Correlating Particle Acceleration Rates with Plasma Conditions in Colliding Wind Binaries

Technical Summary: Correlating Particle Acceleration Rates with Plasma Conditions in Colliding Wind Binaries

Problem Statement
While Supernova Remnants (SNe) are traditionally considered the primary accelerators of Galactic cosmic rays up to $\sim$1 TeV, recent observations by the LHAASO experiment have identified extended very-high-energy (VHE) emission from star-forming regions (SFRs) not associated with SNe. Colliding-wind binaries (CWBs) have been proposed as potential sources of these particles, capable of accelerating hadrons to TeV and PeV energies. However, the specific physical mechanisms driving this acceleration in the complex, turbulent environments of CWBs remain unclear. Previous studies have established that CWBs produce non-thermal emission, but they have not sufficiently disentangled the contributions of classical Diffusive Shock Acceleration (DSA) versus turbulent, magnetically driven processes (such as stochastic acceleration, magnetic reconnection, and the mirror effect). This work aims to identify which plasma conditions and physical quantities most strongly correlate with particle acceleration rates to determine the dominant acceleration mechanisms in CWBs.

Methodology
The authors performed high-resolution, three-dimensional magnetohydrodynamic (MHD) simulations of a CWB system, modeled after the $\eta$ Carinae binary at periastron passage. The simulation setup included:

• MHD Modeling: The system was evolved using the ideal compressible MHD equations solved with the AMUN code. The binary consisted of a primary and secondary star with distinct wind parameters (velocities of 500 km s$^{-1}$ and 2500 km s$^{-1}$, respectively). The simulations explored different plasma compressibility regimes by varying the effective polytropic index ($\Gamma_{\text{eff}}$) from adiabatic (5/3) to strongly cooling (1.1).
• Resolution: Adaptive Mesh Refinement (AMR) was employed, achieving a maximum effective resolution of $960 \times 1920 \times 1920$ cells to accurately capture shocks, turbulence, and magnetic reconnection structures.
• Test Particle Integration: Following the MHD evolution, $10^5$ test particles (representing various ion species, primarily Fe$^+$ for computational efficiency, with results generalized to H$^+$) were injected into the turbulent region. Their trajectories were integrated using the relativistic equation of motion, coupling them to the simulated velocity and magnetic fields.
• Correlation Analysis: The study focused on correlating the local particle acceleration rate ($dE/dt$) with four specific plasma quantities extracted from the simulation snapshots: magnetic field strength ($|B|$), current density ($|J|$), vorticity ($|\nabla \times \mathbf{u}|$), and velocity divergence ($|\nabla \cdot \mathbf{u}|$). Statistical analysis was performed using 2D probability distribution functions (2D-PDFs) and Pearson correlation coefficients, considering only regions where the acceleration rate exceeded 100 eV/s.

Key Results

1. Acceleration Mechanisms: The analysis reveals that particle acceleration is dominated by turbulent, magnetically driven processes rather than classical DSA.

   • Magnetic Field and Vorticity: Strong positive correlations were found between the acceleration rate and both magnetic field strength ($|B|$) and vorticity ($|\nabla \times \mathbf{u}|$). Pearson correlation coefficients ranged from 0.46 to 0.55 for these variables. The regression slopes were superlinear (up to 1.79), indicating that stronger magnetic fields and higher turbulence significantly enhance energy gain.
   • Velocity Divergence: In contrast, the correlation between acceleration rate and velocity divergence (a proxy for shock compression) was weak to non-existent (Pearson coefficients 0.18–0.33, with negative or near-zero slopes). This suggests that well-defined, planar shocks play a limited role in the observed acceleration.
   • Current Density: Current density showed moderate correlation (0.44–0.51) but lower regression slopes. The authors interpret this as an indirect tracer of magnetic field complexity rather than a direct driver, as resistive effects are negligible in the ideal MHD limit.

2. Energy Distributions:

   • Particles evolved from thermal distributions to relativistic power-laws within hours to days, reaching energies of hundreds of TeV.
   • The maximum energy and the position of the "knee" in the energy spectrum depend on the surface magnetic field strength ($B_*$), consistent with the Hillas confinement criterion.
   • The polytropic index ($\Gamma_{\text{eff}}$) significantly influenced the spectral shape. Models with stronger cooling (lower $\Gamma_{\text{eff}}$) produced harder spectra and higher acceleration efficiencies due to increased turbulence and magnetic field complexity. Adiabatic models ($\Gamma_{\text{eff}} = 5/3$) showed steeper slopes and less efficient acceleration.

3. Spatial Distribution: Regions of highest acceleration did not strictly coincide with the strongest shock fronts (velocity divergence) but rather with the turbulent downstream regions where magnetic fields are amplified and complex.

Significance and Claims
The paper claims to provide a definitive link between local plasma conditions and particle acceleration efficiency in CWBs, moving beyond steady-state assumptions. The primary significance lies in demonstrating that turbulence and magnetic field complexity are the dominant drivers of particle acceleration in these systems, overshadowing classical diffusive shock acceleration.

The authors conclude that:

• CWBs can accelerate hadronic particles to hundreds of TeV or even PeV energies, with more than 1% of particles reaching the VHE range, provided moderate magnetization ($>1$ G).
• The acceleration process is inherently stochastic and magnetically driven, relying on the interaction of particles with turbulent structures (Alfvén waves, magnetic mirrors, reconnection) rather than simple shock crossing.
• These findings have direct implications for interpreting future high-sensitivity $\gamma$-ray observations (e.g., by LACT and CTAO), suggesting that the detectability of CWBs as cosmic ray sources depends heavily on the level of turbulence and magnetic field complexity, which are modulated by cooling efficiency and orbital phase.

The work does not propose new experimental setups but rather provides a theoretical framework to interpret existing and future observational data, emphasizing that the absence of strong gamma-ray detection in some CWBs may be due to lower turbulence or magnetization rather than a lack of acceleration capability.