Elemental Spectra of Cosmic Rays: Two Galactic Sources and the Extragalactic Transition Beyond the Knee ✦
The energy spectra of individual elements in cosmic rays in the range from 100×Z GeV to 1000×Z PeV exhibit two 'knee' breaks with rigidity dependence: at energies of about 15×Z TeV and 3×Z PeV, as well as three 'ankle' flattenings. The first ankle (≈500×Z GeV) also depends on rigidity, but the second (≈150 TeV for protons, ~1 PeV for helium) and, probably, the third (≈100 PeV for protons) do not. Thanks to the high measurement precision in the DAMPE and LHAASO experiments, as well as IceTop, it was possible for the first time to establish that all spectral features are well described by the sum of three components: a low-energy galactic component (with a convex spectral shape, fading after the TeV knee), a high-energy galactic component (including the PeV knee), and an extragalactic component. No additional galactic component is required in the 10 PeV – 1 EeV interval.
Context
The nature of the knee in the spectrum of galactic cosmic rays at ~3 PeV remains one of the central puzzles of particle astrophysics. Historically, it was thought it could be linked to the maximum energy of accelerators, propagation effects, or photonuclear processes. Already in the 1990s, hints emerged that the spectrum is shaped by multiple source populations. The classic idea of Fritz Zwicky (1934) that supernova remnants are the main accelerators remains dominant, but the precise configuration of sources and their contributions across different energy ranges were unclear until the advent of next-generation precision measurements.
Methods
The study used spectral data on protons, helium, and iron from the DAMPE (direct satellite measurements), LHAASO, and IceTop (ground-based extensive air shower arrays) experiments. High-resolution spectroscopy enabled the isolation of individual elements. The fit function was a sum of three components (two galactic and one extragalactic). To describe the spectrum of a single source, a new phenomenological form—'double log-parabola with exponential cutoff' (EDLP)—was proposed, which reproduces the observed flattening before the cutoff. The source population was modeled by convolving the single-source spectrum with a luminosity function, after which parameters were fitted to data using MINUIT (MIGRAD) and MCMC (emcee) methods. Fragmentation losses for heavy nuclei were additionally accounted for using a model dependence of the path length on rigidity.
Results
It is shown that the spectra of hydrogen, helium, and iron in the 40 GeV – 500 PeV range are satisfactorily described by the three-component model. The first galactic component has a convex shape with a low-energy spectral index of 2.76 (the same for all nuclei) and ends after the TeV knee at a rigidity of ~15×Z TeV. The second component dominates up to the PeV knee (3×Z PeV) and is responsible for the main break. The third, extragalactic, component appears at ~100 PeV for protons and becomes dominant above 1 EeV. It was found that the so-called 'ankles'—breaks toward flattening at 500×Z GeV and ~150 TeV (protons) or ~1 PeV (helium)—result from the intersection of different components and do not always obey a rigidity dependence due to differences in the chemical composition of the populations. The formal fit of the helium spectrum requires a high-energy cutoff of the first component at 700 TeV, but this detail is not mandatory when accounting for systematic uncertainties. Importantly, no additional third galactic population is needed to describe data in the 10 PeV – 1 EeV region.
Implications
The proposed interpretation means that the transition from galactic to extragalactic cosmic rays occurs 'seamlessly' just beyond the PeV knee, rather than at the traditionally considered 'ankle' in the all-particle spectrum at 4 EeV. This supports the 'disappointing model' for ultra-high energies and underscores that the Galaxy ceases to be an efficient proton accelerator already at energies of ~100 PeV. The convex shape of the first component may indicate nonlinear effects in acceleration processes (diffusive shock acceleration) or a contribution from local sources in the late stages of supernova evolution.
Future development
Future precision measurements of the spectra of CNO group nuclei (carbon, oxygen) and iron in the knee region by LHAASO and the planned SWGO will allow independent testing of the three-component model predictions and the rigidity dependence of breaks. Joint analysis of anisotropy and spectra in the 0.1–10 PeV range, especially accounting for the dipole phase, will help isolate the contribution of nearby sources. Numerical simulations of acceleration in supernova remnants with realistic nonlinear feedback and variable ambient composition will be an important step toward a theoretical foundation of the observed picture.
Impact
The results will impact cosmic-ray physics, high-energy gamma-ray astronomy (search for galactic PeVatrons), and the interpretation of neutrino telescope data, as the sharp transition to extragalactic protons alters expectations for the diffuse neutrino background.
Next steps
Verification of the predicted dependence of the 'running' transition on nuclear charge using new data on CNO and iron from LHAASO; refinement of fragmentation cross sections of heavy nuclei in the TeV region; modeling of secondary nuclei spectra (boron, beryllium) within the three-component model.
Key open problems
The proposed model directly links the origin of the knee to the exhaustion of galactic accelerator capabilities and the immediate appearance of extragalactic sources, necessitating a revision of traditional ideas about the 'galactic' range up to 1 EeV. Moreover, the convex shape of the first component and its possible connection to nonlinear acceleration at shock fronts in supernova remnants remains a subject of active debate, linking the knee problem to the intricacies of plasma physics.
🎯 A proton with an energy of 1 EeV in a typical galactic magnetic field (3 μG) has a Larmor radius of about 300 parsecs—almost half the disk thickness, so the Galaxy cannot confine such particles, and they almost certainly come from beyond its boundaries.
Key numbers
- TeV knee energy for protons: 15 TeV
- PeV knee energy for protons: 3 PeV
- first break (ankle) energy for protons: 500 GeV
- second break energy for protons: 150 TeV
- low-energy spectral index of the first component: 2.76