RE-ACCELERATION MODEL FOR THE 'SAUSAGE' RADIO RELIC

Abstract

The Sausage radio relic is the arc-like radio structure in the cluster CIZA J2242.8+5301, whose observed properties can be best understood by synchrotron emission from relativistic electrons accelerated at a merger-driven shock. However, there remain a few puzzles that cannot be explained by the shock acceleration model with only in-situ injection. In particular, the Mach number inferred from the observed radio spectral index, M_radio ≈ 4.6, while the Mach number estimated from X-ray observations, M_X−ray ≈ 2.7. In an attempt to resolve such a discrepancy, here we consider the re-acceleration model in which a shock of M_s ≈ 3 sweeps through the intracluster gas with a pre-existing population of relativistic electrons. We find that observed brightness profiles at multi frequencies provide strong constraints on the spectral shape of pre-existing electrons. The models with a power-law momentum spectrum with the slope, s ≈ 4.1, and the cutoff Lorentz factor, γ_e,c ≈ 3−5×10⁴, can reproduce reasonably well the observed spatial profiles of radio fluxes and integrated radio spectrum of the Sausage relic. The possible origins of such relativistic electrons in the intracluster medium remain to be investigated further.

1. INTRODUCTION

Giant radio relics such as the Sausage and the Toothbrush relics exhibit elongated morphologies, spectral steepening across the relic width, integrated radio spectra of a power-law form with spectral curvature above ~ 2 GHz, and high polarization level (van Weeren et al. 2010, 2012; Feretti et al. 2012; Stroe et al. 2013, 2016). They are thought to be synchrotron radiation emitted by GeV electrons, which are (re)-accelerated at structure formation shocks in the intracluster medium (ICM) (e.g., Ensslin et al. 1998; Brüggen et al. 2012; Brunetti & Jones 2014). It is now well established that nonthermal particles can be (re)-accelerated at such shocks via diffusive shock acceleration (DSA) process (e.g., Ryu et al. 2003; Vazza et al. 2009; Skillman et al. 2011; Kang & Ryu 2011).

In the simple DSA model of a steady planar shock, the synchrotron radiation spectrum at the shock becomes a power-law of jν (rs)∝ν−αsh with the shock index, while the volume-integrated radio spectrum becomes Jν∝ν−αint with the integrated index, αint = αsh + 0.5, above the break frequency νbr (e.g., Drury 1983; Ensslin et al. 1998; Kang 2011). Here Ms is the shock sonic Mach number. If the shock acceleration duration is less than ~ 100 Myr, however, the break frequency, νbr ~ 1 GHz, falls in the typical observation frequencies and the integrated spectrum steepens gradually over the frequency range of (0.1 − 10)νbr (Kang 2015b). Moreover, additional spectral curvatures can be introduced in the case of a spherically expanding shock (Kang 2015a). On the other hand, in the re-acceleration model, the re-accelerated electron spectrum and the ensuing radio spectrum should depend on the shape of the preexisting electron spectrum as well as Ms.

Recently, Kang (2016) (Paper I) explored the observed properties of the Tooth brush relic, and also reviewed some puzzles in the DSA origin of radio gischt relics such as: (1) the discrepancy between Mradio inferred from the radio index αsh and MX−ray estimated from the X-ray temperature discontinuities in some relics, (2) low DSA efficiency expected for weak shocks with Ms ≲ 3 that form in the hot ICM, (3) low frequency of merging clusters with detected radio relics, compared to the expected occurrence of ICM shocks, and (4) shocks detected in X-ray observations without associated radio emission. In Paper I, it was suggested that most of these puzzles can be explained by the reacceleration model in which a radio relic lights up only when a shock propagates in the ICM thermal plasma that contains a pre-existing population of relativistic electrons (see also Kang et al. 2012; Pinzke et al. 2013; Shimwell et al. 2015; Kang & Ryu 2015).

The so-called Sausage relic is a giant radio relic in the outskirts of the merging cluster CIZA J2242.8+5301, first detected by van Weeren et al. (2010). They interpreted the observed radio spectrum from 150 MHz to 2.3 GHz as a power-law-like synchrotron radiation emitted by shock-accelerated relativistic electrons. So they inferred the shock Mach number, Mradio ≈ 4.6, from the spectral index at the hypothesized shock location, αsh ≈ 0.6, and the postshock magnetic field strength, B2 ≈ 5 or 1.2 μG from the relic width of 55 kpc. Although this shock interpretation was strongly supported by observed downstream spectral aging and high polarization levels, the requirement for a relatively high Mach number of Ms = 4.6 in the ICM called for some concerns. Based on structure formation simulations, the shocks in the ICM are expected to have low Mach numbers, typically Ms < 3 (e.g., Ryu et al. 2003; Vazza et al. 2009).

Stroe et al. (2014b) reported that the integrated spectrum of the Sausage relic steepens toward 16 GHz with the integrated index increasing from αint ≈ 1.06 to αint ≈ 1.33 above 2.3 GHz. They noted that such a curved integrated spectrum cannot be consistent with the simple DSA model for a steady planar shock with Ms ≈ 4.6, suggested by van Weeren et al. (2010). Later, Stroe et al. (2014a) suggested, using a spatially resolved spectral fitting, that the the injection index could be larger, i.e., αsh ≈ 0.77, implying Ms ≈ 2.9. In fact, this lower value of Ms is more consistent with temperature discontinuities detected in X-ray observations by Ogrean et al. (2014) and Akamatsu et al. (2015).

In order to understand the spectral curvature in the integrated spectrum reported by Stroe et al. (2014b), Kang & Ryu (2015) considered various shock models including both an in-situ injection model without pre-existing relativistic electrons and a re-acceleration model with pre-existing relativistic electrons. It was shown that shock models with Ms ≈ 3, either the in-situ injection or the re-acceleration models can reproduce reasonably well the radio brightness profile at 600 MHz and the curved integrated spectrum of the Sausage relic except the abrupt increase of the spectral index above 2 GHz. The authors concluded that such a steep increase of the spectral index cannot be explained by the simple radiative cooling of postshock electrons. On the other hand, it was pointed out that the Sunyaev-Zeldovich effect may induce such spectral steepening by reducing the the radio flux at high frequencies by a factor of two or so (Basu et al. 2015).

Recently, Stroe et al. (2016) presented the integrated spectrum spanning from 150 MHz up to 30 GHz of the Sausage relic, which exhibits a spectral steepening from αint ≈ 0.9 at low frequencies to αint ≈ 1.71 above 2.5 GHz. Kang & Ryu (2016) attempted to reproduce this observed spectrum by a re-acceleration model in which a spherical shock of Ms ≈ 2.7 − 3.0 sweeps through and then exits out of a finite-size cloud with pre-existing relativistic electrons. Since the reacceleration stops after the shock crosses the cloud, the ensuing integrated radio spectrum steepens much more than predicted for aging postshock electrons alone, resulting in a better match with the observed spectrum.

On the other hand, Donnert et al (2016) proposed an alternative approach to explain the spectral steepening of the Sausage relic. In order to match the observed radio brightness profiles, it was assumed that the postshock magnetic field strength increases first with the downstream distance, d, from the shock, peaks around d ≈40 kpc, and then decreases exponentially at larger distances. In this model, the magnetic field strength is lower at the immediate postshock region, compared to the model with constant postshock magnetic field, while the spatial distribution of the highest energy electrons peaks at the shock. As a result, the integrated radio spectrum steepens at high frequencies, leading to the curved spectrum consistent with the observation by Stroe et al. (2016).

Donnert et al (2016) presented beam convolved brightness profiles, Sν(R), at several radio frequencies from 153 MHz to 30 GHz in their Figure 5 and the spectral index, between 153 and 608 MHz in Figure 6. We notice that Sν at 153−323MHz extends well beyond 150 kpc away from the relic edge, and increases from ~ 0.6 at the position of the putative shock to ~ 1.9 at 200 kpc south of the shock. Considering the shock compression ratio, these downstream length scales imply that the shock has swept through a region of at least 450 kpc in the case of a plane shock. Note that Kang & Ryu (2016) considered a much smaller cloud size, ~ 130 kpc, since the observed profile of Sν(R) only at 610 MHz with a FWHM of ~ 55 kpc was available for comparison. Thus in this study, we attempt to explain the much broader profiles of Sν at 153 MHz and by adopting a new set of model parameters.

In the next section, we explain basic physics of the DSA model and review the observed properties of the Sausage relic. In Section 3 the numerical simulations and the shock models are described. The comparison of our results with observations is presented in Section 4, followed by a brief summary in Section 5.

 

2. MODEL

2.1. Physics of DSA

The basic physical features of the DSA model for radio relics were described in detail in Paper I. Some of them are repeated here in order to make this paper sufficiently self-contained.

In the limit where the electrons that are injected in situ at a shock with Ms dominate over the reaccelerated electrons, the radio synchrotron spectrum at the shock position can be approximated by a powerlaw of jν(rs) ∝ ν−αsh with the ‘shock’ index

In the other limit where the re-accelerated electrons dominate over the injected electrons, the radio synchrotron spectrum is not a simple power-law, but depends not only on the shock Mach number but also on the spectral shape of pre-existing electrons, for instance,

where γe is the Lorentz factor. Then the radio spectral index at low frequencies (~ 100 MHz) emitted by reaccelerated electrons can be approximated by αsh = (s − 3)/2, independent of Ms.

Synchrotron and inverse Compton (iC) energy losses introduce gradual spectral steepening in the volume-integrated radio spectrum behind the shock, leading to the increase of the ‘integrated’ index, αint from αsh to αsh + 0.5 over ~ (0.1 − 10)νbr. Here the break frequency depends on the magnetic field strength and the shock age as follows:

The factor Q is defined as

where Brad = 3.24 μG(1 +z)2 takes account for energy losses due to iC scattering off the cosmic background radiation and B is expressed in units of μG (Kang 2011). Figure 1 shows that Q evaluated for z = 0.188 has the maximum value, Q = 0.65 at B ≈ 2.5 μG.

Figure 1.The factor Q(B, z) for z = 0.188 given in Equation (4).

The profiles of observed radio flux at multi frequencies can provide strong constraints on the model parameters for radio relics. For instance, the width of radio relics observed at low frequencies, whose radiation comes mainly from uncooled low energy electrons, should be similar to the advection length:

where u2,3 = u2/103 km s−1 is the downstream flow speed and tage is the duration of the shock acceleration. The factor Wl ~ 1.1 − 1.2 reflects the fact that the downstream flow speed in the shock rest frame increases behind the shock in the case of a spherically expanding shock (see Figure 2 below). For planar shocks, Wl ≈ 1.

Figure 2.Flow velocity, u(r), in units of 103 km s−1 (top panels), and the magnetic field strength, B(r), in units of microgauss, plotted as a function of the radial distance from the cluster center, r(Mpc), for the M3.3 model at tage = 124 (red dotted line), 144 (black solid), and 165 (blue dashed) Myr (left panels) and for the M3.8 model at tage = 125 (red dotted line), 143 (black solid), and 160 (blue dashed) Myr (right panels).

On the other hand, the relic width at high frequencies due to cooled electrons becomes similar to the cooling length:

where νobs is the observation frequency and the factor Wh ~ 1.2 − 1.3 takes account for both the downstream speed effects described above and the synchrotron radiation emitted by low energy electrons. Note that the synchrotron emission decreases smoothly behind the shock, so Equations (5) and (6) give only characteristic length scales.

2.2. Injection-Dominated versus Re-Acceleration Dominated Model

The intracluster space may contain fossil relativistic electrons accelerated by the structure formation shocks, since the ICM is expected to go through such shocks twice or so on average (Ryu et al. 2003). In addition, there could be numerous radio galaxies and radio ghosts in the ICM (Slee et al. 2001). As a result, some cluster shocks could encounter a cloud with pre-existing relativistic electrons. In reality, the DSA process is expected to operate simultaneously on both the injected and re-accelerated populations, but the relative importance of the two populations is unknown. Thus one can construct two limiting scenarios to explain the observations of the Sausage relic: the relic is produced (1) by a shock with Ms ≈ 4.6 without pre-existing electrons (insitu injection model), or (2) by a shock with Ms ≈ 3 with pre-existing electrons with the power-law index, s ≈ 4.2 (re-acceleration model).

In the case where in-situ injection dominates over re-acceleration, the shock Mach number should be Ms ≈ 4.6 to explain the spectral index, αsh ≈ 0.6 at the edge of the Sausage relic. Then, the postshock temperature would be kT2 = 20.2 keV (for kT1 = 2.7 keV), which is well above the observed postshock temperature, keV (Akamatsu et al. 2015). So we disfavor this in-situ injection model mainly because of this discrepancy in the postshock temperature. Moreover, Ms = 4.6 is rather high for typical shocks (Ms ≲ 3) that are expected to form in the hot ICM during the course of cluster mergers (e.g., Ryu et al. 2003).

On the other hand, in the re-acceleration model, one can adjust the pre-existing electron spectrum in order to reproduce the observed radio brightness profiles and the integrated spectrum. For example, the powerlaw index should be s ≈ 4.2 to match the observed value, αsh ≈ 0.6. Then we can adopt a low Mach number (Ms ≲ 3), which is more compatible with X-ray observations. The cutoff energy, γe,c, also controls how fast the electron spectrum steepens behind the shock, so it can be adjusted to reproduce the observed spectral aging in the downstream region.

As pointed out in Paper I, the ubiquitous presence of radio galaxies, AGN relics and radio phoenix implies that the ICM may contain radio-quiet fossil electrons (γe,c ≲ 102) or radio-loud live electrons (γe,c ≲ 104) (e.g., Slee et al. 2001). Fossil electrons with γe ~ 100 provide mainly seed electrons that can be injected to the DSA process, resulting in enhancement of the acceleration efficiency at weak ICM shocks. On the other hand, radio-loud electrons with a power-law spectrum and a cutoff at γe,c ~ 7 − 8 × 104 is required to explain the broad relic width of ~ 150 − 200 kpc in the case of the Toothbrush relic (Paper I).

In our re-acceleration model of the Sausage relic, considered in Kang & Ryu (2015, 2016), the pressure of pre-existing relativistic electrons is assumed to be dynamically insignificant. Note that here the preshock medium is not a bubble of hot buoyant relativistic plasma, which is different from the models studied previously by Ensslin & Gopal-Krishna (2001), Ensslin & Brüggen (2002), and Pfrommer & Jones (2011). Thus the presence of pre-existing electrons does not affect the dynamics of the shock. However, it is not clear how relativistic electrons can be mixed with the thermal ICM gas, if they were to originate from radio jets and lobes ejected from AGNs. On the other hand, such a mixture of thermal gas and relativistic electrons can be understood more naturally, if they were to be produced by previous episodes of structure formation shocks and turbulence generated by merger-driven activities in the ICM (e.g., Brunetti & Jones 2014).

2.3. Observed Properties of the Sausage Relic

The cluster CIZA J2242.8+5301 that hosts the Sausage relic is estimated to be located at the redshift, z = 0.188 (Dawson et al. 2015). According to the Suzaku X-ray observations, the ICM temperature drops from kT2 = keV to kT1 = keV across the relic with the inferred shock Mach number, Ms = (Akamatsu et al. 2015). If we take the mean observed values, the sound speed of the preshock gas with 2.7 keV is cs,1 = 8.4 × 102 km s−1, so the shock speed is us = 2.3 × 103 km s−1 for a Ms = 2.7 shock. With the compression ratio of σ = u1/u2 = 2.83, the downstream flow speed becomes u2 = 8.0×102 km s−1, which is probably too slow to explain the observed relic widths (Donnert et al 2016).

The radio observations of the Sausage relic from 150 MHz to 30 GHz using various radio telescopes were reported by Stroe et al. (2016). The brightness profiles of the relic at several radio frequencies based on the observations in Stroe et al. (2016) have been published recently in Donnert et al (2016). The FWHMs of the relic at 153 MHz and 608 MHz, measured from their Figure 5, are 100 kpc and 55 kpc, respectively. These radio data can be used to infer the shock parameters such Ms, us and B2 according to Equations (1)-(6). According to Equation (6) with Wh = 1.2, u2,3 = 0.8, and Q = 0.65 (B2 = 2.5 μG), for example, the relic width at 608 MHz is estimated to be Δlhigh ≈ 58 kpc, which seems to be consistent with the observed profile. On the other hand, the observed brightness profile at 153 MHz extends up to 200 kpc, while the predicted cooling length is only Δlhigh ≈ 116 kpc. Thus the shock model parameters can be determined more accurately through the detail comparisons between the predicted and observed brightness profiles.

According to Figure 1 of Stroe et al. (2013), there are at least four radio galaxies, sources B, C, D, and H, within 1 Mpc of the Sausage relic, which might provide relativistic electrons to the surrounding ICM. In particular, the source H located at the eastern edge of the relic might be feeding relativistic electrons to the upstream region of the relic shock. It is not clear, however, how those electrons can be mixed with the thermal background gas as we conjecture in our model.

 

3. NUMERICAL CALCULATIONS

The numerical setup for our DSA simulations was described in detail in Kang (2015b). So only basic features are given here.

3.1. DSA Simulations for 1D Spherical Shocks

We follow time-dependent diffusion-convection equation for the pitch-angle-averaged phase space distribution function for CR electrons, fe(r, p, t) = ge(r, p, t)p−4, in the one-dimensional (1D) spherically symmetric geometry:

where u(r, t) is the flow velocity, y = ln(p/mec), me is the electron mass, and c is the speed of light (Skilling 1975). Here r is the radial distance from the center of the spherical coordinates, which is assumed to coincide with the cluster center. We assume a Bohm-like spatial diffusion coefficient, D(r, p) ∝ p/B. The cooling term b(p) = −dp/dt = −p/trad accounts for electron synchrotron and iC losses. The test-particle version of CRASH (Cosmic-Ray Amr SHock) code in a comoving 1D spherical grid is used to solve Equation (7) (Kang & Jones 2006).

3.2. Shock Parameters

We assume that the shock dynamics can be approximated by a self-similar blast wave that propagates through the isothermal ICM with the density profile of ρ ∝ r−2. So the shock radius and velocity evolves roughly as rs ∝ t2/3 and us ∝ t−1/3, respectively (e.g., Ryu & Vishniac 1991).

Donnert et al (2016) chose the following shock parameters to explain the observed profiles of Sν and : kT1 = 3.0 keV, Ms = 4.6, and us = 4.1 × 103 km s−1. Then σ = ρ2/ρ1 = 3.5, kT2 = 22.4 keV, and u2 = 1.2 × 103 km s−1. Although the downstream temperature is well above the observed values, kT2 = keV (Ogrean et al. 2014) or kT2 = keV (Akamatsu et al. 2015), they adopted the high value of Ms = 4.6, because it is consistent with αsh = 0.6 in the in-situ injection model, and because the observed flux profiles require the downstream flow speed as large as 1.2 × 103 km s−1.

Here we adopt a different set of shock parameters that may be more consistent with X-ray observations. The preshock temperature is chosen as kT1 = 3.4 keV (cs,1 = 9.4 × 102 km s−1). Then we choose two values of the initial Mach number (see Table 1):

Table 1.Ms,i: initial shock Mach number at the onset of the simulations kT1: preshock temperature B1: preshock magnetic field strength s: power-law slope in Equation (2) γe,c: exponential cutoff in Equation (2) texit: time when the shock exits the cloud with pre-existing electrons Lcloud: size of the cloud with pre-existing electrons tobs: shock age when the simulated results match the observations Ms,obs: shock Mach number at tobs kT2,obs: postshock temperature at tobs us,obs: shock speed at tobs The subscripts “1” and “2” indicate the preshock and posthoock quantities, respectively.

M3.3 model: Ms,i = 3.3, u2,i = 0.99 × 103 km s−1, and γe,c = 5 × 104.

M3.8 model: Ms,i = 3.8, u2,i = 1.1×103 km s−1, and γe,c = 3 × 104.

Note that the downstream flow speed in these models is smaller than the minimum value required in the in-situ injection model considered by Donnert et al (2016). So we consider several models with a range of parameters for the preshock magnetic field strength and the cutoff Lorentz factor. The preshock magnetic field strength is assumed to be B1 = 1 μG in both models, resulting in the postshock strength, B2 ≈ 2.3 − 2.5 μG, since the factor Q = 0.65 has the greatest value for B2 = 2.5 μG. The postshock magnetic field strength is assumed to scale with the gas pressure, as in Paper I.

We find that the models with B1 = 1 μG and γe,c ≈ 3−5×104 produce the profiles of Sν and that are consistent with the observed profiles given in Donnert et al (2016). If we take smaller values for γe,c or larger values for B1, for example, the simulated spectral index increases too fast in the postshock region, compared to the observed profile of .

If we take the mean value of the observed value, kT1 = keV (Akamatsu et al. 2015), instead of 3.4 keV, then the relevant velocities such as us, u2, and the downstream length scale of radio emitting region will decrease by a factor of

In order to reproduce the steep spectral curvature above 2 GHz, we assume that the cloud with preexisting electrons has a finite size, as in Kang & Ryu (2016). Table 1 also lists the cloud size, Lcloud, and the time when the shock exits the cloud, texit. As mentioned in the Introduction, considering that the observed observed downstream length scale is greater than 150 kpc, Lcloud ≳ 370 kpc is considered.

We find that both the simulated brightness profiles and the integrated spectra become consistent with the observations at the shock age of tobs ≈ 144 Myr in M3.3 model and at tobs ≈ 143 Myr in M3.8 model. The degree of spectral steepening above ~ 2 GHz is controlled by the time elapsed between texit and tobs. At tobs the spherical shock slows down to Ms,obs ≈ 2.7 with kT2,obs = 10.7 keV in M3.3 model, and Ms,obs ≈ 3.1 with kT2,obs = 12.9 keV in M3.8 model.

Kang & Ryu (2016) were able to reproduce the observed Sν at 610 MHz and the integrated spectrum of the Sausage relic using re-acceleration models with kT1 = 3.35 keV, Ms,i ≈ 3.0 − 3.3, s = 4.2, γe,c = 104, B1 = 2.5 μG, Lcloud ≈ 130 kpc, and tobs ≈ 55 Myr. As mentioned in the Introduction, such models cannot be consistent with the broad flux profiles at lower frequencies (e.g., 150 MHz) that extend beyond 150 kpc behind the shock. This demonstrates that multi-frequency brightness data is necessary to constrain the DSA models besides integrated spectrum information.

 

4. RESULTS OF DSA SIMULATIONS

Figure 2 shows the flow speed, u(r), and the magnetic field strength, B(r), at three epochs: tage = 124, 144, 165 Myr for M3.3 model and tage = 125, 143, 160 Myr for M3.8 model. Here r is the radial distance from the cluster center. Note that the downstream flow speed in the shock rest frame increases behind the shock, while the magnetic field strength decreases in the downstream region. As a result, the cooling length, Δlhigh, is somewhat longer than estimated for 1D planar shocks with uniform u(r) and B(r) in the postshock region.

Figure 3 shows the synchrotron emissivity, jν (r) at 153 MHz and 608 MHz in arbitrary units, and the spectral index, in the two models. In both models, the shock is about to exit the cloud with pre-existing electrons at texit = 124 − 125 Myr (red dotted lines), while the predicted quantities become consistent with the observations at tobs = 143 − 144 Myr (black solid lines). So the edge of the radio relic is located slightly behind the shock for the second (black solid) and third (blue dashed) epochs. For example, the relic edge is at 1.162, 1.212, and 1.242 Mpc, while the shock position is at 1.162, 1.22, and 1.28 Mpc in M3.3 model.

Figure 3.Time evolution of the synchrotron emissivity, jν (r) at 153 MHz (top panels) and 608 MHz (middle panels), and the spectral index, between the two frequencies (bottom panels), plotted as a function of the radial distance from the cluster center, r(Mpc), for the M3.3 model (left panels) and the M3.8 model (right panels). The line types and the corresponding epochs are the same as in Figure 2. The black dot-dashed lines show the results of the same models except higher magnetic field strength, B1 = 2.5 μG, at the same tobs.

From the profiles of jν(r) at 153 MHz, one can see that the advection length is 140 − 160 kpc, consistent with Equation (5). Due to faster cooling of the higher energy electrons, jν(r) at 608 MHz decreases much faster than jν(r) at 153 MHz. Since the shock weakens in time, the spectral index, , at the relic edge increases from 0.55 to 0.7 during 125−165Myr. In fact, the gradient of (r) depends on the parameters, s and γe,c as well as Ms(t) and B1. For smaller values of γe,c or larger values of B1, (r) would increase behind the shock faster than that in the fiducial models shown in Figure 3. For example, the black dot-dashed lines show the results of the same models at the same tobs except higher magnetic field strength, B1 = 2.5 μG and B2 ≈ 5.8 − 6.3 μG.

4.1. Surface Brightness and Spectral Index Profiles

The radio surface brightness, Iν (R), is calculated from the emissivity jν (r) by adopting the same geometric volume of radio-emitting electrons as in Figure 1 of Kang (2015b). Here R is the distance behind the projected shock edge in the plane of the sky.

Figure 4 shows Iν (R) at 153 MHz and 608 MHz in arbitrary units at the same time epochs as in Figure 3. Here the projection angle ψ = 10° is adopted. In addition, the spectral index, is calculated from the projected Iν (R) at 153 and 608 MHz and shown in the bottom panels. Note that the relic edge is located 20− 40 kpc behind the shock at the two later epochs (black solid and blue dashed lines), since the shock breaks out of the cloud at 124 − 125 Myr (red dotted lines).

Figure 4.Time evolution of the surface brightness Iν(R) at 153 MHz (top panels) and at 608 MHz (middle panels), and the spectral index between the two frequencies (bottom panels) plotted as a function of the projected distance behind the shock, R(kpc) for the M3.3 model (left panels) and the M3.8 model (right panels). The line types and the corresponding time are the same as in Figure 2. The projection angle, ψ = 10° is adopted. Note that, for the second and third epochs, the relic edge is located behind the shock (at R = 0).

The observed profiles should be compared with the brightness profiles convolved with telescope beam profile with a finite width. So we present the brightness profiles, Sν(R), smoothed by Gaussian smoothing with 50 kpc width (a beam of 16 arcsec) at tobs = 143 − 144 Myr in Figure 5. Note that the normalization of radio brightness is arbitrary in Figures 4 and 5. The magenta dots are observational data read from Figures 5 and 6 in Donnert et al (2016). The simulated profiles of S153MHz(R) seem compatible with the observed data up to 150 kpc, beyond which they could be contaminated by the contributions from the radio galaxies (B, C, and D) in the downstream region. For 608 MHz, S608MHz smoothed with 25 kpc width seems to match much better the observed profile. In the bottom panels, we also present calculated with Sν at the two frequencies, smoothed with 50 kpc width. The figure demonstrates that our model predictions convolved with appropriate beam widths are in reasonable agreement with the observations.

Figure 5.Beam convolved brightness profile Sν (R) at 153 MHz (top panels) and at 608 MHz (middle panels), and the spectral index between the two frequencies (bottom panels) at tobs = 144 Myr for the M3.3 model (left panels) and at tobs = 143 Myr for the M3.8 model (right panels). The simulated brightness profiles, Iν(R), shown in Figure 4 are smoothed with Gaussian smoothing with 50 kpc width in order to emulate radio observations. The brightness profile at 608 MHz smoothed with 25 kpc width is also shown by the red long dashed lines in the middle panels.

Figure 6.Time evolution of volume-integrated radio spectrum for the M3.3 model with Ms,i = 3.3 (top panel) at 131 (black solid line), 138 (red long-dashed), 144 (blue dashed), and 151 Myr (magenta dotted), and for the M3.8 model with Ms,i = 3.8 (bottom panel) at 131 (black solid line), 137 (red long-dashed), 143 (blue dashed), and 149 Myr (magenta dotted). The magenta dots are the observational data taken from Stroe et al. (2016).

4.2. Integrated Spectrum

According to Equation (3), the break frequency at tobs = 143−144Myr is νbr ≈ 130 MHz. So in the in-situ injection model without pre-existing electrons, the integrated radiation spectrum is expected to steepen from αsh to αsh+0.5 gradually over 13 MHz−1.3GHz, which is in contradiction with the observed spectrum shown in Figure 6. In the re-acceleration model, however, the integrated spectra depends also on the spectral shape of the preshock electron population. In the case of the cloud with pre-existing electrons with a finite-size, the degree of the spectral steepening also depends on the time lapse, tobs −texit. So the spectral curvature of the observed spectrum can be reproduced by adjusting the set of model parameters, i.e., Ms, B1, texit, tobs, s, and γe,c in our models.

Figure 6 shows how the integrated spectrum, Jν, steepens at higher frequencies due to lack of preexisting seed electrons as well as radiative cooling during 131−151 Myr in the two shock models. Note again texit = 124 − 125 Myr. The magenta dots show the observational data taken from Table 3 of Stroe et al. (2016), which are rescaled to fit the simulated spectrum near 1 GHz by eye. We find that γe,c = 3 − 5 × 104 is needed in order to reproduce Jν both near 1 GHz and 16 − 30 GHz simultaneously at tobs.

 

5. SUMMARY

The Sausage radio relic is unique in several aspects. Its thin arc-like morphology and uniform surface brightness along the relic length over 2 Mpc could be explained by a re-acceleration model in which a spherical shock sweeps through an elongated cloud of the ICM gas with pre-existing relativistic electrons (Kang & Ryu 2015). Moreover, the re-acceleration model can resolve the discrepancy between Mradio ≈ 4.6 inferred from the radio spectral index (van Weeren et al. 2010) and MX−ray ≈ 2.7 estimated from X-ray temperature discontinuities (Akamatsu et al. 2015). Note that in this model the spectral index at the relic edge, αsh ≈ 0.6, can be controlled by the power-law index of the preexisting electron population, s ≈ 4.1−4.2, independent of the shock Mach number. The sharp steepening of the spectrum above ~ 2 GHz (Stroe et al. 2016) could be understood, if we assume that the cloud with preexisting electrons has a finite width and the shock has exited the cloud about 10 − 20 Myr ago (Kang & Ryu 2016).

In this study, we attempt to reproduce the observed profiles of the surface brightness at 153 and 608 MHz and the spectral index between the two frequencies presented in Donnert et al (2016), using the same reacceleration model but with a set of shock parameters different from Kang & Ryu (2016). Since Kang & Ryu (2016) attempted to reproduce the surface brightness only at 610 MHz, the models with s = 4.2, γe,c = 104, and Lcloud ≈ 130 kpc were adequate. However, in order to explain the observed broad profiles of S153MHz and beyond 150 kpc downstream of the shock (Donnert et al 2016) and the degree of spectral steepening of the integrated spectrum at high frequencies (Stroe et al. 2016), here we adopt the following parameters: s = 4.1, γe,c = 3 − 5 × 104, and Lcloud ≳ 370 kpc (see Table 1). Since the re-accelerated electron spectrum depends on the pre-existing electron population as well as the shock Mach number, we find that the cutoff Lorentz factor should be fine tuned in order to match the observations.

We assume the shock has exited the cloud with preexisting electrons at texit ≈ 124−125 Myr after crossing the length of the cloud, Lcloud = 367 kpc in M3.3 model and Lcloud = 419 kpc in M3.8 model. Although both M3.3 and M3.8 models produce the results comparable to the observations as shown in Figures 5 and 6, M3.3 model seems more consistent with X-ray observations: Ms,obs = 2.7, kT2,obs = 10.7 keV, and us,obs = 2.6 × 103 km s−1 at the time of observation, tobs ≈ 144 Myr.

This study illustrates that it is possible to explain most of the observed properties of the Sausage relic including the surface brightness profiles and the integrated spectrum by the shock acceleration model with pre-existing electrons. If the shock speed and Mach number are specified by X-ray temperature discontinuities, the other model parameters such as magnetic field strength and the spectral shape of pre-existing electrons can be constrained by the radio brightness profiles at multiple frequencies. Moreover, the degree of spectral steepening in the integrated spectrum at high frequencies can be modeled with a finite-sized cloud with preexisting electrons.

However, it is not well understood how an elongated cloud of the thermal gas with such pre-existing relativistic electrons could be generated in the ICM of CIZA J2242.8+5301. It could be produced by strong accretion shocks or infall shocks (Ms ≳ 5) in the cluster outskirts (Hong et al. 2014) or by turbulence induced by merger-driven activities (Brunetti & Jones 2014). Alternatively, it could originate from nearby radio galaxies, such as radio galaxy H at the eastern edge of the relic or radio galaxies B, C, and D downstream of the relic. Again it is not clear how relativistic electrons contained in jets/lobes of radio galaxies are mixed with the background gas instead of forming a bubble of hot buoyant plasma. Note that the shock passage through such relativistic plasma is expected to result in a filamentary or toroidal structure (Ensslin & Brüggen 2002; Pfrommer & Jones 2011), which is inconsistent with the thin arc-like morphology of the Sausage relic. In conclusion, despite the success of the re-acceleration model in explaining many observed properties of the Sausage relic, the origin of pre-existing relativistic electrons needs to be investigated further.

On the other hand, the in-situ injection model for radio relics has its own puzzles: (1) Mradio > MX−ray in some relics, (2) low DSA efficiency expected for weak shocks with Ms < 3, (3) relatively low fraction of merging clusters with detected radio relics, compared to the theoretically expected frequency of shocks in the ICM, and (4) some observed X-ray shocks without associated radio emission. In particular, the generation of suparthermal electrons via wave-particle interactions, and the ensuing enhancement of the injection to the DSA process in high beta ICM plasma should be studied in detail by fully kinetic plasma simulations.