Particle Acceleration in Relativistic Plasma Turbulence

Due to its ubiquitous presence, turbulence is often invoked to explain the origin of nonthermal particles in astrophysical sources of high-energy emission. With particle-in-cell simulations, we study decaying turbulence in magnetically dominated (or, equivalently, “relativistic”) pair plasmas. We find that the generation of a power-law particle energy spectrum is a generic by-product of relativistic turbulence. The power-law slope is harder for higher magnetizations and stronger turbulence levels. In large systems, the slope attains an asymptotic, system-size-independent value, while the high-energy spectral cutoff increases linearly with system size; both the slope and the cutoff do not depend on the dimensionality of our domain. By following a large sample of particles, we show that particle injection happens at reconnecting current sheets; the injected particles are then further accelerated by stochastic interactions with turbulent fluctuations. Our results have important implications for the origin of nonthermal particles in high-energy astrophysical sources.

Figure 1

Developed turbulence from a 2D simulation with ${\sigma }_0=10$, $\delta B_{\mathrm{rms}0}/B_0=1$, and $L/d_{e0}=3280$ (with $l=L/8$). Top: Current density $J_z$ at $ct/l=5.5$ (normalized to $e{n}_{0}c$) indicating the presence of coherent structures like current sheets, plasmoids, and vortices (see inset). Bottom: Magnetic power spectra computed at different times, as indicated by the vertical dashed lines (same color coding) in the inset, which also shows the time evolution of $\delta B_{\mathrm{rms}}^2=⟨\delta B^2⟩$ normalized to $B_0^2$.

Figure 2

Top: Particle spectrum evolution for the simulation in Fig. 1. At late times, the spectrum displays an extended power-law tail with slope $p=-d\log N/d\log (\gamma -1)\sim 2.9$. The inset shows the dependence of $p$ on $\delta B_{\mathrm{rms}0}^2/B_0^2$ and ${\sigma }_0$. Bottom: Particle spectra at late times ($ct/l=12$) for simulations with fixed ${\sigma }_0=10$, $\delta B_{\mathrm{rms}0}/B_0=1$, and $l=L/8$, but different system sizes $L/d_{e0}\in \{410,820,1640,3280,6560\}$. The insets show the dependence of $p$ (dashed line is the asymptotic slope $p=2.9$) and the cutoff Lorentz factor ${\gamma }_c$ [dashed line is the predicted scaling ${\gamma }_c\sim \sqrt{{\sigma }_z}{\gamma }_{th0}(l/d_{e0})$] on the system size.

Figure 3

Top: Current density $J_z$ at $ct/l=4$ from a 3D simulation with ${\sigma }_0=10$, $\delta B_{\mathrm{rms}0}/B_0=1$, $L/d_{e0}=820$, and $l=L/4$, showing the copious presence of current sheets. Bottom: Time evolution of the corresponding particle spectrum. The inset shows for two different box sizes that the time-saturated particle spectra are almost identical between 2D (blue) and 3D (red).

Figure 4

Top: Evolution of the Lorentz factor for 10 representative particles selected to end up in different energy bins at $ct/l=12$ (matching the different colors in the colorbar on the right). Middle: PDFs of $|J_{z,p}|/J_{z,\mathrm{rms}}$ experienced by the injected particles at their $t_{\text{inj}}$ (red circles) and by all tracked particles at $ct/l=4$ (blue diamonds). Bottom: Zoom of $J_z$ at $ct/l=4$ with circles indicating the positions of the particles that are injected around this time.

Figure 5

Evolution of the mean Lorentz factor of different generations of particles. The initial energy gain, due to the reconnection electric field, can be modeled as in Eq. (1) with ${\beta }_R=0.05$ (dashed lines), while the subsequent evolution, governed by stochastic interactions with the turbulent fluctuations, follows Eq. (3) (dot-dashed line).