The Coma Cluster at LOw Frequency ARray Frequencies. I. Insights into Particle Acceleration Mechanisms in the Radio Bridge

The discovery of large-scale radio bridges connecting A399 and A401 (Govoni et al. 2019) calls for the existence of different (re)acceleration scenarios than those used to explain radio halos and relics. Recently, Brunetti & Vazza (2020) proposed that stochastic acceleration of relativistic electrons by turbulence could explain the observed emission. According to this model, a key role is played by the complex dynamics along the filament connecting the two clusters. Indeed, cosmological simulations indicate that the chaotic accretion of matter along the filament could generate turbulence. The resulting radio emission is expected to be volume filling and with a steep spectrum (α < −1.3).

The Coma radio bridge has a different spatial scale and is located in a different environment than the large-scale bridge found between A399 and A401. However, several optical, millimetric, and X-ray, studies indicate that the bridge is located in a very active dynamical region. Hence, we will investigate in this work if turbulence could also play a role in the post-merging Coma bridge.

Malavasi et al. (2020) recently analyzed the large-scale structure around the Coma cluster. They found that Coma is located at the center of a network of three filaments, one of which connects Coma to the cluster A1367, a massive (M_SZ = 1.7 ± 0.1 × 10¹⁴ Planck Collaboration et al. 2014), dynamically young cluster, that is undergoing a merger between its two subgroups (Ge et al. 2019).

The group NGC 4839 and the radio-galaxy NGC 4789 are located at the periphery of Coma, toward A1367, and close to the filament of galaxies and groups that connect the two clusters. Lyskova et al. (2019) analyzed the interaction between the NGC 4839 group and the Coma cluster using X-ray observations from XMM-Newton and Chandra and comparing their analysis with numerical simulations. They observed a tail of cold gas to the southwest of the group and a "sheath" region of hotter gas that envelops the group. They concluded that the most likely scenario is the one where the group is at its second infall toward the Coma cluster after passing the apocenter. The shock that is responsible for the relic would be due to the first infall of the group.

More recently, Mirakhor & Walker (2020) analyzed XMM-Newton and Planck Sunyaev Zel'dovich data of the Coma cluster. They found a significant entropy deficit in the southwest (SW) sector of Coma, toward A1367. This entropy deficit is consistent with theoretical expectations for filamentary gas streams from the cosmic web toward the Coma cluster.

Overall, several independent works indicate that the SW region of Coma, where the radio bridge is located, is a dynamically active region where the NGC 4839 group is falling into Coma at its second passage, an out-moving shock wave is likely originating the relic, and filaments of gas are accreting onto the main cluster along the filament that connects Coma with A1367. All these motions could inject turbulence in the bridge region and possibly accelerating a seed population of mildly relativistic electrons to originate the radio bridge.

In this work, we use LOFAR and Westerbork Synthesis Radio Telescope (WSRT) observations to analyze the properties of the Coma bridge. We also use X-ray observations with the ROSAT to understand the connection between thermal and non-thermal plasma in the bridge. A detailed study of the halo and relic will be the subject of a forthcoming paper. This paper is structured as follows: in Section 2, we describe the observations and data reduction techniques, and in Section 3 we analyze the bridge properties and its connections with the cluster thermal plasma. In Section 4, we use the cosmological simulations of a Coma-like cluster to inspect under which conditions the Coma bridge can be powered by turbulence. Finally, we conclude in Section 5. Throughout this paper, we assume a Lambda cold dark matter cosmological model, with H₀ = 69.6 km s⁻¹ Mpc⁻¹ and Ω_M = 0.286, Ω_Λ = 0.714. At the Coma redshift (z = 0.0231) the angular to linear scale is 0.47'' kpc⁻¹.

The data used in this work are part of the LOFAR Two Meter Sky Survey (LoTSS; see Shimwell et al. 2019). They consist of two pointings of 8 hours each that have been calibrated independently following a slightly different procedure than the standard one described in Shimwell et al. (2019) and T. W. Shimwell et al. (2020, in preparation). Specifically, for the pointing P192+27, a larger region of calibration and imaging has been used, to account for a strong off-axis source located outside the beam first side lobe. For both the P195+27 and P192+27 pointings, we kept all baselines for calibration, while the LoTSS images and calibration are done with an inner cut of the recorded visibilities below 0.1 km. This cut allows one to easily eliminate radio frequency interferences on the short baselines, and filters out extended emission (larger than 34' in the sky.), e.g., coming from our Galaxy, that is hard to deconvolve and hence would make the calibration difficult. Hence, it cannot be applied to the Coma cluster, that extends for more than 2°.

In order to image the diffuse emission of the Coma cluster, we performed the following steps:

• 1.  
  Subtraction of sources outside the Coma cluster region: sources outside the inner 1 .5° circle radius were subtracted using killMS and DDF (Smirnov & Tasse 2015) in order to speed up the following imaging steps. Specifically, after imaging the model components were corrupted by the antenna gains and subtracted by the UV visibilities. This procedure was repeated twice with different inner UV range (baselines longer than 0.1 km, and all baselines) in order to remove both compact and diffuse emission outside the Coma cluster. The resulting data set consists of emission from the Coma cluster only.
• 2.  
  The data set has been reimaged using an inner UV-cut of 0.3 km, in order to obtain a model for pointlike and extended sources inside the Coma cluster that we subtracted from the data. The model components corresponding to NGC 4839 and NGC 4789 have been excluded from the subtraction, as the emission from those tails merges with the diffuse emission of the bridge and of the relic. The resulting data sets consist of only diffuse emission in the Coma cluster region. Some residual emission from embedded sources could still be present, but its contribution should be negligible.

Imaging was performed using DDF (Tasse et al. 2018) to apply the directional gains derived during calibration and to take into account the differential beam pattern throughout the Coma field. We used the Sub Space Deconvolution (SSD) algorithm, which is generally robust in the deconvolution of extended emission (Tasse et al. 2018 and references therein). However, the emission of the Coma field is very extended and characterized by a low-surface brightness. Even with SSD, multiple cycles of deconvolution with optimized masks, each covering a different portion of the halo/bridge have been performed. Although this procedure is not optimal, it gives better results than the hybrid matching pursuit (HMP) algorithm, and no other algorithm was implemented in DDF at the time of writing. We checked that the residual images did not present regions of un-deconvolved emission in the region of interest.

The images made from observations of the pointing P192+27 have a higher noise with respect to those from P195+27 (see Table 1). From a comparison between the images presented in this work and those of the Data Release 2 (T. W. Shimwell et al. 2020, in preparation) we conclude that this is due to interference and/or flags occurring in the short baselines.

The images of the two pointings have been restored to the same beam, corrected for the primary beam response, and finally combined by weighting with the inverse of their noise. The combined image of the Coma cluster diffuse emission (plus the two tails) is shown in Figure 1. Details about the images are given in Table 2. The absolute flux scale has been checked to ensure consistency with the LOFAR DR2 images (T. W. Shimwell et al. 2020, in preparation). The error on the LOFAR absolute flux scale is assumed to be 15%. Details of the observations are listed in Table 1.

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We analyzed the data from four ROSAT Position Sensitive Proportional Counter observations of the Coma cluster extending out to radii of approximately 60'–70'. These observations, with identifiers rp800006N00, rp800013N00, rp800005N00, and rp800009N00, were carried out between 1991 June 16 and 18 and their combined clean exposure time is 78 ks. The data were reduced using the Extended Source Analysis Software (Snowden et al. 1994). Background and exposure-corrected images in three energy bands (0.7–0.9, 0.9–1.3, and 1.3–2.0 keV, respectively) were combined, after removing artifacts associated with the detector edges.

In order to account for the Galactic foreground, whose intensity may be non-negligible in the low-surface brightness bridge, we fitted a β-model (Cavaliere & Fusco-Femiano 1976) plus a constant to the ROSAT mosaic. We excluded from the fit the central 3' region of the Coma ICM, the contaminating pointlike sources, and the region contained between the azimuths −78° to 2° from the W axis, which correspond to the bridge region and the NGC 4789 sub-halo. The best-fit constant thus determined has been subtracted from the exposure-corrected image to eliminate the Galactic foreground and emission from unresolved distant active galactic nuclei (AGNs).

We used WSRT archival data (Venturi et al. 1990) to create new images of the Coma cluster in order to derive spectral index information in the bridge region. This is a fundamental quantity to understand the origin of the emission. WSRT data were reimaged to achieve a resolution of 150'' ×  100'', which is the best compromise between resolution and sensitivity to extended emission. LOFAR data have been reimaged accordingly, matching the UV range of WSRT observations (minimum baseline ∼40λ) and convolved to the same restoring beam. The WSRT image is shown in Figure 4, left panel, and the properties of the bridge as derived from this image are listed in Table 3.

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As mentioned above, the observations P195+27 has higher sensitivity than P192+27, and is less affected by artifacts from the subtraction of sources close to the bridge. When the images are convolved to a resolution of 150'' × 100'', the quality of the two images differs even more, indicating that the short baselines of P195+27 are better calibrated. Hence, in the following analysis of the spectral index properties, we only use the observations from the pointing P195+27.

For a better comparison between the two images, we identified and masked the sources in the WSRT and LOFAR at low resolution. Ideally, contaminating sources should be identified in high-resolution images and either masked or subtracted from the UV data. However, LOFAR and WSRT data have a very different resolutions, and the weak sources identified in the high-resolution LOFAR image are blended and confused with the bridge diffuse emission once the LOFAR image is convolved to the WSRT restoring beam. One possibility would be to identify the sources in the high-resolution LOFAR image, rescale their flux density and subtract it from the WSRT visibilities. We decided against this approach, as the spectral index of sources is uncertain, and not constant throughout the sources when they are unresolved. In addition, we cannot exclude that they could have an inverted spectrum between 144 and 323 MHz. Hence, we decided to adopt a simpler approach, and identify and mask only the sources that are visible at low resolution. The WSRT image is shown in Figure 4. Since the brightness at 144 MHz is rather uniform (see Figure 2), this means that we can measure the spectral index between 144 and 326 MHz only in the flattest spectral regions, and obtain limits in the remaining regions. The average spectral index in the area where WSRT observations show emission above 2σ_rms is α_patches = −1.3 ± 0.2. This value can be contaminated by the presence of pointlike sources that are not perfectly masked out from the bridge emission, and that usually have a spectrum flatter than −1.2.

To provide conservative constraints to the spectrum of the bridge, we can assume two extreme scenarios:

• 1.  
  The emission of the bridge at 326 MHz is completely detected by WSRT observations above 2σ_rms, i.e., no significant further emission would be detected by deeper observations at 326 MHz. We can compute the average spectrum of the bridge considering the emission observed at 144 MHz above 2 σ_rms, and the emission observed at 326 MHz above 2σ_rms. In this case, we obtain α_L = −1.6 ± 0.2. This value must be considered a lower limit to the real spectrum of the bridge.
• 2.  
  The emission at 326 MHz falls just below 2σ_rms. Hence, we can consider the emission at 326 MHz as the sum of two terms: ${S}_{\mathrm{WSRT}}={S}_{\mathrm{meas}}+2\,{\sigma }_{\mathrm{rms}}\,({N}_{\mathrm{beam}}^{L}-{N}_{\mathrm{beam}}^{W})$, where S_meas is the measured flux density above 2 σ_rms, N_beam^L indicates the bridge area detected above 2σ_rms by LOFAR in beam units, and similarly, N_beam^W indicates the bridge area detected above 2σ_rms by WSRT in beam units. In this case, we obtain α_U = −1.4 ± 0.2, that is an upper limit to the spectral index of the bridge.

These two scenarios are both extreme and allow us to constrain the average spectrum of the bridge as detected by LOFAR: −1.4 ± 0.2 ≤ α ≤ −1.6 ± 0.2. The spectral indexes derived in this section are listed in Table 4.

In Figure 4, the spectral index trend in the region of the bridge is shown. We can see that the spectral index steepens from the relic's outer edge toward the bridge, and does not show any particular trend within the bridge itself. In the region between the source NGC 4839 and the Coma halo, it has an average value of −1.3 ± 0.2.

Also these values are representative of the flattest regions of the bridge that are detected at both 144 and 326 MHz. We can place a lower limit on the spectral index in each box considering the emission detected in the LOFAR image above 2σ_rms. These values are shown with upwards arrows in Figure 4.

Radio and X-ray emission have been found to be spatially correlated in some radio halos (e.g., Govoni et al. 2001; Hoang et al. 2019; Botteon et al. 2020a). The existence or absence of this correlation can provide important insights into the spatial distribution of the magnetic field and emitting particles with respect to the thermal gas. Here, we compute this correlation in the Coma bridge using LOFAR data at 144 MHz. Since we used the radio image at 45'' resolution, we smoothed the X-ray image accordingly.

Let I_R and I_X be the radio and X-ray surface brightness computed in each cell of a regular grid that we used to analyze the bridge (see Figure 5). The grid is composed of a large number of cells of different sizes, as explained below and specified in Table 5. Cells intercepting residual emission from AGNs and radio sources have been identified and excluded from the analysis.

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To search for a possible correlation, we performed a fit in log–log space:

$$\begin{eqnarray}&&\mathrm{log}({I}_{{\rm{R}}})=\beta \mathrm{log}({I}_{{\rm{X}}})+{\alpha }_{c}.\end{eqnarray} \tag{ 1 }$$

We used a hierarchical Bayesian model (Kelly 2007), that allowed us to perform linear regression of I_R on I_X accounting for intrinsic scatter in the regression relationship, censored data, possibly correlated measurement errors, and selection effects (e.g., Malmquist bias). This method derives a likelihood function for the data, hence, we consider the mean of the posterior distribution as the best-fit slope. Following Botteon et al. (2020a), we considered as upper limits the radio values that are below 2σ_rms. The upper limits are properly taken into account in the fit. The fit has been made with different grids, with cells of 90'' × 90'' and 135'' × 135'', corresponding to 7 and 16 times the beam area, respectively.

In Figure 5, the data are shown together with the best-fit line, and the 25th and 75th percentile lines of the fitted posterior regression slope distribution β. In Table 5, the results of the fits are listed. I_R and I_X show a moderate positive correlation (Pearson correlation coefficient equal to 0.67). A positive correlation is expected if the emission originates from the same physical volume, i.e., the radio emission is as volume filling as the X-ray emission, and we are not observing radio and X-ray emission in the same region because of projection effects.

Turbulence mechanisms for particle acceleration predict a high-frequency cutoff of the radiating electrons (e.g., Brunetti & Lazarian 2016). Specifically, super-Alfvénic solenoidal turbulence has been used to explain the emission in large-scale radio bridges, and according to this model the cutoff synchrotron frequency depends on the turbulent energy density and on turbulent energy in the bridge. Hence, particles that emit radiation at high frequencies are only those accelerated in the most turbulent regions. It follows that the properties of the radio emission in the bridge change with frequency, and specifically the emission at lower frequencies should be more volume filling than the emission at higher frequencies. We can test this prediction for the case of the Coma bridge, using LOFAR and WSRT data.

To quantify the volume-filling factor of the emission in the bridge, we used the same images used for the spectral index analysis. The area emitting above 2σ_rms at 326 MHz is 60% of the area above 2σ_rms at 144 MHz. However, this value could be affected by the different sensitivity of the two observations.

To quantify the spatial distribution of the emission at the two frequencies and evaluate whether its distribution is clumpy or smooth, we convolved our images I(x, y) with Gaussian kernels of increasing width (G_X), where X indicates the Gaussian FWHM in arcseconds. We then used a normalized unsharp-masking filter, computing

$$\begin{eqnarray}&&{I}_{\nu }^{\mathrm{clumps},{\rm{X}}}(x,y)=\displaystyle \frac{I(x,y)-I(x,y)\ast {G}_{X}(x,y)}{I(x,y)\ast {G}_{X}(x,y)},\end{eqnarray} \tag{ 2 }$$

where ν indicates the observing frequency, and (x,y) indicate the pixels in the image. ${I}_{\nu }^{\mathrm{clumps},{\rm{X}}}(x,y)$ enhances the emission on angular scale X convolved with the restoring beam of the image. By comparing the distribution of ${I}_{\nu }^{\mathrm{clumps},{\rm{X}}}$ at different frequencies, we could estimate the level of clumping in the radio emission as a function of frequency. We computed ${I}_{\nu }^{\mathrm{clumps},X}(x,y)$ for X = 120'', 225'', and 315''. For each ${I}_{\nu }^{\mathrm{clumps},{\rm{X}}}(x,y)$, we computed the distribution of the flux density at 326 and 144 MHz in the bridge area, limited to cells where radio emission above 2σ was observed over a region larger than the restoring beam. Two example distributions are plotted in Figure 6 for X = 225''. The distribution of the WSRT data shows a long tail at high values, possibly indicating that the emission is clumpier at 326 MHz than at 144 MHz. Similar trends are obtained with X = 120'' and X = 315''. However, the different sensitivity of the two observations could play a role here, as we note that the LOFAR image at high resolution is also quite filamentary. Deeper and higher resolution observations at frequencies higher than 144 MHz would be required to derive robust conclusions from this analysis.

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The cosmological simulation that we use in the following analysis is taken from the suite of MHD non-radiative high-resolution simulations presented in Vazza et al. (2018) and Domínguez-Fernández et al. (2019). The simulations were made with Enzo (Bryan et al. 2014) and include eight levels of adaptive mesh refinement to increase the spatial and force resolution in a large fraction of the simulated cluster volume, up to Δx = 3.95 kpc/cell. One of the simulated clusters has properties similar to Coma (see Vazza et al. 2018 for further details). In particular, the simulation evolves a uniform primordial magnetic field of B₀ = 10⁻¹⁰ G (co-moving) and the resulting Faraday RM profile of the cluster is in agreement with the one derived from observations (Bonafede et al. 2010; Vazza et al. 2018).

Hence, this Coma-like cluster is suitable for comparison with LOFAR observations and is used in the calculations below.

We focus on a region at the periphery of the simulated cluster, where matter is accreted onto the main cluster, and where it has been found that two satellites have an additional velocity component perpendicular to the cluster radius (see Figure 8). Hence, even if the specific dynamical status of the simulated cluster is not exactly the same as the observed bridge, the simulation represents a realistic environment to investigate the origin of the radio emission. In the following, we refer to the region in the simulation where matter is accreting onto the main cluster as to the simulated bridge.

In order to study the plasma conditions that could lead to the (re)acceleration of radio-emitting electrons in the bridge, we separated turbulent fluctuations, δV, from bulk motions on larger scales, using an iterative three-dimensional filtering technique. Specifically, we extracted solenoidal, ∇ · v = 0, and compressive, ∇ × v = 0, turbulent components using the Hodge–Helmholtz projection in Fourier space (e.g., Vazza et al. 2017). This allowed us to compute the local turbulent energy flux, F, due to solenoidal and compressive modes (e.g., Angelinelli et al. 2020).

Here, we compute the electron acceleration rate due to turbulence using the cosmological simulations presented above. In principle, the evolution of radio-emitting particles should be computed with a Lagrangian approach (e.g., Wittor et al. 2017), and then integrating in time their spectral energy evolution using Fokker–Planck equations (e.g., Donnert & Brunetti 2014). However, this approach is numerically challenging and beyond the scope of this paper. Hence, we adopt a simplified approach to estimate the radio emission as a function of frequency from the simulated bridge. Following Brunetti & Vazza (2020), we measured the distribution function of acceleration times in boxes of ${\left(500\mathrm{kpc}\right)}^{3}$ , i.e., 30³ cells, in the simulated bridge. The acceleration time is

$$\begin{eqnarray}&&{\tau }_{\mathrm{acc}}=1.25\cdot {10}^{5}(\mathrm{Myr})\frac{{L}_{0.5}{B}_{\mathrm{\mu G}}^{* }}{\sqrt{{n}_{\mathrm{th}3}}\delta {V}_{7}^{3}},\end{eqnarray} \tag{ 6 }$$

where n_th3 is the thermal particle number density normalized by 10⁻³ cm³, δV₇ is the turbulent velocity (in units of 10⁷ cm s⁻¹) measured on a scale of L_0.5 in each cell, and is the magnetic field strength in microgauss. Following Brunetti & Vazza (2020), we account for the effect of unresolved dynamo amplification in the simulation by setting in every cell: ${B}^{* }=\max \{{B}_{\mathrm{sim}},\sqrt{4\pi \rho {\eta }_{B}\delta {V}^{2}\}}$, where ${B}_{\mathrm{sim}}$ is the magnetic field measured in the simulation, and η_B is the fraction of turbulent kinetic energy flux that is converted into magnetic fields. As discussed in Brunetti & Vazza (2020), we assumed η_B = 0.03. Then, we sampled the distribution function of acceleration times with a step of ${\rm{\Delta }}\mathrm{log}({\tau }_{\mathrm{acc}})=0.2$, obtaining 10 values for $\mathrm{log}({\tau }_{\mathrm{acc}})$. For each τ_acc, we evolved an initial spectrum of relativistic electrons solving the Fokker–Planck equations for a turbulent eddy-turnover time and assuming a single-zone model, i.e., using the average thermal densities and magnetic field in the simulated box. We have considered the combined effect of acceleration, radiative, and Coulomb losses.

Finally, we computed the integrated synchrotron spectrum combining the contributions from the spectra derived from different acceleration times and assuming a single value for the magnetic field (〈B^*〉 = 0.5 μG). These spectra have been weighted according to the distribution functions of acceleration times, and are shown in Figure 7.¹²

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The synchrotron luminosity produced by a region of 1 Mpc³ is shown in Figure 7. We have to assume that the fossil population of electrons have an initial energy budget that is 3 · 10⁻⁴ with respect to the thermal gas in order to match the observed luminosity.¹³ This value is almost a factor 10 higher than the one used Brunetti & Vazza (2020) for the bridge in A399-A401.

This discrepancy can be due to several reasons, such as the magnetic field properties in the bridge, as the simulation is only a rough approximation of the physical conditions in the Coma bridge. Another possibility is that the model proposed by Brunetti & Vazza (2020) can only explain a fraction of the observed emission. More interestingly, this discrepancy could suggest that a key role is played by the large number of seed electrons released in the medium, either by the NGC 4839 tail during its motion or by other radio galaxies present in the bridge region (see, e.g., Venturi et al. 1990; Giovannini et al. 1991). In this case, we can conclude that the Coma bridge is radio bright thanks to a combination of turbulent motions and to the presence of tailed radio galaxies that populate the medium with a high density of mildly relativistic electrons.

The spectrum of the radio luminosity between 150 and 330 MHz is α = −1.5, within the range allowed by our observational constraints.

Using some further assumptions, we can investigate whether the properties of the bridge resulting from turbulent reacceleration would be in agreement with observations. We obtained the synchrotron emissivity from each cell along the simulated bridge, assuming that a fraction of the turbulent energy flux is converted into particle energy flux, and then into synchrotron, and inverse Compton radiation:

$$\begin{eqnarray}&&\epsilon (\nu )\propto {n}_{\mathrm{th}3}\frac{\delta {V}_{7}^{3}}{{L}_{0.5}}\frac{{\left({B}_{\mathrm{\mu G}}^{* }\right)}^{2}}{{\left({B}_{\mathrm{\mu G}}^{* }\right)}^{2}+{\left[3.25{\left(1+z\right)}^{2}\right]}^{2}}{\rm{\Theta }}({\nu }_{s}).\end{eqnarray} \tag{ 7 }$$

Here, ν_s is the maximal frequency for which synchrotron radiation is emitted in each cell. We put Θ(ν_s) = 1 for ν_s > ν₀ and  = 0 otherwise, in order to select only the cells that contribute to the synchrotron radiation observed at the frequency ν₀.

Following Cassano et al. (2010), we assumed ν_s ∼ 7ν_c, ν_c being the critical synchrotron frequency, ${\nu }_{c}\sim 4.6{\gamma }_{\max }^{2}{B}_{\mathrm{\mu G}}^{* }\,\mathrm{Hz}$, where ${\gamma }_{{\rm{\max }}}$ is the Lorentz factor of electrons when radiative losses balance the acceleration rate. Hence, the critical frequency is

$$\begin{eqnarray}&&{\nu }_{c}(\mathrm{MHz})=1.4\cdot {10}^{-3}\frac{{{\rm{\Phi }}}_{7}{n}_{\mathrm{th}3}\delta {V}_{7}^{6}}{{L}_{05}^{2}{B}_{\mathrm{\mu G}}^{* }{\beta }_{B,z}^{2}}{\left(1-\frac{{\tau }_{\mathrm{acc}}}{{\tau }_{\mathrm{adv}}}\right)}^{2},\end{eqnarray} \tag{ 8 }$$

where β_B,z is defined as

$$\begin{eqnarray}&&{\beta }_{B,z}={\left(\frac{{B}_{\mathrm{\mu G}}^{* }}{3.25}\right)}^{2}\times {\left(1+z\right)}^{4},\end{eqnarray} \tag{ 9 }$$

and τ_adv is the expansion/compression timescales, defined as

$$\begin{eqnarray}&&{\tau }_{\mathrm{adv}}=7720\,\mathrm{Myr}{\left(8.11\cdot {10}^{16}{\rm{\nabla }}\cdot {\boldsymbol{v}}\right)}^{-1},\end{eqnarray} \tag{ 10 }$$

where v is the velocity field measured in km s⁻¹, and Φ₇ ∼ 0.71–1.14. We refer the reader to Brunetti & Vazza (2020) and Brunetti & Lazarian (2016) for further details.

Figure 8 shows the simulated radio emission derived from the above formula, for ν = 140 MHz, overlayed onto the X-ray emission contours in the [0.5–2.0] keV energy range (obtained assuming a uniform 0.3Z_⊙ metallicity and using the Broadened Astrophysical Plasma Emission Code (B-APEC) model). The map zooms in on a ${\left(2\mathrm{Mpc}\right)}^{2}$ region in the cluster periphery, where also the profile of Faraday rotation is compatible with the trend derived by Bonafede et al. (2013) for the SW sector of Coma (the simulated bridge, see Vazza et al. 2018). The simulated radio emission shows an excess spanning of ∼2 Mpc along the simulated bridge, with a surface brightness of ∼10–10² lower than what the same model predicts for the central radio halo.

We find that the parts of the bridge that are most suitable for the turbulent acceleration mechanism have a thermal gas density in the range of n ∼ 1–3 · 10⁻²⁸ g cm⁻³, a gas temperature of T ∼ 4–7 keV, and a (solenoidal) turbulent velocity dispersion of σ_v ∼ 500–700 km s⁻¹. The gas density and temperature are in general agreement, though slightly lower than the estimates from X-ray analysis (see Simionescu et al. 2013, and Section 3).

For patches with lower densities or velocities, the model predicts ν_s < ν₀. Denser substructures in this gaseous bridge produce more emission with a relatively steep spectrum (α ≈  −1.2 to 1.3). However, such self-gravitating halos might in reality turn into galaxies, causing us to overestimate their gas content.

Finally, we derived from the simulation the I_X–I_R correlation, using a region comparable to the one used in Figure 5 and the same method described in Section 3. We find a moderate correlation in simulated quantities (Pearson correlation coefficient  = 0.5), with a sublinear scaling of β = 0.80. The 5th and 95th percentiles of the posterior distribution of β are 0.82 and 0.73, respectively, which are larger than the observed value (∼0.5, see Section 3.2). Even though the moderate correlation and the sublinear scaling found in simulations are in rough agreement with the observed ones, our simulation neglects important physical details: we considered a single-zone model, assumed that the amplification of magnetic fields via (unresolved) small-scale dynamo happens instantaneously, and that relativistic electrons are not subject to spatial diffusion on scales larger than the cell size. All these assumptions are clearly violated in the real ICM, and only with more sophisticated approaches (e.g., using tracer particles, Donnert et al. 2013) a quantitative three-dimensional model can be obtained.