Structured Jet Model in Relativistic Outflows

• Structured Jet Model is a framework describing relativistic outflows where energy per solid angle and Lorentz factor vary with angle, improving fits to broadband observations.
• It employs various parameterizations—such as cored power laws, Gaussian profiles, and two-component models—to model the angular dependence and dynamics of jet emissions.
• Integration of forward and reverse shock contributions in differing media enhances predictions of light curves, spectra, and secondary signals like neutrino fluxes.

Searching arXiv for recent and foundational papers on structured jet models across TDEs, GRBs, and related jet geometries. A structured jet model describes a relativistic outflow whose energy per unit solid angle, Lorentz factor, or both vary with angle from the jet axis, in contrast to a uniform or “top-hat” jet with constant properties inside a fixed aperture and an abrupt edge. In the literature represented here, the term encompasses several geometries: angularly structured gamma-ray burst outflows, two-component jets with a narrow core and broader wing, Gaussian and power-law profiles, spine-sheath configurations in active galactic nuclei, and multizone jet models for tidal disruption events. Across these applications, the structured jet model is used to connect viewing angle, jet composition, shock physics, and external-medium interaction to broadband light curves, spectra, image morphology, and population statistics (Pescalli et al., 2014, Duffell et al., 2013, Troja et al., 2018, Yuan et al., 2024, Boughelilba et al., 2023).

1. Conceptual definition and relation to uniform-jet models

In gamma-ray burst work, the basic distinction is between a jet that is uniform within its opening angle and a jet in which “energy and velocity depend on the angular distance from the axis” (Pescalli et al., 2014). The structured alternative is motivated observationally by missing or subdued jet breaks, off-axis afterglows, late achromatic peaks, and cases in which radio, optical, X-ray, GeV, and TeV data cannot be fit with a single sharp-edged outflow (Duffell et al., 2013, O'Connor et al., 2023, Troja et al., 2018).

One important early formulation is the “boosted fireball” model of Duffell & MacFadyen, in which an isotropically expanding fireball in its center-of-momentum frame appears as a beamed outflow in the lab frame. In that construction, the flow is jet-like with $\Gamma \theta_0 \sim 2 \eta_0$, the angular structure is fixed by the two parameters $\eta_0$ and $\gamma_B$, and the afterglow jet break is “greatly subdued” relative to the top-hat case because the edge is smooth rather than sharp (Duffell et al., 2013).

Population arguments also constrain what a structured jet can be. Analytical luminosity-function modeling shows that a structured jet can fit current gamma-ray burst data only if the angular energy distribution is relatively strongly structured, with $E \propto \theta^{-k}$ and $k>4$; the classical $E \propto \theta^{-2}$ structured jet model is excluded by the current data (Pescalli et al., 2014). This indicates that “structured jet model” is not a single universal profile, but a class of models whose admissible forms are limited by afterglow fitting, imaging, and population statistics.

2. Mathematical parameterizations

Most structured jet models specify an angular dependence for the isotropic-equivalent kinetic energy, the energy per solid angle $\epsilon(\theta)=dE/d\Omega$, the initial Lorentz factor $\Gamma_0(\theta)$, or some combination of these quantities. The most frequently used parameterizations in the present literature are summarized below.

Model family	Representative expression	Representative context
Cored power law	$$E_{k,\rm iso}(\theta)\propto \left[1+\left(\frac{\theta}{\theta_c}\right)^2\right]^{-a/2}$$	GRB 221009A afterglow (Gill et al., 2023)
Piecewise power law	$\epsilon(\theta)=\epsilon_c$ for $\eta_0$0 and $\eta_0$1 for $\eta_0$2	Luminosity-function studies (Pescalli et al., 2014)
Gaussian angular structure	$\eta_0$3 and $\eta_0$4	Orphan afterglows and short-GRB populations (Xu et al., 2023, Guo et al., 2020)
Two-component core + wing	$\eta_0$5 for $\eta_0$6 and $\eta_0$7 for $\eta_0$8; similarly for $\eta_0$9	GRB 221009A narrow core plus structured wing (Zhang et al., 2023)

The cored power-law profile is especially common in afterglow modeling. In GRB 221009A, the angular kinetic-energy profile was written as $\gamma_B$0, with the best-fitting afterglow solution requiring a narrow core with $\gamma_B$1 and a shallow angular structure with $\gamma_B$2 in a stellar wind with $\gamma_B$3 (Gill et al., 2023). A closely related power-law structured jet was later inferred for the orphan afterglow AT 2023sva, with $\gamma_B$4, $\gamma_B$5, and $\gamma_B$6 (Srinivasaragavan et al., 6 Jan 2025).

Gaussian profiles are used when a smooth decline away from the core is preferred. In the short-GRB luminosity-distribution work, both luminosity and Lorentz factor were taken as Gaussian functions of angle, $\gamma_B$7 and $\gamma_B$8 (Guo et al., 2020). The same class of profile was used to model AT2021any, where the core Lorentz factor was about $\gamma_B$9, the half-opening angle was $E \propto \theta^{-k}$0, and the viewing angle was $E \propto \theta^{-k}$1 (Xu et al., 2023).

Not all structure is purely angular. In low-luminosity AGNs, a structured jet may be radially divided into a fast inner spine and a slower outer sheath, modeled as two coaxial cylindrical zones with different bulk velocities (Boughelilba et al., 2023). This radial structure introduces inter-zone radiative feedback and shear acceleration, so the term “structured jet model” includes both angular stratification and multi-zone velocity stratification.

3. Dynamics, shocks, and radiative calculation

The dynamical core of most afterglow-oriented structured jet models is a set of angular elements that evolve as local blast waves and whose emission is integrated over equal-arrival-time surfaces. In the GRB 221009A modeling, the jet was treated as an angular grid, with each $E \propto \theta^{-k}$2 evolving independently as a spherical outflow with energy $E \propto \theta^{-k}$3 and Lorentz factor $E \propto \theta^{-k}$4, while the observed emission was obtained by integrating over the equal-arrival-time surface (Gill et al., 2023). This construction makes the observed light curve sensitive to the entire angular profile rather than only to the line-of-sight core.

Forward shock and reverse shock contributions are frequently both required. For GRB 221009A, the forward shock dominates the optical and X-ray flux, while the reverse shock produces the radio afterglow; the radio is reverse-shock dominated at all times, the optical is initially reverse-shock dominated and later forward-shock dominated, and the X-ray is forward-shock dominated throughout (Gill et al., 2023). In the analytical model of the matter-dominated structured wing of the BOAT burst, forward and reverse shock emission are both derived during the deceleration phase, and the reverse shock can provide significant early radio emission whose spectral and temporal behavior is sensitive to the angular profile (Zhang et al., 2023).

The dynamics depend on both the external medium and the angularly weighted energy budget. In a constant-density ISM, the bulk Lorentz factor scales as $E \propto \theta^{-k}$5 (Zhang et al., 2023). For a structured wing, an “effective” isotropic energy within the visible angular region is required rather than the energy in a single ring, and light-curve closure relations depend on the angular structure indices, the electron index $E \propto \theta^{-k}$6, and the external density profile (Zhang et al., 2023).

The AT 2022cmc model extends this framework to a tidal disruption event by introducing multiple zones with different Lorentz factors and continuous energy injection tied to time-dependent accretion before and after the mass fallback time. In that model, the X-ray spectra and light curves can be described by electron synchrotron emission from the reverse shock of the faster jet, while the radio observations can be interpreted as synchrotron emission from the forward shock region of the slower jet; the late-time X-ray upper limits extending to $E \propto \theta^{-k}$7 days after disruption could be interpreted as jet-break steepening (Yuan et al., 2024). This is a structured-jet interpretation in which the structure is resolved into dynamically distinct faster and slower components.

In AGN spine-sheath calculations, the dynamical and radiative coupling is different but conceptually analogous. The CR-ENTREES-based model includes photo-meson production, Bethe-Heitler pair-production, inverse-Compton scattering, $E \propto \theta^{-k}$8-$E \propto \theta^{-k}$9 pair production, decay of unstable particles, synchrotron radiation from electrons, protons, and secondaries, and particle escape, together with feedback between two zones of different bulk velocities (Boughelilba et al., 2023). The main inter-zone acceleration process is shear acceleration, described by a Fokker-Planck equation, while each zone’s radiation field acts as an external target photon field for the other zone’s interactions (Boughelilba et al., 2023). In this setting, the structured jet model is not only geometric; it is a coupled transport problem.

4. Empirical diagnostics and benchmark systems

The most direct empirical support for a structured jet came from GW170817. Troja et al. modeled the broadband afterglow with a Gaussian structured jet seen at an angle of about $k>4$0 from the axis, with jet width $k>4$1, blastwave energy $k>4$2 erg, and ambient density $k>4$3 (Troja et al., 2018). The afterglow showed an initial shallow rise $k>4$4, peaked at about $k>4$5 days, and then declined as $k>4$6, a temporal evolution that challenged most choked-jet or cocoon models and was instead consistent with the emergence of a relativistic structured jet (Troja et al., 2018). Independent VLBI observations constrained the apparent source size to be smaller than $k>4$7 milliarcseconds at the $k>4$8 confidence level, excluding the isotropic outflow scenario and indicating that GW170817 produced a structured relativistic jet; the same rate analysis implied that at least $k>4$9 of neutron star mergers produce such a jet (Ghirlanda et al., 2018).

GRB 221009A became the principal laboratory for structured-jet phenomenology at extreme luminosity. One analysis found that the X-ray brightness decays as a power law with slope $E \propto \theta^{-2}$0, “not consistent with standard predictions for jetted emission,” and attributed this behavior to a shallow energy profile of the relativistic jet (O'Connor et al., 2023). A more detailed afterglow fit required a narrow core, $E \propto \theta^{-2}$1, and a shallow angular structure, $E \propto \theta^{-2}$2, expanding into a stellar wind, with a shallow achromatic break at about $E \propto \theta^{-2}$3 days and $E \propto \theta^{-2}$4 (Gill et al., 2023). A separate interpretation argued for two jet components: a narrow $E \propto \theta^{-2}$5 degree pencil-beam, Poynting-flux-dominated jet and a broader matter-dominated structured jet with angular structure, in which the wing’s forward shock dominates the late afterglow and the wing’s reverse shock explains the early radio light curve (Zhang et al., 2023). Subsequent VHE-oriented studies used Gaussian structured jets to reproduce the GeV–TeV output without the extreme energy requirements of a top-hat jet, both in a uniform-density medium and in a wind-driven medium (Mondal et al., 17 Nov 2025, Mondal et al., 4 Dec 2025).

Orphan afterglows provide a complementary test because the prompt $E \propto \theta^{-2}$6-ray signal may be absent or strongly suppressed. For AT2021any, multi-wavelength fitting favored the structured Gaussian jet over top-hat and isotropic fireball models, with a best-fit Lorentz factor of about $E \propto \theta^{-2}$7, half-opening angle $E \propto \theta^{-2}$8, viewing angle $E \propto \theta^{-2}$9, GRB trigger time about $\epsilon(\theta)=dE/d\Omega$0 s before the first detection, and an upper limit of $\epsilon(\theta)=dE/d\Omega$1 on the radiative efficiency (Xu et al., 2023). For AT 2023sva, Bayesian modeling favored a shallow power-law structured jet viewed slightly off-axis, just outside the core opening angle, which was identified as the likely reason for the lack of a detected GRB counterpart (Srinivasaragavan et al., 6 Jan 2025). These systems show that the structured jet model is now used not only to explain bright on-axis events but also to interpret optically discovered afterglows with no prompt high-energy trigger.

AT 2022cmc broadens the empirical domain beyond GRBs. There the structured interpretation is a multizone TDE jet with different Lorentz factors, distinct reverse- and forward-shock origins for X-ray and radio emission, and continuous energy injection driven by accretion history (Yuan et al., 2024). This suggests that the same modeling vocabulary can be transferred from collapsar and merger outflows to jetted tidal disruption events when the data require spectrally and temporally distinct emission zones.

5. Physical origins and variant structured outflows

Structured jets need not be imposed purely phenomenologically; several works derive them from jet propagation. Geng et al. performed relativistic magnetohydrodynamics simulations of jets propagating through neutron-star-merger ejecta and found that “regardless of the jet launching delay time, a structured jet with an angle-dependent luminosity and Lorentz factor is always formed after the jet breaks out the ejecta” (1904.02326). The launching delay $\epsilon(\theta)=dE/d\Omega$2 affects the nature of the wings: for relatively short delays, the structured wing is dominated by jet material, whereas for relatively long delays, such as $\epsilon(\theta)=dE/d\Omega$3 s, the line-of-sight material has a dominant contribution from the cocoon (1904.02326). In this sense, the structured jet may be an outcome of breakout and cocoon interaction rather than a direct imprint of the central engine alone.

A different route to structure is precession. Huang et al. argued that for a precessing jet the structure relevant to the prompt-emission phase and the afterglow phase may differ, and that a narrow uniform jet in the prompt phase can produce a non-uniform afterglow-phase structure (Huang et al., 2019). Depending on the precession angle, the effective afterglow structure can resemble a narrow uniform core with power-law wings and sharp cut-off edges, a Gaussian profile, a ring shape, or more complex forms (Huang et al., 2019). They further found that intrinsic kinetic energy, electron index, and jet opening angle inferred from afterglow fitting may be incorrect if the precession-generated structure is ignored (Huang et al., 2019).

The surrounding medium can also modify the emerging structure. A semi-analytical model for a non-magnetized relativistic jet propagating in a static magnetized medium with a tangled field showed that the jet and cocoon properties may be affected by high magnetic fields, $\epsilon(\theta)=dE/d\Omega$4 G, and by a mixing parameter describing how much of the ambient field is entrained into the cocoon (Garcia-Garcia et al., 2024). Relative to the non-magnetized case, the evolution may vary up to $\epsilon(\theta)=dE/d\Omega$5: low mixing may produce a slower-broader jet with a broader and more energetic cocoon, whereas high mixing could produce a faster-narrower jet with a narrow and less-energetic cocoon (Garcia-Garcia et al., 2024).

Prompt-emission structure can be even more elaborate. In the 2026 shear-acceleration study, the structured jet consists of an ultra-relativistic uniform core, a structured cocoon, a thin jet-cocoon interaction layer, and a mixed jet-cocoon region. In the weak-scattering regime, thermalized electrons in the mixed jet-cocoon region produce a quasi-thermal component, while in the strong-scattering regime shear-accelerated electrons produce broader non-thermal spectra (Wang et al., 24 May 2026). This use of the structured jet model links macroscopic geometry to prompt spectral diversity.

6. Interpretive tensions, selection effects, and observational prospects

Several persistent debates define the current status of the structured jet model. One is the relation between structured jets and top-hat jets. Some events can be fitted by both, but the structured model is often preferred because it yields smoother jet breaks, more natural off-axis behavior, or lower energy requirements (Duffell et al., 2013, Xu et al., 2023). Another is the role of cocoons. For GW170817, broadband light curves alone were not sufficient to discriminate, whereas high-resolution imaging and source compactness were decisive; the VLBI result that the source remained smaller than $\epsilon(\theta)=dE/d\Omega$6 milliarcseconds excluded the isotropic outflow scenario (Ghirlanda et al., 2018). A further tension concerns universal angular profiles: the luminosity-function analysis disfavors the classical $\epsilon(\theta)=dE/d\Omega$7 model and instead requires a much steeper structure (Pescalli et al., 2014), while some individual bursts such as GRB 221009A and AT 2023sva favor shallow angular wings (Gill et al., 2023, Srinivasaragavan et al., 6 Jan 2025). A plausible implication is that no single analytic profile is likely to describe all jet-producing systems.

Selection effects are central to population inference. Under a structured Gaussian jet scenario for short GRBs, nearby events within luminosity distance less than $\epsilon(\theta)=dE/d\Omega$8 Mpc typically have gamma-ray luminosities around $\epsilon(\theta)=dE/d\Omega$9–$\Gamma_0(\theta)$0 erg s$\Gamma_0(\theta)$1, naturally explaining the low luminosity of GRB 170817A, while the expected detection rate by Fermi-GBM is only about $\Gamma_0(\theta)$2 yr$\Gamma_0(\theta)$3 within several hundred Mpc (Guo et al., 2020). The same work found that the merger-time power-law index $\Gamma_0(\theta)$4 has little effect on the observed luminosity and viewing-angle distributions but strongly affects the number-redshift distribution, and that $\Gamma_0(\theta)$5 may be identified when the number of observed short GRBs with known redshifts is more than $\Gamma_0(\theta)$6 (Guo et al., 2020). For orphan afterglows, AT 2023sva explicitly motivated “broadening orphan afterglow search strategies to a diverse range of GRB jet angular energy profiles” (Srinivasaragavan et al., 6 Jan 2025).

The multi-messenger outlook is correspondingly geometry dependent. In the Gaussian structured-jet modeling of GRB 221009A, the predicted neutrino flux inferred from multi-wavelength data lies below the sensitivities of IceCube Gen2 and GRAND200k, and even highly optimistic microphysical conditions yield an expected number of events of order $\Gamma_0(\theta)$7 for upcoming GRAND200k; jet orientation alone can introduce nearly an order-of-magnitude variation in the signal (Mondal et al., 17 Nov 2025). In a wind-driven medium, only about ten per cent of simulated TeV events exceed CTA sensitivity, with detectability favored by near core-aligned views, high kinetic energy and wind density, moderate initial Lorentz factor and downstream magnetic field, and a relatively large fraction of energy in nonthermal electrons (Mondal et al., 4 Dec 2025). For merger events, future GW/GRB associations can help differentiate cocoon-dominated and jet-dominated wing scenarios (1904.02326). For AT 2022cmc, current and future observations were explicitly identified as a means of testing the multizone structured interpretation (Yuan et al., 2024).

Taken together, these results define the structured jet model as a broad but technically specific framework: an angularly or radially stratified relativistic outflow whose observable properties depend on how energy, Lorentz factor, composition, and radiative coupling are distributed across the flow. Its current form is not a single canonical profile, but a hierarchy of models constrained by afterglow fitting, VLBI imaging, luminosity functions, orphan searches, and increasingly by VHE and neutrino data (Pescalli et al., 2014, Ghirlanda et al., 2018, Mondal et al., 17 Nov 2025).