The β -dependence of Particle Spectra in Relativistic Turbulent Reconnection

Abstract We perform numerical simulations of particle acceleration in relativistic, self-driven turbulent magnetic reconnection using the magnetohydrodynamic–particle-in-cell method. We systematically investigate the dependence of the nonthermal particle spectral exponent on the plasma β . We find that particle acceleration proceeds in two stages: an initial, efficient, first-order Fermi phase, where momentum gains are comparable in parallel and perpendicular directions, followed by a slower drift-dominated phase. The power-law slope of the nonthermal spectrum is established during the Fermi phase, as found in previous studies. Our results demonstrate a systematic steepening of the accelerated particle energy spectrum with increasing β . We derive empirical scaling relations: the spectral exponent α ∝ β 0.5 in the relativistic regime, compared to α ∝ β 0.3 in the nonrelativistic case. This marked difference is rooted in relativistic physics: the increased inertial mass density ( ρh ) in high- β plasmas acts as an energy sink, reducing the Alfvén velocity and thereby altering the dynamics of the magnetic energy release and its partition efficiency. The derived scaling provides a unified physical framework for interpreting the diversity of the nonthermal radiation spectra observed in astrophysical sources, including black hole corona X-ray flares, gamma-ray bursts, and active galactic nucleus jets.