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Beyond Fermi-II: Intermittent Particle Acceleration by Relativistic Turbulence in Astrophysical Plasmas

Anton Dmytriiev, Frans van der Merwe, Markus Böttcher

Abstract

Stochastic particle acceleration in turbulent plasmas plays a key role in shaping high-energy emission from relativistic outflows, such as those in Active Galactic Nuclei (AGN) and microquasars. While traditional Fermi-II models provide a foundational framework, they often oversimplify the complex nature of realistic magnetohydrodynamic (MHD) turbulence, especially in high-amplitude ($δB/B_0 \sim 1$) and relativistic regimes. Recent plasma simulations for these conditions have revealed highly non-linear energization effects, such as sudden, large momentum jumps, that remain largely unexplored in astrophysical applications. We present a novel Monte Carlo framework STRIPE that models particle acceleration as a continuous-time random walk (CTRW), capturing both intermittent energy gains and radiative losses. The stochastic evolution of particle momenta is driven by jumps with random magnitudes determined by a distribution of magnetic-field-line velocity gradients, with synchrotron and inverse Compton cooling incorporated self-consistently. Using STRIPE, we explore particle acceleration under physical conditions characteristic of TeV-PeV $γ$-ray emitting microquasars recently identified by Large High Altitude Air Shower Observatory (LHAASO). We find that relativistic, high-amplitude turbulence naturally produces particle spectra with steep low-energy cutoffs, and hard extended power-law high-energy tails reaching tens of PeV. These features differ markedly from standard quasi-linear theory and are well suited to explaining the unexpectedly hard TeV-PeV spectra of LHAASO-detected microquasars. These results highlight turbulent acceleration in the relativistic regime as a promising mechanism for particle energization in microquasar systems, as well as potentially other extreme astrophysical environments.

Beyond Fermi-II: Intermittent Particle Acceleration by Relativistic Turbulence in Astrophysical Plasmas

Abstract

Stochastic particle acceleration in turbulent plasmas plays a key role in shaping high-energy emission from relativistic outflows, such as those in Active Galactic Nuclei (AGN) and microquasars. While traditional Fermi-II models provide a foundational framework, they often oversimplify the complex nature of realistic magnetohydrodynamic (MHD) turbulence, especially in high-amplitude () and relativistic regimes. Recent plasma simulations for these conditions have revealed highly non-linear energization effects, such as sudden, large momentum jumps, that remain largely unexplored in astrophysical applications. We present a novel Monte Carlo framework STRIPE that models particle acceleration as a continuous-time random walk (CTRW), capturing both intermittent energy gains and radiative losses. The stochastic evolution of particle momenta is driven by jumps with random magnitudes determined by a distribution of magnetic-field-line velocity gradients, with synchrotron and inverse Compton cooling incorporated self-consistently. Using STRIPE, we explore particle acceleration under physical conditions characteristic of TeV-PeV -ray emitting microquasars recently identified by Large High Altitude Air Shower Observatory (LHAASO). We find that relativistic, high-amplitude turbulence naturally produces particle spectra with steep low-energy cutoffs, and hard extended power-law high-energy tails reaching tens of PeV. These features differ markedly from standard quasi-linear theory and are well suited to explaining the unexpectedly hard TeV-PeV spectra of LHAASO-detected microquasars. These results highlight turbulent acceleration in the relativistic regime as a promising mechanism for particle energization in microquasar systems, as well as potentially other extreme astrophysical environments.
Paper Structure (6 sections, 4 equations, 3 figures)

This paper contains 6 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Stochastic Acceleration Model
  3. The STRIPE code
  4. Application to Microquasars
  5. Conclusions
  6. Acknowledgments

Figures (3)

  • Figure 1: Electron spectra for the baseline parameter set $B = 10 \mu$G, $l_c = 1$ pc, $\beta_a = 0.41$, $\gamma_0 = 10^5$. Left: time evolution of the electron spectrum in the SPI mode (with synchrotron cooling included). Right: corresponding steady-state electron spectra in the CI mode for different values of $t_{\rm esc}$.
  • Figure 2: Asymptotic electron spectra in the SPI mode for the baseline parameter set $B = 10 \mu$G, $l_c = 1$ pc, $\beta_a = 0.41$, $\gamma_0 = 10^5$. Top left: comparison of spectra with synchrotron cooling enabled and disabled. Top right: dependence on turbulent Alfvén speed $\beta_a$. Bottom left: dependence on magnetic field strength $B$ (cooling enabled). Bottom right: dependence on coherence length $l_c$.
  • Figure 3: Comparison of the MC-simulated electron spectra for baseline parameter set with cooling disabled (solid lines) to corresponding solutions of Fokker-Planck equation at different evolution times (dashed lines).