may represent the electron populations that were produced earlier and then have cooled down to . SC1pex1 model with a weaker shock () reproduces better the characteristics of the observed integrated spectrum. But the steepening of the integrated spectrum due to radiative cooling alone may not explain the strong spectral curvature above 1.5 GHz toward 16 GHz.

6) This strong curvature at GHz may imply that the downstream electron energy spectrum is influenced by some additional physical processes other than radiative losses, because the integrated spectrum of radiatively cooled electrons steepens with the frequency only gradually. This conclusion is likely to remain unchanged even in the case where the observed spectrum consists of the synchrotron emission from multiple shocks with different Mach numbers, as long as the postshock electrons experience only simple radiative cooling.

Other models that may explain the curved spectrum will be further explored and presented elsewhere.

The authors thank the anonymous referee for his/her thorough review and constructive suggestions that lead to a significant improvement of the paper. HK was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2057940). DR was supported by the National Research Foundation of Korea through grant NRF-2014M1A7A1A03029872 and NRF-2012K1A3A7A03049606.

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