Late-time structure and electron acceleration in Weibel-mediated relativistic jet–ambient double shocks: forward-shock saturation, ion thermalization, and the transition to diffusive shock acceleration

When a relativistic, unmagnetized plasma jet bores into an unmagnetized electron–ion ambient medium — the configuration of a gamma-ray-burst external shock — the interaction is mediated by the Weibel (current-filamentation) instability and produces a double shock: a reverse shock in the jet, a forward shock in the ambient medium, separated by a contact discontinuity. The 3D particle-in-cell simulation of Ardaneh, Cai, Nishikawa and Lembège (2015), the source of this problem, captured this system only in its early stage (jet Lorentz factor 10, reduced ion-to-electron mass ratio 16, run time $t^*=500$ in units of the inverse electron plasma frequency, $\omega_{pe}^{-1}$): the forward shock had not reached the predicted density jump, the two ion populations had not fully thermalized, electrons were accelerated by an early shock-surfing-like process without any diffusive-shock-acceleration signature, and the tenuous high-energy electron population escaping into the upstream was described only qualitatively. The problem is to determine the late-time, fully developed state of this double-shock system — whether the forward shock saturates at the magnetohydrodynamic jump conditions, how ion thermalization completes, what accelerates electrons in the weak-field forward-shock region, how the early acceleration gives way to diffusive shock acceleration, and what becomes of the tenuous electron population escaping into the upstream — with the sources' own caveat that the reduced mass ratio overweights Weibel-like modes and speeds ion isotropization, so late-time conclusions drawn at $m_i/m_e=16$ may qualitatively misrepresent the developed system. A later simulation by the same group and a body of single-shock particle-in-cell literature inform several of these questions but none resolves them for the jet–ambient double-shock configuration.

Background

A gamma-ray burst (GRB) afterglow is produced when the relativistic ejecta expelled by the burst decelerates by sweeping up the surrounding medium; in hydrodynamic terms the interaction forms a double shock consisting of a reverse shock (RS) propagating back into the jet, a forward shock (FS) propagating into the ambient medium, and a contact discontinuity (CD) separating the two shocked plasmas. Because astrophysical plasmas are effectively collisionless, these shocks cannot form through particle collisions; instead, counterstreaming plasmas drive the Weibel instability (also called the current-filamentation instability), which breaks the flow into current filaments and amplifies transverse magnetic fields that provide the scattering and dissipation needed to thermalize the plasma. The natural theoretical benchmark for the resulting compressions is the set of Rankine–Hugoniot (jump) conditions — the conservation laws for mass, momentum and energy across a one-dimensional magnetohydrodynamic (MHD) discontinuity — which predict, for the simulation parameters below, an ambient-frame density jump $9.4\,n_1 \le n_{21} \le 16.0\,n_1$ across a fully developed FS (the range reflects the uncertain downstream adiabatic index $4/3\le\tilde\Gamma\le5/3$).

The tool of choice for this physics is the particle-in-cell (PIC) method, which follows the full electromagnetic dynamics of computational particles on a grid. Ardaneh, Cai, Nishikawa and Lembège (2015, ApJ 811, 57) performed a 3D PIC simulation of an unmagnetized relativistic jet (Lorentz factor $\gamma_{41}=10$, jet-to-ambient density ratio 1.7, reduced ion-to-electron mass ratio $m_i/m_e=16$) injected into an unmagnetized electron–ion ambient plasma. The jet Lorentz factor of 10 corresponds to the deceleration phase relevant to early GRB afterglows. The simulation found: a Weibel-mediated double shock whose RS was already fully developed and consistent with the analytic jump conditions, while the FS was still developing with a measured compression $n_{21}/n_1\approx6$, about a factor of 1.5–2.5 below the analytic range; jet and ambient ions in the shocked region still distinguishable, i.e. not yet thermalized or mixed; electrons heated to a maximum energy fraction $\epsilon_{\rm ele}\approx\sqrt{\epsilon_b}$, where $\epsilon_b$ and $\epsilon_{\rm ele}$ are the magnetic-field and electron energy densities normalized to the total incoming energy (fractions known as equipartition parameters, the standard inputs of afterglow radiation modeling); and, behind the RS, a non-thermal electron power-law tail with index $\alpha=2.2$ over momenta $p^*\approx100$–360 (momenta normalized to $m_ec$). Tracing individual trajectories, the authors attributed this acceleration not to diffusive shock acceleration (DSA, the standard first-order Fermi process in which particles scatter back and forth across the shock and gain energy on each crossing) but to a process analogous to shock-surfing acceleration (SSA), in which electrons are trapped by the longitudinal ambipolar electric field behind the RS — the field arising from the density gradient and the different mobilities of electrons and ions — and accelerated by the transverse electric fields of the ion current filaments, with a tenuous population of the accelerated electrons convected back into the jet upstream. Established results on unmagnetized relativistic shocks (Spitkovsky 2008; Sironi, Spitkovsky and Arons 2013) indicate that DSA eventually dominates at late times, but the 2015 run ended at $t^*=500$ (in units of the inverse electron plasma frequency) before any DSA signature appeared. The simulation's reduced mass ratio — 16 versus the real proton-electron value 1836 — is known to distort the instability spectrum: it overweights the Weibel-like modes relative to oblique modes and accelerates ion isotropization, so late-time conclusions drawn at this reduced ratio carry an acknowledged uncertainty. The 2016 follow-up by the same group (ApJ 827, 124) extended the analysis, identified electrostatic double layers — narrow, charge-separating structures sustaining a net potential drop — as the injection agent, and compared 2D with 3D behavior, but used the same reduced mass ratio and comparable run times, leaving the late-time regime unexplored.

Problem Statement

Determine, with long-duration self-consistent 3D particle-in-cell simulations of an unmagnetized relativistic electron–ion jet propagating into an unmagnetized ambient plasma — the gamma-ray-burst external-shock configuration, at jet Lorentz factor of order 10 and jet-to-ambient density ratio of order 1.7, with $m_i/m_e=16$ as the required reference case (this single reduced mass ratio is the case on which the deliverables below are defined; additional, more realistic mass ratios would only serve as a robustness check on the reduced-ratio caveat and are not part of the required deliverable) — the late-time, fully developed structure of the Weibel-mediated double shock and the electron-acceleration picture within it that the source simulations (Ardaneh et al. 2015, 2016; $\gamma=10$, density ratio 1.7, $m_i/m_e=16$, $t^*\le560$ in units of $\omega_{pe}^{-1}$) captured only in their developing stage, the required extension being run times of several thousand $\omega_{pe}^{-1}$ — at least an order of magnitude beyond the source runs, $t^*\gtrsim5\times10^{3}$ — with the developed state certified by a convergence criterion (the ambient-frame forward-shock compression and the downstream ion energy distribution stationary to within about 10% over a window of at least several hundred $\omega_{pe}^{-1}$) rather than read off a single fixed-time snapshot. The single overarching objective is a quantitative characterization of the developed double shock and of electron acceleration within it, requiring answers to: (1) does the forward-shock density compression, measured in the ambient frame, asymptotically reach the magnetohydrodynamic Rankine–Hugoniot predictions (for the source parameters, $9.4\,n_1 \le n_{21} \le 16.0\,n_1$ versus the early-time values $\approx6$–$6.5$), or does the Weibel-mediated three-dimensional structure genuinely depart from the one-dimensional jump conditions at late times? (2) how do the jet and ambient ion populations complete their thermalization, isotropization and mixing downstream of the reverse and forward shocks — including the 2016 expectation that, in longer runs, the significantly decelerated jet ions come to contribute to both shock structures? (3) what is the resulting late-time electron-acceleration picture in the developed double shock: the mechanism, efficiency and energy spectrum in the forward-shock transition region, where the self-generated magnetic fields are weaker than behind the reverse shock; whether and how the early shock-surfing-like (or double-layer-mediated) injection gives way to diffusive shock acceleration at late times; and what becomes of the tenuous high-energy electrons convected back into the jet upstream, the population relevant to precursor emission (radiation from the region ahead of the shock, perturbed by particles escaping into it)? These deliverables are scientifically linked: each is a diagnostic of the same developing shock system observed at successively later times, and together they decide whether the early-time picture of Weibel-mediated GRB external shocks survives into the asymptotic regime that afterglow models assume. The sources' own caveat applies to the interpretation of any answer: at the reduced mass ratio $m_i/m_e=16$ the Weibel-like modes are overweighted relative to oblique modes, ion isotropization is sped up, and the shocks reach their steady state earlier than a realistic-ratio ($m_i/m_e=1836$) system would, so late-time conclusions drawn at reduced mass ratio carry this acknowledged uncertainty.

References

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