The all-particle cosmic-ray energy spectrum steepens at the knee (about 3 PeV), and why it does so is still unsettled. Bijay and Bhadra (2016) proposed that the knee is not a rigidity cutoff of individual sources but a population effect: Galactic cosmic rays up to the ankle are produced by black-hole-forming gravitational explosions whose maximum attainable energy EmaxE_{max} varies from source to source because the released energy scales with the zero-age mass of the progenitor star, so that the Salpeter initial mass function directly imprints a steepening of about 0.35 in spectral index above the minimum EmaxE_{max}, identified with the knee. The model, however, was left schematic and untested: the quantitative mapping from progenitor mass through collapse and jet physics to EmaxE_{max} and injected composition was never built, the source class was never identified, and its distinguishing prediction — a nearly unaltered or only slightly heavier composition across the knee — has never been quantitatively confronted with the knee-region composition measurements that have since appeared. The open problem is to decide whether the progenitor-mass mechanism is a viable origin of the knee or is excluded in favor of rigidity-ordered scenarios, completing the population model quantitatively where that is needed and confronting its predictions with modern spectral and mass-composition data.

Background

Cosmic rays are charged nuclei arriving at Earth with a power-law energy spectrum that spans from below a GeV to beyond 102010^{20} eV. Two spectral features frame the problem. At the knee, around 3 PeV, the spectrum steepens from index about −2.7-2.7 to about −3.1-3.1; at the ankle, around 3 EeV, it flattens again, usually interpreted as the emergence of an extragalactic component. Extensive air showers (EAS) — cascades initiated when a primary particle strikes the atmosphere — are the only observation channel above direct-detection energies: experiments infer the primary energy from the shower particles, Cherenkov or fluorescence light, or radio emission, and infer the primary mass either from XmaxX_{max}, the atmospheric depth where the cascade is largest (lighter nuclei penetrate deeper), or from the muon content, summarised as the mean logarithmic mass ⟨ln⁡A⟩\langle\ln A\rangle. Both mass proxies depend on hadronic interaction models that remain uncertain at PeV energies and beyond.

In the standard picture, Galactic cosmic rays are accelerated by diffusive shock acceleration in supernova remnants (SNR), the Fermi-type process in which particles repeatedly cross a shock front and gain energy. The maximum energy a source can reach is limited by confinement, roughly Emax∼ZeBβsRsE_{max}\sim ZeB\beta_s R_s for charge ZeZe, magnetic field BB, flow speed βsc\beta_s c, and size RsR_s (the Hillas condition). A widespread family of knee explanations is rigidity-dependent: each nuclear species cuts off (or escapes the Galaxy) at a rigidity-dependent energy — rigidity being momentum per unit charge, so the cutoff scales as E∝ZE\propto Z for relativistic nuclei — meaning protons steepen first at the knee, the composition grows heavier with energy, and an iron knee appears near Z×Z\times (proton knee) ∼80\sim 80 PeV — the pattern reported by the KASCADE-Grande air-shower experiment.

Bijay and Bhadra (2016) proposed instead a source-population origin. They conjectured that (i) the dominant Galactic sources up to the ankle are gravitational explosions or accretion events that form black holes, and (ii) the maximum energy varies across sources because the released energy tracks the zero-age (i.e., birth) mass of the progenitor star: black holes, unlike neutron stars or white dwarfs, have no mass ceiling, so collapse energy grows with progenitor mass. For a relativistic shock with Lorentz factor Γs\Gamma_s, size RsR_s, speed βsc\beta_s c, and field BB, they adopt

Emax=ZeB ΓsβsRs,E_{max}=ZeB\,\Gamma_s\beta_s R_s,

and, taking the explosion energy and hence Γs\Gamma_s to scale with progenitor mass, the Salpeter initial mass function (dn/dM∝M−2.35dn/dM\propto M^{-2.35}, the empirical distribution of stellar birth masses) implies a power-law distribution of EmaxE_{max} across sources. Above the minimum EmaxE_{max}, identified with the knee, only the rarer, more massive progenitors still contribute, steepening the all-particle spectrum by α−2≈0.35\alpha-2\approx 0.35 — close to what is observed. The energetic budget works for a black-hole-formation fraction ρBH≈0.5\rho_{BH}\approx 0.5 above about 20 M⊙20\,M_\odot and an acceleration efficiency ζ∼0.01\zeta\sim 0.01–0.10.1. The model's sharpest discrimination from rigidity-type knee scenarios is compositional: since the knee is set by the population of source EmaxE_{max} values rather than by species-dependent cutoffs, the composition should stay almost unaltered across the knee or become only slightly heavier — unlike the rigidity picture's pronounced transition to heavy nuclei between the knee and an iron knee near 80 PeV.

What the 2016 paper explicitly left undone — and what later work has and has not delivered — defines the problem. The mapping from progenitor mass through collapse, fallback accretion (explosion ejecta falling back onto the newborn black hole), and jet formation (the collapsars picture of black-hole-forming stellar collapse) to the actual distribution of EmaxE_{max} and to the injected nuclear composition was asserted through linear scalings, never modeled. No source class (type Ib/Ic supernovae, hypernovae (exceptionally energetic stellar explosions), gamma-ray bursts, or active galactic nuclei) was identified as the dominant population. Meanwhile the data side has advanced sharply: LHAASO-KM2A measured the spectrum and ⟨ln⁡A⟩\langle\ln A\rangle from 0.3 to 30 PeV and concluded the all-particle knee is the light-component knee with ⟨ln⁡A⟩\langle\ln A\rangle rising as a power law above it; TALE measured XmaxX_{max} from 1016.510^{16.5} to 1018.510^{18.5} eV and found composition growing heavier to about 101710^{17} eV and then lightening toward the second knee; KASCADE-Grande's post-LHC reanalyses show composition already getting heavier through the break with a slope step near 0.5. Yet composition inference in this energy range is still hadronic-model dependent — simulations show a persistent muon deficit relative to data (the muon puzzle) — and different experiments' composition results have not converged, a state of affairs documented by Falalaki and Pavlidou (2026), who place Bijay and Bhadra's model within the broader, still-unsettled question of what the knees reveal about cosmic-ray source populations.

Problem Statement

Determine whether the steepening of the cosmic-ray spectrum at the knee originates from the mass distribution of the progenitors of black-hole-forming cosmic-ray sources, by completing and decisively testing the progenitor population model of Bijay and Bhadra (2016). The overarching objective — establishing or excluding this mechanism as the origin of the knee — involves two lines of inquiry: (1) confront the model's distinguishing composition prediction — approximately unaltered or only mildly heavier across the knee — with the measured spectral and composition data (for example KASCADE-Grande, LHAASO-KM2A, TALE, and radio-based XmaxX_{max} measurements), with a statistical assessment of whether the progenitor-mass mechanism can be discriminated from or is excluded by rigidity-ordered knee scenarios, taking into account the hadronic-model dependence of air-shower mass inference; and (2) identify or constrain, within the model's energetic budget and EmaxE_{max} distribution, which source class (type Ib/Ic supernovae, hypernovae (exceptionally energetic stellar explosions), gamma-ray bursts, active galactic nuclei, or other black-hole-forming events) could dominate Galactic cosmic-ray production up to the ankle. Make the model quantitatively testable by building the quantitative mapping from progenitor zero-age mass through stellar collapse, fallback accretion (explosion ejecta falling back onto the newborn black hole), and jet or shock dynamics to each source's maximum energy (based on Emax=ZeB ΓsβsRsE_{max}=ZeB\,\Gamma_s\beta_s R_s with explicit, physically derived scalings for the explosion energy, the shock Lorentz factor Γs\Gamma_s, magnetic-field amplification, and shock size, replacing the paper's assumed linear scalings) and to the composition of the injected nuclei; and by performing the associated population synthesis — convolving the stellar initial mass function with the black-hole-formation fraction (including its metallicity dependence), the explosion rate, and the acceleration efficiency — demonstrating consistency with the Galactic cosmic-ray luminosity, reproducing the observed spectral steepening of about 0.35 above the knee, and predicting the mean logarithmic mass ⟨ln⁡A⟩\langle\ln A\rangle as a function of energy from the knee through the iron-knee region to the ankle. The mechanism may also be shown to be excluded if the model's assumptions fail, with the analysis identifying which of them fail. Line of inquiry (2) presupposes that the mechanism holds.

References

  1. Biplab Bijay and Arunava Bhadra, 'Progenitor model of cosmic ray knee', Research in Astronomy and Astrophysics 16, 006 (2016), DOI 10.1088/1674-4527/16/1/006, arXiv:1412.0818, https://arxiv.org/abs/1412.0818
  2. Biplab Bijay, Prabir Banik, and Arunava Bhadra, 'The knee in the cosmic ray energy spectrum from the simultaneous EAS charged particles and muon density spectra', Astrophysics and Space Science 361, 311 (2016), DOI 10.1007/s10509-016-2897-7, arXiv:1511.05739, https://arxiv.org/abs/1511.05739
  3. Prabir Banik, Biplab Bijay, Samir K. Sarkar, and Arunava Bhadra, 'Probing the cosmic ray mass composition in the knee region through TeV secondary particle fluxes from solar surroundings', Physical Review D 95, 063014 (2017), DOI 10.1103/PhysRevD.95.063014, arXiv:1703.05083, https://arxiv.org/abs/1703.05083
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  10. Myrto Falalaki and Vasiliki Pavlidou, 'What can cosmic-ray knees reveal about source populations?', Astronomy & Astrophysics 706, A101 (2026), DOI 10.1051/0004-6361/202450483, arXiv:2404.15097, https://arxiv.org/abs/2404.15097