[Paper Review] Nonthermal radiation from relativistic electrons accelerated at spherically expanding shocks

Hyesung Kang|arXiv (Cornell University)|Nov 27, 2014

Astrophysics and Cosmic Phenomena4 references3 citations

TL;DR

This paper investigates nonthermal synchrotron radiation from relativistic electrons accelerated at spherically expanding shocks in galaxy clusters using time-dependent diffusive shock acceleration (DSA) simulations. It shows that evolving shock parameters and spatially varying magnetic fields lead to nonlinear spectral evolution, causing deviations from standard test-particle DSA predictions in volume-integrated spectra, especially in models with constant magnetic fields.

ABSTRACT

We study the evolution of the energy spectrum of cosmic-ray electrons accelerated at spherically expanding shocks with low Mach numbers and the ensuing spectral signatures imprinted in radio synchrotron emission. Time-dependent simulations of diffusive shock acceleration (DSA) of electrons in the test-particle limit have been performed for spherical shocks with parameters relevant for typical shocks in the intracluster medium. The electron and radiation spectra at the shock location can be described properly by the test-particle DSA predictions with instantaneous shock parameters. However, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws, because the shock compression ratio and the flux of injected electrons at the shock gradually decrease as the shock slows down in time.So one needs to be cautious about interpreting observed radio spectra of evolving shocks based on simple DSA models in the test-particle regime.

Motivation & Objective

• Understand the nonthermal radiation properties of relativistic electrons accelerated at weak, spherically expanding shocks in galaxy cluster outskirts.
• Address the limitations of standard test-particle DSA models when applied to time-evolving, non-planar shocks relevant to radio relics.
• Investigate how time-dependent shock parameters and spatially varying magnetic fields affect electron energy spectra and synchrotron emission.
• Evaluate the impact of inverse-Compton cooling and magnetic field decay on spectral curvature and emission profiles.
• Assess the reliability of interpreting observed radio spectra using simple DSA models in the test-particle regime.

Proposed method

• Performed time-dependent DSA simulations for cosmic-ray electrons at planar and spherically expanding shocks using a one-dimensional spherical version of the CRASH code with a co-expanding spherical grid.
• Adopted a self-similar Sedov-Taylor blast wave solution as initial postshock conditions, with shock speed declining as $u_s(t) \propto t^{-3/5}$.
• Solved the diffusion-convection equation for the pitch-angle-averaged electron distribution function $f_e(r,p,t)$ alongside gasdynamic conservation equations in the test-particle regime.
• Modeled three magnetic field configurations: constant $B(r)$, and two decaying profiles with scale heights of 100–150 kpc behind the shock.
• Calculated synchrotron emission spectra $J_\nu(\nu,t)$ and spatial profiles $j_\nu(r)$ using the electron energy spectrum $N_e(r,\gamma_e)$ from DSA simulations.
• Compared results with standard test-particle DSA predictions, focusing on volume-integrated spectral indices $A_\nu = \alpha_{\rm inj} + 0.5$ and spectral curvature.

Experimental results

Research questions

• RQ1 How do time-evolving shock parameters (e.g., decreasing Mach number and shock speed) affect the electron energy spectrum and synchrotron emission in spherical shocks?
• RQ2 What is the impact of spatially varying postshock magnetic fields on the morphology and spectral properties of synchrotron emission?
• RQ3 How do inverse-Compton losses relative to synchrotron cooling influence the electron energy distribution and resulting radiation spectra?
• RQ4 To what extent do nonlinear features in the electron distribution and magnetic field profile manifest in volume-integrated spectra?
• RQ5 Can standard test-particle DSA models accurately describe observed radio relic spectra when applied to evolving, non-planar shocks?

Key findings

• The electron energy spectrum at the shock reaches a steady state determined by instantaneous shock parameters, supporting the validity of DSA predictions at the shock front.
• Volume-integrated electron momentum spectra $F_e(p)$ and synchrotron emission spectra $J_\nu$ exhibit nonlinear evolution due to decreasing particle injection flux and declining Mach number.
• Deviations from the standard test-particle DSA prediction $A_\nu = \alpha_{\rm inj} + 0.5$ are most pronounced in models with constant magnetic fields (MF1), indicating that time evolution alters spectral steepening.
• In models with decaying magnetic fields ($B(r)$ decreasing over 100–150 kpc), the impact on $j_\nu(r)$ and $\alpha_\nu(r)$ is minimal due to broad electron energy contributions to emission at each frequency.
• Any nonlinear features from spatial variations in $f_e(r,p)$ and $B(r)$ are largely averaged out in volume-integrated quantities, leaving only subtle signatures in $J_\nu$ and $A_\nu$, especially due to dominant inverse-Compton cooling.