RE-ACCELERATION OF FOSSIL ELECTRONS BY SHOCKS ENCOUNTERING HOT BUBBLES IN THE OUTSKIRTS OF GALAXY CLUSTERS

Abstract

Galaxy clusters are known to host many active galaxies (AGNs) with radio jets, which could expand to form radio bubbles with relativistic electrons in the intracluster medium (ICM). It has been suggested that fossil relativistic electrons contained in remnant bubbles from extinct radio galaxies can be re-accelerated to radio-emitting energies by merger-driven shocks via diffusive shock acceleration (DSA), leading to the birth of radio relics detected in clusters. In this study we assume that such bubble consist primarily of thermal gas entrained from the surrounding medium and dynamically-insignificant amounts of relativistic electrons. We also consider several realistic models for magnetic fields in the cluster outskirts, including the ICM field that scales with the gas density as $B_{ICM}{\infty}n^{0.5}_{ICM}$. Then we perform time-dependent DSA simulations of a spherical shock that runs into a lower-density but higher-temperature bubble with the ratio $n_b/n_{ICM}{\approx}T_{ICM}/T_b{\approx}0.5$. We find that inside the bubble the shock speed increases by about 20 %, but the Mach number decreases by about 15% in the case under consideration. In this re-acceleration model, the observed properties of a radio relic such as radio flux, spectral index, and integrated spectrum would be governed mainly by the presence of seed relativistic electrons and the magnetic field profile as well as shock dynamics. Thus it is crucial to understand how fossil electrons are deposited by AGNs in the ICM and how the downstream magnetic field evolves behind the shock in detailed modeling of radio relics.

Figure 1. Schematic diagram showing a shock encountering a hot bubble (green) with low-energy fossil relativistic electrons in the outskirts of a galaxy cluster. Radio-emitting region (pink) behind the shock contains high-energy cosmicray electrons.

Figure 2. Left and Middle panels: Time evolution of a planar shock with M_s,i = 3.0 running in to a hotter bubble with b = nb/n_ICM = T_ICM/T_b = 0.5. Here the shock is moving to the right and the boundary between the ICM nd the bubble changes gradually with a hyperbolic tangent profile. The shock speed begins to increase at the third time epoch (dashed lines), but the shock Mach number decreases due to higher sound speed as it enters the hotter bubble. Right panel: Shock Mach number inside the bubble, M_s,b, estimated from Equation (1), versus the initial shock Mach number, M_s,i for b = 1 (green long-dashed line), b = 0.5 (black solid), b = 0.25 (red dashed), b = 0.1 (blue dotted), and b = 0.01 (magenta dotted). The values of M_s,b inferred from 1D hydrodynamic simulations are shown with black filled circles for b = 0.5, red open circles for b = 0.25, and blue filled triangles for b = 0.1.

Figure 3. Initial conditions for the gas density, fossil CR electron density, and magnetic field strength for the model that contains a hotter bubble with b = 0.5. The shock is located at r/r₀ = 1.0 at t_age = 0. See Section 2.2.2 for the details of Model A (black solid line), B (red dotted line), and C (blue dashed line) for the magnetic field profiles.

Figure 4. Time evolution of M_s,i = 4.0 shocks without (b = 1.0, upper panels) and with (b = 0.5, lower panels) a bubble. The gas density (left panels) and X-ray emissivity (middle panels) are shown at six different time epochs. Right panels: Evolution of the shock speed, u_s(t)/u₀ (black lines, the left-hand axis), and Mach number, M_s(t) (red lines, the right-hand axis) in the two models. Here, r₀ = 0.8 Mpc, t₀ = 176 Myr, and u₀ = 4.4 × 10³ km s ⁻¹. Inside the bubble the shock speeds up but the Mach number decreases due to the higher sound speed.

Figure 5. Time evolution of M_s,i = 4.0 shocks without (b = 1.0, upper panels) and with (b = 0.5, lower panels) a bubble. Magnetic field strength (left panels), synchrotron emissivity at 150 MHz (middle panels), spectral index, α$\frac{608}{153}$ between 153 and 608 MHz (right panels) are shown at six different time epochs, t_age = 88, 106, 123, 141, 159, and 179 Myr (the shock expands radially outward). Here r is the radial distance from the cluster center. The shock is located inside the bubble at t_in (black solid lines), at the boundary zone at t_at (red solid lines), and outside the bubble at t_out (blue solid lines) in both models.

Figure 6. Postshock profiles of the surface brightness, I_ν(R) at 153 MHz (top panels) in arbitrary units, the spectral index α$\frac{608}{153}$(R) between 153 MHz and 608 MHz (middle panels), and the magnetic field (bottom panels) for the M4.0noA, M4.0bA, M4.0bB, and M4.0bC models. For some models, I_ν’s are scaled by the numerical factors specified in the figure. Here R is the distance in units of kpc behind the projected shock edge (R = 0) in the plane of the sky. The extension angle, ψ = 10°, is adopted. As in Figure 5, the shock is located inside the bubble at t_in (black), in the boundary zone at t_at (red), and outside the bubble at t_out (blue) in all the models. Three shock ages are t_in,at,out = 123, 141, 159 Myr for M4.0noA and t_in,at,out = 106, 123, 141 Myr for M4.0bA, M4.0bB, and M4.0bC. The initial profile of B(r) for each model is shown in the right panel of Figure 3.

Figure 7. Time evolution of volume-integrated radio spectrum for M4.0noA, M4.0Ba, M4.0Bb, and M4.0Bc models are shown at six different time epochs, t_age = 88, 106, 123, 141, 159, and 179 Myr as in Figure 5. The three shock ages, t_in,at,out, are the same as in Figure 6.

Table 1 Model Parameters for Spherical Shocks