Magnetic Field Amplification and Particle Acceleration in Weakly Magnetized Trans-relativistic Electron-ion Shocks

TL;DR

This study investigates magnetic-field amplification and particle acceleration in weakly magnetized, quasi-parallel trans-relativistic electron–ion shocks using long-duration 2D PIC simulations. A magnetization-dependent transition from Bell-dominated to Weibel-dominated upstream instabilities is identified, with Bohm-like ion acceleration (E_{max,i} ∝ t) in Bell-dominated shocks and a slower E_{max,i} ∝ t^{1/2} in Weibel-dominated shocks; electron acceleration is efficient primarily in the Weibel regime (ε_e ≈ ε_i ≈ 0.1) while Bell-dominated shocks yield ε_e ≪ 0.1. The results provide a coherent picture of how upstream CR currents shape magnetic-field amplification, energy partition, and maximum particle energies, with direct implications for the afterglows of GRBs, FBOTs, and jet termination shocks. The work furnishes microphysical parameters (ε_B, ε_i, ε_e) for trans-relativistic shocks and highlights how deceleration can trigger a transition between acceleration regimes, affecting nonthermal luminosity over time.

Abstract

We investigate the physics of quasi-parallel trans-relativistic shocks propagating in weakly magnetized plasmas by means of long-duration two-dimensional particle-in-cell simulations. The structure of the shock precursor is shaped by a competition between the Bell instability and the Weibel instability. The Bell instability is dominant at relatively high magnetizations $(σ\gtrsim10^{-3})$, whereas the Weibel instability prevails at lower magnetizations $(σ\lesssim10^{-4})$. Bell-dominated shocks efficiently accelerate ions, converting a fraction $\varepsilon_{\mathrm{i}}\sim0.2$ of the upstream flow energy into downstream nonthermal ion energy. The maximum energy of nonthermal ions exhibits a Bohm scaling in time, as $E_{\max}\propto t$. A much smaller fraction $\varepsilon_{\mathrm{e}}\ll0.1$ of the upstream flow energy goes into downstream nonthermal electrons in the Bell-dominated regime. On the other hand, Weibel-dominated shocks efficiently generate both nonthermal ions and electrons with $\varepsilon_{\mathrm{i}}\sim\varepsilon_{\mathrm{e}}\sim0.1$, albeit with a slower scaling for the maximum energy, $E_{\mathrm{max}}\propto t^{1/2}$. Our results are applicable to a wide range of trans-relativistic shocks, including the termination shocks of extragalactic jets, the late stages of gamma-ray burst afterglows, and shocks in fast blue optical transients.

Figures (13)

• Figure 1: Schematics of the setup. Panel (a) corresponds to the upstream frame, in which $\sigma$ and $\theta_B$ are defined. Panel (b) is the downstream frame, in which the PIC simulations are performed. We work in the $x-y$ plane with an in-plane background magnetic field.
• Figure 2: Snapshot of the $\sigma=10^{-3}$ shock, taken at $\omega_{\mathrm{pi}}t=7000$. Panel (a) is the plasma density $N_{\mathrm{e}}$ normalized by the upstream value. Panels (b-d) are the magnetic field components in units of the equipartition magnetic field, Equation (\ref{['eq:b_eq']}). The colorbars are in a symmetric log scale, in which the range $[-0.01, 0.01]$ is in a linear scale, and values outside this range are in a log scale. Panels (e, f) are the phase space densities $f(x, p_x)$ of ions and electrons, respectively.
• Figure 3: Snapshot of the $\sigma=10^{-3.5}$ shock, taken at $\omega_{\mathrm{pi}}t=7000$. The format is the same as in Figure \ref{['fig:1e-3']}.
• Figure 4: Snapshot of the $\sigma=10^{-4}$ shock, taken at $\omega_{\mathrm{pi}}t=7000$. The format is the same as in Figure \ref{['fig:1e-3']}. Compared to Figure \ref{['fig:1e-3']}, the shock structure is notably changed, as the precursor is now shaped by the Weibel instability.
• Figure 5: Time evolution of the upstream cosmic ray current density for various $\sigma$. Panel (a) shows the raw $\eta=J_{\mathrm{CR}}/en_0c$, and panel (b) the normalized $\eta/\eta_{\mathrm{crit}}$. Dark teal, orange, dark green, and turquoise curves represent $\sigma=10^{-3}, 10^{-3.5}, 10^{-4}$, and $10^{-4.5}$, respectively. Cosmic ray current is measured using only ions.