Ballistic Surfing Acceleration as a Coherent Mechanism for Electron Acceleration in Galaxy Cluster Shocks

TL;DR

This work tests ballistic surfing acceleration (BSA) as a physically grounded alternative to diffusive shock acceleration (DSA) for electrons in galaxy cluster merger shocks. By formulating BSA under typical cluster conditions and balancing the coherent acceleration with radiative losses, the authors derive a macroscopic limit on $\gamma_{\max}$ and a corresponding steady-state electron spectrum, then forward-model the resulting synchrotron emission. Forward comparison with the Sausage and Toothbrush relics shows that the observed spectral curvature and high-frequency steepening can be reproduced with a very small ensemble BSA efficiency $\eta_{\rm BSA} \sim 10^{-9}-10^{-8}$, while still achieving $\gamma \sim 10^{4}-10^{5}$. The results imply that radio relics can probe coherent electrodynamic acceleration and that BSA can account for relativistic electrons in cluster shocks within energy budgets and inverse-Compton constraints.

Abstract

Radio relics in merging galaxy clusters are widely interpreted as synchrotron emission from relativistic electrons accelerated at large-scale shocks. However, the efficiency of diffusive shock acceleration (DSA) is expected to be reduced in the low-Mach-number, weakly turbulent environments characteristic of cluster merger shocks, and recent results suggest that DSA itself may not constitute a viable physical mechanism. In this work, we investigate ballistic surfing acceleration (BSA) as an electrodynamically grounded mechanism for electron energization that does not rely on prescribed diffusion coefficients. We formulate BSA under typical cluster shock conditions and derive the balance between coherent acceleration by the shock convection electric field and radiative losses due to synchrotron and inverse-Compton cooling. This balance determines both the maximum electron energy and the resulting steady-state spectrum. By forward-modeling the associated synchrotron emission and comparing it with integrated radio observations of the Sausage and Toothbrush relics, we find that the observed spectral curvature and high-frequency steepening can be reproduced when only a very small fraction ($\sim 10^{-9} - 10^{-8}$) of the available BSA acceleration capacity contributes to systematic electron energization. Despite this extremely small efficiency, it is sufficient to accelerate electrons to Lorentz factors $γ\sim 10^4 - 10^5$ under cluster conditions. These results suggest that radio relics provide a promising astrophysical laboratory for probing coherent acceleration, and that the BSA framework may account for the production of relativistic electrons in cluster shocks.

Figures (4)

• Figure 1: Comparison between the acceleration timescale $t_{\rm acc}=\gamma/\dot{\gamma}_{\rm acc}$ and the radiative cooling timescale $t_{\rm cool}=\gamma/|\dot{\gamma}_{\rm loss}|$ as functions of the electron Lorentz factor for representative cluster merger shock parameters. The intersection of the two curves defines the maximum attainable Lorentz factor $\gamma_{\max}$, illustrating that the high-energy cutoff arises naturally from the balance between ballistic surfing acceleration and radiative losses.
• Figure 2: Mapping between ballistic surfing acceleration and observable radio emission in galaxy cluster merger shocks. The upper panels show the maximum electron Lorentz factor $\gamma_{\max}$, determined by the balance between BSA and radiative losses, as a function of upstream velocity $V_u$ and magnetic field strength $B$ for two representative values of $\eta_{\rm BSA}$. The lower panels show the corresponding maximum synchrotron frequency $\nu_{\max}$ inferred from $\gamma_{\max}$, with the characteristic observing bands of LOFAR (HBA) vanHaarlem2013 and the VLA (L-band) Perley2009 indicated.
• Figure 3: Steady-state electron energy distributions $N(\gamma)$ produced by ballistic surfing acceleration in galaxy cluster merger shocks. Different curves illustrate the dependence on magnetic field strength, BSA efficiency, and injection spectral index, while the high-energy steepening near $\gamma_{\max}$ reflects radiative cooling.
• Figure 4: Synchrotron spectra produced by BSA–accelerated electrons in galaxy cluster shocks, compared with integrated radio relic observations. The upper and lower panels show the Sausage and Toothbrush relics, respectively. Solid curves denote model spectra for different values of the BSA efficiency parameter $\eta_{\rm BSA}$, while points indicate observed integrated flux densities Stroe2016.