Constraining the magnetic field structure in collisionless relativistic shocks with a radio afterglow polarization upper limit in GW 170817

ABSTRACT Gamma-ray burst (GRB) afterglow arises from a relativistic shock driven into the ambient medium, which generates tangled magnetic fields and accelerates relativistic electrons that radiate the observed synchrotron emission. In relativistic collisionless shocks the post-shock magnetic field $\boldsymbol {B}$ is produced by the two-stream and/or Weibel instabilities on plasma skin-depth scales (c/ωp), and is oriented predominantly within the shock plane (B⊥; transverse to the shock normal, $\hat{\boldsymbol {n}}_{\rm {sh}}$), and is often approximated to be completely within it ($B_\parallel \equiv \hat{\boldsymbol {n}}_{\rm {sh}}\, \cdot \, \boldsymbol {B}=0$). Current 2D/3D particle-in-cell simulations are limited to short time-scales and box sizes ≲104(c/ωp) ≪ R/Γsh much smaller than the shocked region’s comoving width, and hence cannot probe the asymptotic downstream $\boldsymbol {B}$ structure. We constrain the latter using the linear polarization upper limit, $\vert \Pi \vert \lt 12{{\ \rm per\ cent}}$, on the radio afterglow of GW $170817$ / GRB 170817A. Afterglow polarization depends on the jet’s angular structure, our viewing angle, and the $\boldsymbol {B}$ structure. In GW $170817$ / GRB 170817A the latter can be tightly constrained since the former two are well-constrained by its exquisite observations. We model $\boldsymbol {B}$ as an isotropic field in 3D that is stretched along $\hat{\boldsymbol {n}}_{\rm {sh}}$ by a factor ξ ≡ $B_\parallel $/B⊥, whose initial value ξf ≡ $B_\parallel,$f/B⊥, f describes the field that survives downstream on plasma scales ≪R/Γsh. We calculate Π(ξf) by integrating over the entire shocked volume for a Gaussian or power-law core-dominated structured jet, with a local Blandford-McKee self-similar radial profile (used for evolving ξ downstream). We find that independent of the exact jet structure, $\boldsymbol {B}$ has a finite, but initially sub-dominant, parallel component: 0.57 ≲ ξf ≲ 0.89, making it less anisotropic. While this motivates numerical studies of the asymptotic $\boldsymbol {B}$ structure in relativistic collisionless shocks, it may be consistent with turbulence amplified magnetic field.