Gamma-ray burst (GRB) afterglow polarimetry is a promising probe of the axial symmetry and angular structure of relativistic jets, but current predictions rest on idealized toy models. Baxter and Kobayashi (2025) showed, for jets with elliptical (non-axisymmetric) heads, that the polarization degree curve shows two peaks (top-hat jets) or a single peak (structured jets), an incomplete polarization drop between peaks, and polarization-position-angle rotations that deviate from the 90 degrees expected for axisymmetric jets — yet their calculation assumed a circular visible region of radius Γ−1\Gamma^{-1}, uniform ISM (interstellar medium) dynamics, and tangled shock-generated magnetic fields. This problem asks for the missing quantitative theory and observational framework: compute the polarization of realistic non-axisymmetric structured jets including the deformation of the visible region by lateral angular Lorentz-factor gradients (in both uniform and wind-stratified media), determine which symmetry diagnostics survive, distinguish the two candidate magnetic-field amplification mechanisms through their wavelength- and time-dependent polarization signatures, and disentangle geometric forward-shock polarization from refreshed-shock or central-engine contamination. The literature through August 2026 leaves this core open: the deformed-visible-region calculation has not been published, the shock magnetic-field amplification mechanism remains unsettled, and existing polarization models fail to reproduce recent multi-band observations such as those of GRB 221009A.

Background

Gamma-ray bursts (GRBs) are produced by ultra-relativistic, collimated jets launched by compact engines. As a jet sweeps up the surrounding medium, its forward shock accelerates electrons that radiate synchrotron afterglow emission across the electromagnetic spectrum. Relativistic beaming confines the observable emission at early times to a small patch of the shock front, the visible region, of angular size roughly Γ−1\Gamma^{-1} around the point where the observer's line of sight meets the shock, where Γ\Gamma is the bulk Lorentz factor. Synchrotron radiation is linearly polarized, and although the shock-generated magnetic field is tangled, individual fluid elements emit polarized light; the net polarization observed is set by the degree to which the visible region is asymmetric about the line of sight, since a symmetric emitting pattern cancels.

A top-hat jet is one with uniform energy per unit solid angle inside a sharp edge and zero outside; a structured jet has energy and Lorentz factor that vary smoothly with angle from the jet axis (e.g., Gaussian or power-law profiles). As the jet decelerates, Γ\Gamma falls and the visible region grows until the jet edge becomes visible, producing an achromatic steepening of the light curve called the jet break. The classical predictions for an off-axis observer viewing a top-hat jet are two peaks in the polarization degree (PD) straddling the jet-break time, with a 90-degree rotation of the polarization position angle (PPA) between them (Ghisellini and Lazzati 1999; Sari 1999). For axisymmetric structured jets the PD curve instead typically shows a single peak with an approximately constant PPA (Rossi et al. 2002). The statistical polarization properties also depend on how the shock magnetic field is amplified. In the standard picture, kinetic plasma instabilities of the Weibel type amplify fields with a coherence length comparable to the plasma skin depth, the intrinsic microphysical length scale of a plasma (Medvedev and Loeb 1999), producing tangled microscopic fields; more recently, magnetohydrodynamic (MHD) turbulence behind the shock has been proposed to generate fields coherent over the hydrodynamic thickness of the blast wave (Kuwata et al. 2022), which predicts qualitatively different behavior, such as radio polarization exceeding optical polarization (Kuwata et al. 2024).

Baxter and Kobayashi (2025) revisited these questions for non-axisymmetric jets. Motivated by three-dimensional hydrodynamic simulations of jets from neutron-star mergers showing that the jet head is generally elliptical rather than circular, they adopted a toy model in which the jet head is an ellipse of eccentricity ee, lateral expansion is neglected, the dynamics follow Γ∝t−3/8\Gamma\propto t^{-3/8} for a uniform interstellar medium (ISM), and the visible region is a thin ring of radius Γ−1\Gamma^{-1} with emission polarized radially. They found that for such non-axisymmetric jets the PD curve still shows two peaks for top-hat jets or one peak for structured jets, but the PD does not drop to zero between the peaks and the PPA rotation between peaks can deviate substantially from 90 degrees; in single-peak cases the PPA evolves gradually rather than staying constant. These signatures — the peak-to-trough PD ratio and the PPA rotation — were proposed as diagnostics of jet-head eccentricity and applied to GRB 121024A, GRB 091018, GRB 020813, and GRB 210610B. However, the authors explicitly flag the limitations that define the present problem: the circular-ring visible region ignores the deformation produced by lateral angular Lorentz-factor gradients in realistic structured jets (toward the jet core, higher Γ\Gamma and higher luminosity bend the beaming pattern toward the jet center); the dynamics are idealized top-hat ISM evolution, with wind media and smooth angular gradients untreated; the detectability of the two separate geometric breaks expected for an elliptical jet (with break-time ratio of order 0.55 for e=0.6e=0.6, i.e., blending into one smooth transition) is unquantified; the magnetic-field amplification mechanism is unsettled and controls the wavelength dependence of the PD; and early-time polarization can be contaminated by refreshed-shock emission (produced when slower shells launched later by the central engine catch up with and re-energize the decelerating forward shock) or by central-engine-related emission carrying large-scale magnetic fields, which must be disentangled from forward-shock geometric polarization. Related but distinct threads exist: polarization curves for axisymmetric structured jets (Rossi et al. 2002; Birenbaum et al. 2024, 2025), polarization from non-axisymmetric jets built from discrete azimuthal sectors of different Lorentz factor and energy (Li et al. 2024), numerical-simulation studies showing that lateral expansion qualitatively changes off-axis polarization (Medina Covarrubias et al. 2023), and stratified-medium polarization including the equal-arrival-time-surface (EATS) effect — the fact that photons arriving simultaneously were emitted from different radii and times (Lan et al. 2023; Birenbaum et al. 2026). None of these combines smooth non-axisymmetric structure, deformed visible regions, and realistic dynamics and media.

Problem Statement

The problem, as posed by Baxter and Kobayashi (2025) through their acknowledged model limitations, is to convert afterglow polarimetry into a robust, quantitative probe of the axial symmetry and angular structure of gamma-ray burst jets. Its common objective is a validated mapping between measured polarization observables and jet geometry, and it requires, as jointly supporting deliverables: (1) a self-consistent calculation of the time-dependent linear polarization (Stokes parameters, polarization degree, and polarization position angle) of the forward-shock afterglow of realistic non-axisymmetric structured jets, in which lateral angular gradients of the Lorentz factor and energy per solid angle deform the visible region away from the circular ring of radius Γ−1\Gamma^{-1} assumed in the toy model, for both uniform (ISM) and wind-stratified circumburst media; (2) a determination of which diagnostic signatures of non-axisymmetry — the incomplete drop of polarization between the two peaks, the deviation of the position-angle rotation from 90 degrees, the single-peak curves with gradually evolving position angle, and the presence of two separate geometric jet breaks — survive this more realistic treatment, and with what dependence on jet-head eccentricity, viewing offset, and azimuthal viewing angle; (3) wavelength- and time-dependent polarization predictions that distinguish the two candidate magnetic-field amplification mechanisms in relativistic shocks, namely kinetic (Weibel-type) instabilities producing fields coherent only over the plasma skin depth versus magnetohydrodynamic turbulence producing fields coherent over the hydrodynamic thickness of the blast wave (including whether the radio polarization degree can exceed the optical one); and (4) a quantitative framework for disentangling forward-shock geometric polarization from early-time contamination by refreshed-shock or central-engine-related emission carrying large-scale magnetic fields, using multi-epoch, multi-wavelength polarimetry. A complete answer establishes whether, and under what observing conditions, afterglow polarimetry can constrain the axial symmetry and structure of GRB jets.

References

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