Particle injection in three-dimensional relativistic magnetic reconnection

Abstract

Relativistic magnetic reconnection has been proposed as an important nonthermal particle acceleration (NTPA) mechanism that generates power-law spectra and high-energy emissions. Power-law particle spectra are in general characterized by three parameters: the power-law index, the high-energy cutoff, and the low-energy cutoff (i.e., the injection energy). Particle injection into the nonthermal power law, despite also being a critical step in the NTPA chain, has received considerably less attention than the subsequent acceleration to high energies. Open questions on particle injection that are important for both physical understanding and astronomical observations include how the upstream magnetization~$σ$ influences the injection energy and the contributions of the known injection mechanisms (i.e., direct acceleration by the reconnection electric field, Fermi kicks, and pickup acceleration) to the injected particle population. Using fully kinetic particle-in-cell simulations, we uncover these relationships by systematically measuring the injection energy and calculating the contributions of each acceleration mechanism to the total injected particle population. We also present a theoretical model to explain these results. Additionally, we compare two- and three-dimensional simulations to assess the impact of the flux-rope kink and drift-kink instability on particle injection. We conclude with comparisons with previous work and outlook for future work.

Figures (13)

• Figure 1: Cartoons of several particle injection mechanisms, adapted from French_2023. In each panel, $B_0$ is the reconnecting magnetic field, $E_{\rm rec}$ is the reconnection electric field, and $v_{\rm out}$ is the reconnection outflow speed. (a) Injection by direct acceleration from the reconnection electric field near an X-point. (b) Injection by a Fermi "kick." (c) Injection by the pickup process, wherein $\lvert \textbf{p}_\perp'\rvert$ suddenly increases upon crossing the separatrix and subsequent entry into the downstream region.
• Figure 2: Absolute current density $\lvert J/J_0 \rvert$ at different times after the time of reconnection onset $t_{\rm onset}$. Panels on the left display a 2D simulation ($\sigma = 8$) and the panels on the right display an otherwise identical 3D simulation.
• Figure 3: Reconnection rates for various $\sigma$. (a): Time-dependent reconnection rates of 3D (solid) and a few 2D (dashed) simulations. (b): Peak reconnection rates, with green squares representing 3D and red triangles representing 2D.
• Figure 4: Downstream particle spectra from 3D simulations. Panel (a): Evolving downstream particle spectrum from a $\sigma=32$ 3D simulation fitted at the final time step. The vertical dashed green line indicates the measured injection energy $\gamma_{\rm inj}$, the vertical dashed red line the measured cutoff energy $\gamma_c$, and the dashed black line is $\gamma^{-p}$ with measured power-law index $p$. Solid color lines show particle spectra taken every $(1/8) \, L_x/c$, from $t = t_{\rm onset}$ to $t = t_{\rm onset} + 3\, L_x/c$. Panel (b): Downstream particle spectra of 3D simulations at times $t = t_{\rm onset} + 3\, L_x/c$ for various initial upstream magnetizations $\sigma$. Dashed black lines show $\gamma^{-p}$ for $\gamma \in [\gamma_{\rm inj}, \gamma_c]$ using the measured values of $p, \gamma_{\rm inj}, \gamma_c$ and dotted vertical lines are colored and positioned at $\sigma$ values.
• Figure 5: Spectral parameters measured via fitting procedure (described in Appendix \ref{['sec:Appendix_procedure']}) for various $\sigma$ at each time step for 3D simulations (panels a, c, e) and at $t = t_{\rm onset} + 3\, L_x/c$ for all simulations (panels b, d, f). Dotted colored lines in panels (a, c, e) indicate time steps where the power-law extent is short, i.e., $\gamma_c/\gamma_{\rm inj} < 10$. In panels (b, d, f), red triangles are 2D runs and green squares are 3D runs. Panels (a, b): Power-law indices $p(t)$ and $p(t_{\rm onset} + 3\,L_x/c)$. Panels (c, d): Injection energies $\gamma_{\rm inj}(t)$ and $\gamma_{\rm inj}(t_{\rm onset} + 3\,L_x/c)$. The dashed black line in (d) shows linear scaling, assuming $\gamma_{\rm inj} = 1 + \sigma/4$ and the dashed purple line shows $\gamma_{\rm inj} \simeq \sigma [ \sigma_h^{-1} \, (1 + b_g^2)/2 ]^{1/2}$ [i.e., Eq. \ref{['eqn:ginj_model_soft']}], derived in Section \ref{['ss:injection_energy_model']}. Panels (e, f): High-energy cutoffs $\gamma_c(t)$ and $\gamma_c(t_{\rm onset} + 3\,L_x/c)$. The semi-transparent dashed colored lines in (e) show the fit from Eq. \ref{['eqn:gc_fit']} and the green (red) dashed line in (f) shows it evaluated at $t_{\rm onset} + 3\,L_x/c$, i.e. $\gamma_c(\sigma) = 6\sqrt{3}\sigma$ in 3D ($\gamma_c(\sigma) = 4\sqrt{3}\sigma$ in 2D).