Simulation Study of Binary Mergers of Galaxy Clusters I: Properties of Merger Shocks and Radio Emission

TL;DR

This study develops a high-order, three-dimensional framework to simulate binary galaxy-cluster mergers, coupling magnetohydrodynamics with a Fokker-Planck CR-electron solver to track DSA at merger shocks, post-shock turbulent acceleration, and radiative cooling. By injecting CR electrons at shocks and accounting for fossil populations, the authors generate synthetic synchrotron maps that reveal patchy radio relics shaped by MHD turbulence and curved shock geometry, with spectra deviating from simple planar-shock expectations. They find that axial shocks ahead of the heavier subcluster are typically stronger yet more compact than those ahead of the lighter subcluster, and that turbulent magnetic fields produce fine-scale brightness variations and spectral structure in the relics. The results highlight the coupled roles of merger dynamics, turbulence, and CR physics in shaping cluster outskirts radio emission, and provide a theoretical basis for interpreting observed relic morphologies and spectra while outlining future improvements such as self-consistent gravity and cosmological context.

Abstract

We investigate binary mergers of galaxy clusters and the resulting radio relics using three-dimensional simulations. The initial setup consists of two idealized, spherical subclusters with a mass ratio below three, each permeated by turbulent magnetic fields, and we follow their mergers with a high-order accurate magnetohydrodynamic (MHD) code. In parallel, we track the acceleration of cosmic-ray electrons (CRe) via diffusive shock acceleration (DSA) at merger-driven shocks, together with radiative cooling and Fermi-II (turbulent) acceleration in the postshock region, employing a high-order Fokker-Planck solver. Synchrotron emission is computed from the simulated CRe distribution and magnetic fields. In this paper, we detail these numerical approaches and present the first results obtained with them. Two prominent axial shocks emerge along the merger axis; the shock ahead of the heavier subcluster systematically attains a higher Mach number, although it is more compact, than that ahead of the lighter subcluster. Turbulent magnetic fields--both inherited from the initial condition and amplified during the merger--produce patchy, fine-scale structures in the radio surface brightness. Because of the combined effects of turbulent acceleration, spatially nonuniform magnetic fields, and the curved geometry of merger shocks, the volume-integrated radio spectra show deviations from the canonical power-law steepening expected for a planar shock with a uniform field. Reacceleration of preexisting fossil CRe enhances the surface brightness. Our results highlight the coupled roles of merger dynamics, MHD turbulence, and CRe physics in shaping up the observed properties of radio relics in cluster outskirts.

Figures (12)

• Figure 1: (a) Schematic diagram of a binary merger at $t=0$. The initial separation of the two subclusters is $d_{1,2} = f_d \cdot (r_{200,1}+r_{200,2})$, with $f_d = 1$. The impact parameter is $b = d_{1,2} \sin \theta$, where $\theta$ is the velocity inclination angle. (b) Illustration of the shock geometry in the $x-y$ plane ($z=0$) containing the merger axis at $\sim1$ Gyr after pericenter passage in the m4m2$\theta$10 model. Axial shock 1 (blue) propagates ahead of the heavier subcluster, axial shock 2 (red) propagates ahead of the lighter subcluster, and equatorial shocks (black) expand toward the direction perpendicular to the merger axis. The solid blue (red) curve traces the trajectory of the heavier (lighter) gravity halo. Black dots mark the initial core positions. Pinkish (bluish) regions denote inflows associated with the heavier (lighter) subcluster; the lighter subcluster inflow is faster ($u_{\rm inf,2} > u_{\rm inf,1}$). (c) 2D distribution of $u_x$ at $\sim1$ Gyr after pericenter passage in the m4m2$\theta$10 simulation.
• Figure 2: Initial state at $t=0$ for the m4m2$\theta$0 model simulation with $512^3$ grid zones. (a)--(d) 2D distributions of $\rho_g$, $P_g$, velocity magnitude $u=(u_x^2+u_y^2+u_z^2)^{1/2}$, and magnetic field strength $B=(B_x^2+B_y^2+B_z^2)^{1/2}$ in the $x-y$ merger plane ($z=0$). (e)--(h) Corresponding 1D profiles along the $x$-axis ($y=0$, $z=0$). All quantities are normalized by their respective reference units in Section \ref{['s2.1']}.
• Figure 3: (a) Model spectra of CRe for a shock with $M_s=3.0$. Shown are the postshock Maxwellian distribution (black), the freshly injected CRe spectrum $f_{\rm inj}$ (red), the preexisting fossil CRe spectrum $f_{\rm pre}$ (cyan), and the reaccelerated fossil CRe spectrum $f_{\rm RA}$ (blue). The logarithmic momentum axis covers $p_{\rm min}=10\,m_e c$ to $p_{\rm max}=10^5\,m_e c$. (b) Acceleration efficiency of CRe computed using the injection spectrum $f_{\rm inj}$ given in Equation (\ref{['finj']}).
• Figure 4: (a) Volume rendered images of the 3D Mach number distribution of merger shocks at $t/t_0 = 0.6-1.0$ from the $512^3$m4m2$\theta$0 simulation. Equatorial shocks are present at $t/t_0 = 0.6$, near pericenter passage, after which two axial shocks propagate in opposite directions. The equatorial shocks and the two axial shocks, Shock 1 (ahead of the heavier subcluster) and Shock 2 (ahead of the lighter subcluster) are indicated. (b)--(d) Corresponding distributions of the shock speed $V_s$, Mach number $M_s$, and kinetic energy flux $f_{\phi} = (1/2)\,\rho_1 V_s^3$ (in arbitrary units). Solid and dotted lines show results from the $512^3$ and $256^3$m4m2$\theta$0 simulations, respectively. Blue (red) histograms represent Shock 1 (Shock 2), while green histograms denote equatorial shocks.
• Figure 5: (a) Time evolution of the relative distance, $d_{1,2}$, and velocity, $V_{1,2}$, between the two DM clumps for the cases of the m4m2$\theta$10 model with different velocity factors: $f_V=0.5$ (blue), $0.7$ (green), $0.8$ (red), $1.0$ (black), and $1.5$ (magenta). (b) 2D distributions of the shock Mach number in the $x-y$ merger plane ($z=0$) for four corresponding cases, taken approximately 1 Gyr after pericenter passage. All results are from simulations using $256^3$ grid zones.