Generation of Large-scale Magnetic Fields Upstream of Gamma-Ray Burst Afterglow Shocks

Abstract The origins of the magnetic fields that power gamma-ray burst (GRB) afterglow emission are not fully understood. One possible channel for generating these fields involves the pre-conditioning of the circumburst medium: in the early afterglow phase, prompt photons streaming ahead of the GRB external shock can pair produce, seeding the upstream with drifting electron–positron pairs and triggering electromagnetic microinstabilities. To study this process, we employ 2D periodic particle-in-cell simulations in which a cold electron–proton plasma is gradually enriched with warm electron–positron pairs injected at mildly relativistic speeds. We find that continuous pair injection drives the growth of large-scale magnetic fields via filamentation-like instabilities; the temporal evolution of the field is self-similar and depends on a single parameter, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">pi</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> <mml:msub> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">pi</mml:mi> </mml:mrow> </mml:msub> </mml:math> , where α is the ratio of final pair beam density to background plasma density, t f is the duration of pair injection, and ω pi is the plasma frequency of background protons. Extrapolating our results to parameter regimes realistic for long GRBs, we find that upstream pair enrichment generates weak magnetic fields on scales much larger than the proton skin depth; for bright bursts, the extrapolated coherence scale at a shock radius of R ∼ 10 17 cm is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close="〉" open="〈"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mspace width="0.25em"/> <mml:mo>∼</mml:mo> <mml:mn>100</mml:mn> <mml:mspace width="0.33em"/> <mml:mi>c</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">pi</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and the corresponding magnetization is σ ∼ 10 −8 for typical circumburst parameters. These results may help explain the persistence of magnetic fields at large distances behind GRB shocks.