Particle acceleration up to the synchrotron burn-off limit in relativistic magnetized turbulence

TL;DR

This work shows that in relativistic, magnetized turbulence with moderate radiative cooling, particle spectra extend beyond the naive acceleration-loss balance up to the synchrotron burn-off limit, with a slope of $s\approx3$ in the absence of cooling and $s\approx4$ when cooling is present. Using 2D3V PIC simulations including synchrotron reaction, the authors identify a generalized Fermi acceleration mechanism driven by large-scale gradients of the four-velocity field $\boldsymbol{u_E}$, requiring relativistic fluctuations $\delta u_E \gtrsim c$ on scales near the particle gyroradius. They find a broad, nearly Bohm-like distribution of acceleration rates, explaining rapid, high-energy energization up to $\gamma_{\rm rad}$ and substantial variability in the emitted radiation, with $h\nu_{\rm rad}\approx160$ MeV. Applying these results to pulsar-wind nebulae, especially the Crab, suggests stochastic acceleration in postshock turbulence can account for the observed high-energy spectral components and variability, motivating further simulations across parameter spaces and inclusion of backreaction effects on turbulence.

Abstract

In high-energy astrophysics, interpreting observed spectra hinges on understanding the competition between energy gains and radiative losses. To progress along these lines, we report on particle-in-cell simulations of particle acceleration in relativistic, magnetized turbulent pair plasmas including synchrotron radiative losses. Our key finding is that the particle energy spectrum does not terminate at this maximal energy but extends beyond with a steepened spectrum, up to the synchrotron burn-off limit where particles cool within a gyrotime. For our adopted parameters (magnetization $σ\approx 1 $ and amplitude $δB/B_0\simeq 1$), the particle distribution follows ${\rm d}n/{\rm d}γ\propto γ^{-s}$ with $s\simeq 3$ below the predicted maximal energy, then steepens to $s\simeq 4$ above. The particle distribution and the radiated synchrotron spectra display strong variability near the cutoff energy down to timescales well below the largest eddy turn-around time. We substantiate our results by demonstrating that the acceleration rate itself displays a broken powerlaw-like distribution whose maximal value is the gyrofrequency. The highest energy particles are accelerated by a generalized Fermi process in ideal electric fields, driven by a gradient of the $4$--velocity field $u_E$ of the magnetic field lines of relativistic amplitude, $δu_E \gtrsim c$, ordered on a scale comparable to the particle gyroradius. We contend that this is a generic feature of relativistic, large-amplitude turbulence. Lastly, we apply our results to the Crab nebula, which exhibits a hierarchy of characteristic Lorentz factors similar to that studied here. We conclude that stochastic acceleration in this environment is a promising mechanism for explaining the highest-energy part of the synchrotron spectral energy distribution, and its variability. [Abridged]

Figures (13)

• Figure 1: Log-log sketch without specific scales of the acceleration rate $\nu_{\rm acc}$, the synchrotron loss rate $\nu_{\rm syn}$ and the gyrofrequency $c/r_{\rm g}$ versus particle Lorentz factor. This figure defines and illustrates the ordering of maximal Lorentz factors, see text for details.
• Figure 2: Spectral energy distributions $\gamma^2 {\rm d}n/{\rm d}\gamma$ versus particle Lorentz factor $\gamma$. All four simulations shown here share the same characteristics up to the cooling Lorentz factor $\gamma_{\rm syn}$, the value of which is indicated in the lower left corner. Spectra are plotted in blue colors at different times, with hue from light to dark as time progresses from $t=3\,\ell_{\rm c}/c$ to $5\,\ell_{\rm c}/c$. In each panel, the thick green curve corresponds to the spectrum at $t=4\,\ell_{\rm c}/c$. The dotted gray line indicates the Lorentz factor $\gamma_{\rm c}$ [Eq. \ref{['eq:gammac']}], the dashed blue line the Lorentz factor $\gamma_{\rm D}$ [Eq. \ref{['eq:gammamax']}] and the dashed red line the synchrotron burn-off Lorentz factor $\gamma_{\rm rad}$ [Eq. \ref{['eq:gradgsyn']}]. The short blue dash-dotted lines indicate power-law slopes ${\rm d}n/{\rm d}\gamma \propto \gamma^{-4}$ and $\propto\gamma^{-5}$ when $\gamma_{\rm syn}<\infty$, and $\propto \gamma^{-3}$ for $\gamma_{\rm syn}\rightarrow \infty$. The thin dashed black curves represent Maxwell-Jüttner distributions with $k_B T/m_ec^2 = 3.8$, 3., 2.2 and 1.9 from top to bottom.
• Figure 3: Estimates of the ideal and nonideal energy gains for a large sample of particles in the simulation with $\gamma_{\rm syn}=2000$. For each point, the abscissa indicates the observed jump in Lorentz factor $\Delta\gamma_{\rm obs}$ between two points selected randomly along particle trajectories, while the ordinates show the respective contributions from the nonideal electric field (red symbols) and ideal electric field corrected by the energy lost through radiation (blue symbols). The thick red (respectively, blue) line plots the median of the distribution of red (respectively, blue) symbols at different values of $\Delta \gamma_{\rm obs}$. The thin black line indicates the one-to-one correlation $\Delta\gamma = \Delta\gamma_{\rm obs}$. At large values of $\Delta \gamma_{\rm obs}$, the thin dashed lines indicate the errors resulting from the limited sample size.
• Figure 4: Radiated synchrotron power per logarithmic interval of frequency $\nu L_\nu$ versus energy $h\nu$, for the simulations whose particle energy distributions have been plotted in Fig. \ref{['fig:meanspec']}. The vertical lines retain their meaning from Fig. \ref{['fig:meanspec']}, although translated into frequency space. Here, the dotted gray line indicates the peak synchrotron frequency $\nu_{\rm c}$ for particles with Lorentz factor $\gamma_{\rm c}$; the dashed blue line indicates the peak synchrotron frequency $\nu_D$ corresponding to the Lorentz factor $\gamma_{\rm D}$, and the dashed red line the peak frequency $\nu_{\rm rad}$ corresponding to the Lorentz factor $\gamma_{\rm rad}$. The thin blue curves show spectra taken at different times, ordered as before from light to dark as time proceeds, while the thick green curve shows the spectrum at $t = 4\,\ell_{\rm c}/c$. The thin dash-dotted lines indicate scalings $\nu L_\nu \propto \nu^{-0.5}$ and $\propto \nu^{-1.5}$ when $\gamma_{\rm syn}<\infty$, and $\propto \nu^0$ for $\gamma_{\rm syn}\rightarrow \infty$.
• Figure 5: Light curves of the radiated synchrotron power ($\nu L_\nu$) in different frequency bins versus time, between $3\,\ell_{\rm c}/c$ and $6\,\ell_{\rm c}/c$. Each bin is represented by its central value $\nu/\nu_{\rm rad}$, and indicated by the color code.