A numerical model of the Crab Nebula predicts a sharply different gamma-ray size at 1.1 PeV: its 68% flux-containment radius is about 1.43 arcminutes when hadronic emission is included, compared with about 0.25 arcminutes in a pure leptonic calculation. The hybrid prediction is approximately six times larger.
That contrast is the main result because the study asks whether the nebula's changing gamma-ray size can help distinguish hadronic from leptonic emission at ultra-high energies. The calculation evaluates a spatially resolved lepto-hadronic model against pure leptonic and pure hadronic alternatives. In ordinary terms, the hybrid setup combines electron and proton processes in the same picture.
A model built around the nebula's changing size
The researchers used compiled observations across multiple wavelengths and published measurements of the Crab Nebula's spatial extension as checks on the model. They solved coupled numerical transport equations for electrons, protons and turbulent waves with a finite-difference Crank-Nicolson method. The aim was to follow both the predicted spectrum and the way the source's apparent size changes with energy.
The hybrid calculation included synchrotron radiation, inverse-Compton scattering and emission from proton-proton interactions. It also calculated the source's spatial profile, so the model could be compared with both the amount of gamma-ray light and its apparent extent. The size measure was a 68% flux-containment radius, meaning the radius within which 68% of the modelled gamma-ray flux falls.
Leptonic parameters were taken from an earlier reference. Hadronic parameters were fitted to the TeV-PeV spectrum and spatial extension, while radial indices were adjusted to reproduce the observed energy-dependent extension. The analysis then compared the hybrid result with separate pure leptonic and pure hadronic calculations of that containment radius.
The proton contribution grows with energy
The two pure-component calculations missed the observations in opposite ways. The pure hadronic version predicted a radius larger than observed, while the pure leptonic version fell below the observed extension data at TeV energies. The hybrid version, by contrast, was reported to reproduce the Crab Nebula's multiwavelength spectrum and apparent extension at the same time.
The model's balance changed as the gamma-ray energy rose. Its hadronic neutral-pion-decay contribution was approximately 15% at 1 TeV, approximately 50% at approximately 400 TeV, and approached 100% at PeV energies. Within the calculation, the PeV emission therefore approached near-total hadronic dominance.
The model also allocated about 25% of the pulsar spin-down luminosity to proton components. That allocation sits alongside the fitted hadronic parameters, so the reported PeV picture is a property of the model configuration used to match the spectrum and extension.
A diagnostic tied to the model
The large PeV gap is a model prediction. The hadronic parameters and radial indices were fitted to the spectrum and extension, so the proposed diagnostic depends on that model setup.
That leaves a clear target for future observations. The study proposes comparing energy-dependent gamma-ray extension measurements with the hybrid and pure-component predictions, especially near 1.1 PeV, where the calculated radii differ by about six times. The document is labeled arXiv:2608.25497v1 and dated 26 Aug 2026.
Paper data and sources
Original title: Energy-dependent extension of the Crab Nebula as a diagnostic of ultrahigh-energy hadronic emission
Authors: Fang-Wu Lu, Bo-Tao Zhu, Ji-Yang Ren, Li Zhang
Journal/Repository: Phys. Rev. D 114, 043066 (2026)
Status: Peer-reviewed
First online: 2026-08-26
DOI: 10.1103/dnsv-plc9
Original paper · Full text