Particle injection in three-dimensional relativistic magnetic reconnection

Technical Summary: Particle Injection in Three-Dimensional Relativistic Magnetic Reconnection

Problem Statement
Relativistic magnetic reconnection is a primary mechanism for nonthermal particle acceleration (NTPA) in astrophysical environments, generating power-law spectra and high-energy emissions. While the acceleration of particles to high energies is well-studied, the initial "injection" phase—where thermal particles are energized to a threshold $\gamma_{inj}$ to enter the power-law distribution—remains poorly understood. Critical open questions include how the upstream magnetization $\sigma$ influences the injection energy and the relative contributions of known injection mechanisms (direct acceleration, Fermi kicks, and pickup acceleration) to the injected population. Furthermore, the impact of three-dimensional (3D) effects, such as flux-rope kink instabilities, on these processes has not been systematically quantified.

Methodology
The authors employ fully kinetic particle-in-cell (PIC) simulations using the Zeltron code to study relativistic magnetic reconnection in nonradiative, collisionless pair plasmas.

• Simulation Setup: The study utilizes both 2D and 3D rectangular domains with a force-free current sheet initialized with a weak guide field ($b_g = 0.3$) and sub-relativistic temperature ($\theta_0 = 0.3$).
• Parameter Scan: The cold upstream magnetization $\sigma$ is varied across eight values ($\{8, 12, 16, 24, 32, 48, 64, 96\}$), with the highest values restricted to 2D due to computational constraints.
• Analysis Techniques:
  • Spectral Fitting: An improved spectral fitting procedure (building on French et al. 2023) is used to extract the power-law index $p$, injection energy $\gamma_{inj}$, and high-energy cutoff $\gamma_c$ from downstream particle distributions.
  • Mechanism Attribution: Tracer particles are analyzed to determine their "injection share" (fraction of the total injected population) based on the work done by specific electric field components (parallel vs. perpendicular) and momentum characteristics at the moment they cross the injection threshold.
  • NTPA Correlations: The probability that a particle of a specific energy $\gamma$ was injected by a specific mechanism is calculated to distinguish between injection shares and high-energy correlations.
  • Downstream Isolation: A particle-mixing approach is used to distinguish downstream regions from upstream regions, with specific thresholds applied for 2D and 3D to account for particle escape in 3D.

Key Contributions and Results

1. Injection Criterion and Energy ($\gamma_{inj}$):

   • The authors propose a theoretical model where injection occurs when a particle's gyroradius $r_g$ exceeds half the current sheet thickness $\delta/2$. This yields an injection energy $\gamma_{inj} \simeq \sigma [\sigma_h^{-1}(1+b_g^2)/2]^{1/2}$ in the thermally-dominated regime.
   • 2D Results: Measured $\gamma_{inj}$ values in 2D align well with the theoretical model for the thermally-dominated regime. They are significantly lower than the often-assumed linear relation $\gamma_{inj} = \sigma/4$ for $\sigma \gtrsim 30$.
   • 3D Results: Injection energies in 3D are consistently $\sim 1.5$ times higher than in 2D. The authors attribute this to greater current sheet thicknesses at disruption in 3D caused by flux-rope kink instabilities. The theoretical model underestimates 3D measurements, likely due to these 3D structural differences.

2. Injection Mechanisms and Shares:

   • Temporal Sequence: A distinct activation sequence is observed: (1) Direct acceleration dominates immediately after reconnection onset; (2) Fermi kicks become dominant after $\sim 0.25 L_x/c$; (3) Pickup acceleration activates later ($\sim 0.5 L_x/c$) as outflows establish.
   • Steady-State Shares: In 3D, Fermi kicks contribute $\sim 50\%$ and direct acceleration $\sim 40\%$ to the total injected population (for $\sigma > 8$). In 2D, these roles are reversed (Direct $\sim 50\%$, Fermi $\sim 40\%$). Pickup acceleration is suppressed in 3D ($\sim 5\%$) compared to 2D at lower $\sigma$, though it rises in 2D at very high $\sigma$.
   • NTPA Correlations: Despite direct acceleration injecting only $\sim 40-50\%$ of particles, it shows a strong correlation ($\sim 80\%$) with the highest energy particles ($\gamma \approx \gamma_c$). This is because early-injected particles participate in the continuous Fermi acceleration process for a longer duration. This distinction helps reconcile discrepancies in literature where some studies emphasize direct acceleration (based on high-energy correlations) while others emphasize Fermi kicks (based on injection shares).

3. Efficiencies:

   • Injection Efficiency ($\eta_N$): In 2D, the number efficiency saturates at $\sim 40\%$ for high $\sigma$. In 3D, it is lower ($20-25\%$), likely due to softer power-law indices.
   • Energy Efficiency ($\eta_E$): In 2D, energy efficiency saturates at $\sim 90\%$. In 3D, it grows monotonically with $\sigma$, exceeding $70\%$ at $\sigma=32$, but has not yet converged.

4. High-Energy Cutoff ($\gamma_c$):

   • The evolution of $\gamma_c$ is decomposed into three phases: a transient linear rise ($\gamma_c \propto t$), an intermediate phase, and a quasi-steady phase where $\gamma_c \propto \sqrt{t}$ due to adiabatic compression of plasmoids.
   • In the steady state, $\gamma_c$ scales linearly with $\sigma$. The coefficient is $\sim 4\sigma$ in 2D and $\sim 6\sigma$ in 3D.

Significance and Claims
The paper claims to provide a systematic understanding of particle injection in 3D relativistic reconnection, addressing the gap between 2D and 3D physics.

• Theoretical Validation: The study validates a gyroradius-based injection criterion that successfully predicts $\gamma_{inj}$ in 2D and explains the deviation in 3D via current sheet thickness variations.
• Mechanism Clarification: By distinguishing between "injection shares" (total population contribution) and "NTPA correlations" (contribution to the high-energy tail), the authors resolve apparent contradictions in previous literature regarding which mechanism dominates. They demonstrate that while Fermi kicks inject the majority of particles, direct acceleration is disproportionately responsible for the highest energy particles due to temporal head-starts.
• Dimensionality Effects: The work highlights that 3D instabilities (specifically flux-rope kink modes) fundamentally alter the injection landscape, leading to higher injection energies, lower injection efficiencies, and a shift in the dominance of acceleration mechanisms compared to 2D simulations.

The authors conclude that while their model explains 2D results well, future work is required to systematically explore the transition to ultra-relativistic magnetizations in 3D and to investigate electron-ion plasmas where two-tiered current sheet structures may further complicate injection physics.