Maximum Energy of Particles Accelerated in GRB Afterglow Shocks

TL;DR

This work tackles how maximum electron energies in relativistic GRB afterglow shocks encode particle-acceleration physics. It implements two prescriptions—PIC-inspired small-angle scattering with $\gamma_{\max,\mathrm{PIC}}(t') \propto t'^{1/2}$ and Bohm diffusion with $\gamma_{\max,\mathrm{Bohm}}$—within a one-zone, self-consistent afterglow model that evolves the electron distribution under synchrotron and KN-corrected SSC cooling. A key finding is that a pronounced exponential synchrotron cutoff at $h\nu_{\max}$ is expected for low-energy, low-density bursts during early times, but current LAT data for GRB 190114C and GRB 130427A cannot decisively discriminate between the PIC and Bohm pictures due to SSC dominance and photon statistics. The results underscore the potential of coordinated MeV–GeV–TeV observations to break model degeneracies and place tighter constraints on relativistic-shock particle acceleration. This work thus provides a practical framework for using GRB afterglows to probe fundamental microphysics of collisionless shocks.

Abstract

Particle acceleration in relativistic collisionless shocks remains an open problem in high-energy astrophysics. Particle-in-cell (PIC) simulations predict that electron acceleration in weakly magnetized shocks proceeds via small-angle scattering, leading to a maximum electron energy significantly below the Bohm limit. This upper bound manifests observationally as a characteristic synchrotron cutoff, providing a direct probe of the underlying acceleration physics. Gamma-ray burst (GRB) afterglows offer an exceptional laboratory for testing these predictions. Here, we model the spectral evolution of GRB afterglows during the relativistic deceleration phase, incorporating PIC-motivated acceleration prescriptions and self-consistently computing synchrotron and synchrotron self-Compton emission. We find that low-energy bursts in low-density environments, typical of short GRBs, exhibit a pronounced synchrotron cutoff in the GeV band within minutes to hours after the trigger. Applying our framework to GRB 190114C and GRB 130427A, we find that current observations are insufficient to discriminate between PIC-motivated acceleration and the Bohm limit, primarily due to large uncertainties in the Fermi-LAT band. Nevertheless, future MeV-TeV afterglow observations can break model degeneracies and place substantially tighter constraints on particle acceleration in relativistic shocks.

Figures (4)

• Figure 1: Afterglow spectrum for a burst placed at $D_{\mathrm{L}} = 100$ Mpc at $T_{\mathrm{obs}} = 100$ s. The burst parameters are $\varepsilon_e = 0.1$, $\varepsilon_B = 3.5 \times 10^{-3}$, $E_{\mathrm{iso}} = 10^{54}$ erg, $\Gamma_0 = 500$, $n = 0.5~\mathrm{cm^{-3}}$, and $p = 2.4$. The solid curve shows the numerical spectrum, while the dashed and dotted curves show the analytical synchrotron and SSC components with characteristic frequencies denoted by blue symbols. The red markers indicate $\nu_{\rm gap}$ and $0.1\,\nu_{\rm gap}$, with their vertical separation corresponding to $\log_{10}\mathscr{F}$, which quantifies the prominence of the synchrotron cutoff.
• Figure 2: Flux ratio $\mathscr{F} \equiv \nu F_\nu(0.1\,\nu_{\rm gap}) / \nu F_\nu(\nu_{\rm gap})$ shown as a function of $E_{\rm iso}$ and $n$ at observer times $T_{\rm obs} = 100,\ 250,\ 750\,$ and $3000\,\mathrm{s}$ after trigger. Colors indicate the value of $\mathscr{F}$, as shown by the color bar. All results are computed using the PIC-motivated acceleration prescription with $\varepsilon_e = 0.1$, $\varepsilon_B = 3.5 \times 10^{-3}$, and $p = 2.4$. Dash–dotted curves denote the theoretical prediction corresponding to $\mathscr{F} = 10$, in good agreement with the numerical results. The shaded and dashed contours indicate detectability for bursts placed at redshifts $z = 0.2$ and $z = 0.5$, respectively.
• Figure 3: Spectral fitting of GRB 190114C from X-ray to TeV energies across multiple observation intervals. Different colors denote different time intervals, as indicated in the legend. Circles, squares, and triangles represent data from Swift/XRT–BAT, Fermi-LAT, and MAGIC, respectively, with error bars indicating $1\sigma$ uncertainties. Solid curves are based on the PIC acceleration model, while dashed curves correspond to the Bohm diffusion limit. Both of them adopt $\varepsilon_e = 0.1$, $\varepsilon_B = 10^{-3}$, $E_{\mathrm{iso}} = 1.5 \times 10^{54}\,\mathrm{erg}$, $n = 1\,\mathrm{cm^{-3}}$, $p = 2.4$, and $\Gamma_0 = 500$.
• Figure 4: Spectral fitting of GRB 130427A from X-ray to GeV energies over the interval 138–750 s after the onset of the prompt emission. Circles and squares denote data from Swift/XRT and Fermi-LAT, respectively, with error bars indicating $1\sigma$ uncertainties. The solid curve shows the PIC-motivated acceleration model, while the dashed curve corresponds to the Bohm diffusion limit. Both models adopt $\varepsilon_e = 0.1$, $\varepsilon_B = 3 \times 10^{-4}$, $E_{\mathrm{iso}} = 4 \times 10^{54}$ erg, and $\Gamma_0 = 500$. The PIC model assumes $n = 1.5~\mathrm{cm^{-3}}$ and $p = 2.2$, whereas the Bohm model adopts $n = 1~\mathrm{cm^{-3}}$ and $p = 2.4$.