pith. machine review for the scientific record. sign in

arxiv: 2604.07977 · v1 · submitted 2026-04-09 · 🌌 astro-ph.HE · hep-ph

Recognition: unknown

Oblique Shocks at Supernova Remnants in Massive Star Clusters: A Model for the Cosmic-Ray Knee Observed by LHAASO

Luana N. Padilha , Rita C. Anjos
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:03 UTC · model grok-4.3

classification 🌌 astro-ph.HE hep-ph
keywords cosmic rayscosmic ray kneeoblique shocksmassive star clusterssupernova remnantsparticle accelerationLHAASOrigidity dependence
0
0 comments X

The pith

Oblique shocks in massive star clusters accelerate cosmic rays to multi-PeV energies, explaining the knee as a sequence of rigidity-dependent cutoffs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a model in which oblique shocks at supernova remnants inside massive star clusters act as the main accelerator for cosmic rays up to the knee. The approach combines individual supernova shocks with collective stellar-wind shocks and treats the shock obliquity angle as the factor that sets the highest reachable energy for each particle. It shows that the right obliquity angles raise acceleration efficiency enough to produce multi-PeV particles whose cutoffs depend on rigidity. The resulting spectrum and composition match the all-particle data reported by LHAASO. Readers care because the model supplies a concrete astrophysical site and mechanism that links cluster environments directly to the observed break in the cosmic-ray spectrum.

Core claim

Oblique shocks in massive star clusters significantly enhance acceleration efficiency, allowing particles to reach multi-PeV energies in a rigidity-dependent manner. By incorporating the combined contribution of supernova remnants and collective wind shocks while emphasizing the role of the shock obliquity angle, the preferred model reproduces the all-particle spectrum and composition observed by LHAASO. The knee is interpreted as arising from a sequence of rigidity-dependent cutoffs. The framework also predicts subdominant but detectable gamma-ray and neutrino emissions.

What carries the argument

The obliquity angle of shocks inside massive star clusters, which sets the maximum particle energy by improving acceleration efficiency for particles of given rigidity.

If this is right

  • Cosmic-ray particles reach multi-PeV energies through rigidity-dependent acceleration at oblique shocks.
  • The knee arises from a sequence of rigidity-dependent cutoffs rather than a single abrupt limit.
  • The model simultaneously reproduces the all-particle spectrum and elemental composition reported by LHAASO.
  • Subdominant gamma-ray and neutrino fluxes from these clusters are expected and could be detectable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same obliquity dependence could operate in other dense stellar environments that contain oblique shocks.
  • Targeted gamma-ray observations of individual massive star clusters would provide a direct test of the predicted emission levels.
  • Full hydrodynamic modeling of shock geometry inside clusters would be a logical next step to reduce the current parameterization.

Load-bearing premise

Oblique shocks with suitable angles are common enough in massive star clusters and the combined supernova plus wind contribution can be parameterized to match LHAASO data without full hydrodynamic verification of the shock geometry.

What would settle it

A measurement of cosmic-ray composition near the knee that fails to show cutoffs scaling with rigidity, or the non-detection of the predicted subdominant gamma-ray emission from massive star clusters.

Figures

Figures reproduced from arXiv: 2604.07977 by Luana N. Padilha, Rita C. Anjos.

Figure 1
Figure 1. Figure 1: Three-dimensional distribution of the selected massive star clusters from the N. V. Kharchenko et al. (2013) catalog in Sun-centered Cartesian coordinates (X, Y, Z in kpc). Marker colour encodes the cluster age, and larger markers indicate higher estimated cluster mass Mcl (selection requires Mcl > 103 M⊙); the concentration near Z ≃ 0 traces the Galactic mid-plane. ria we define two categories: Powerful c… view at source ↗
Figure 2
Figure 2. Figure 2: Spatial distribution of massive star clusters in the Galactic plane. Upper panel: combined sample that in￾cludes the simulated population and the observed objects, projected onto the XY plane. The green circle marks the sources inside a radius of 3.0 kpc from the Sun as listed by N. V. Kharchenko et al. (2013). Lower panel: the observa￾tional catalog alone with the same projection. Grey curves indicate the… view at source ↗
Figure 3
Figure 3. Figure 3: shows Eq. (5) as a function of cos θ for repre￾sentative values of η. As η increases, diffusion becomes strongly anisotropic and the dependence on θ steepens: nearly parallel configurations (cos θ → 1) yield larger k1,2 and therefore longer tacc, while more perpendic￾ular geometries favor smaller effective diffusion across the shock and hence faster acceleration. In the opposite limit η ≪ 1, diffusion is n… view at source ↗
Figure 4
Figure 4. Figure 4: All particle cosmic ray spectra for Models A, B, C, and D. Each panel displays the predicted flux J(E) for nuclei up to Z = 40 (solid curve) and the corresponding measurements from LHAASO (Z. Cao et al. 2024), the Pierre Auger Observatory (V. Verzi 2019; P. Abreu et al. 2021), KASCADE Grande (M. Bertaina et al. 2015), IceCube/IceTop (M. G. Aartsen et al. 2019), Tibet ASγ (M. G. Aartsen et al. 2019), Tunka … view at source ↗
Figure 5
Figure 5. Figure 5: Spectral composition for Models B and C. Each panel shows the energy flux for the total all–particle together with the contributions from four charge groups: protons (Z = 1), light nuclei (2 ≤ Z ≤ 5), intermediate nuclei (6 ≤ Z ≤ 24), and heavy nuclei (24 < Z ≤ 40). Solid curves give the model prediction for each group and their sum. Extragalactic component (gray dotted line) from S. Thoudam et al. (2016) … view at source ↗
Figure 6
Figure 6. Figure 6: Cosmic-ray composition in terms of the mean log￾arithmic atomic mass 〈ln A〉, derived from Models A, B and C and compared with data from AUGER (A. Yushkov 2019; B. Pont 2023), TALE (R. U. Abbasi et al. 2021), TUNKA (N. M. Budnev et al. 2022; P. A. Bezyazeekov et al. 2018), IceTop ( IceCube Collaboration et al. 2013), KASCADE (L. G. Sveshnikova et al. 2013), JACEE (Y. Takahashi et al. 1998) and ATIC (A. D. P… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Contribution from MSC to gamma-ray emission, compared with observations of the diffuse Galactic gamma-ray emission (B. Bartoli et al. 2015; D. Grasso et al. 2017; M. Amenomori et al. 2021, 2009; A. Borione et al. 1998; M. G. Aartsen et al. 2017; P. De La Torre Luque et al. 2023; M. Ackermann et al. 2015; Z. Cao et al. 2023). (b) Muon neutrino flux from the combined contribution of the powerful and soft… view at source ↗
Figure 8
Figure 8. Figure 8: Corner plots for Models A to D showing the local exploration of parameter space around the best-fit values, weighted by the χ 2 difference. The diagonal panels display one dimensional posteriors; vertical lines mark the median and the central 68% confidence interval. The off diagonal panels show joint posteriors with contours at 68% and 95% credibility. Parameters include the wind power Pw (in erg s−1 ), t… view at source ↗
read the original abstract

This work establishes oblique shocks in Massive Star Clusters (MSC) as a primary mechanism for accelerating cosmic rays (CR) up to the knee of the energy spectrum. We develop a model that incorporates the combined contribution of supernova and collective wind shocks, emphasizing the critical role of the shock obliquity angle in determining the maximum particle energy. We illustrate, within our model that oblique shocks can significantly enhance acceleration efficiency, allowing particles to reach multi-PeV energies in a rigidity-dependent manner. Our preferred model, which incorporates oblique shocks, reproduces the all-particle spectrum and composition observed by The Large High Altitude Air Shower Observatory (LHAASO), interpreting the knee as arising from a sequence of rigidity-dependent cutoffs. The model also predicts subdominant but detectable gamma-ray and neutrino emissions. This study provides an attempt at building a unified framework connecting MSC particle acceleration to the observed features of the cosmic-ray knee.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes that oblique shocks in massive star clusters are a primary site for accelerating cosmic rays up to the knee, developing a model that combines supernova remnant shocks with collective stellar wind shocks. Shock obliquity is highlighted as critical for setting maximum particle energies, enabling multi-PeV acceleration in a rigidity-dependent manner. A preferred parameterization is presented that reproduces the all-particle spectrum and composition measured by LHAASO, interpreting the knee as arising from a sequence of rigidity-dependent cutoffs, while also predicting subdominant gamma-ray and neutrino fluxes.

Significance. If the key assumptions about shock geometry and parameter values can be independently justified, the work would provide a coherent framework connecting massive star cluster environments to the cosmic-ray knee, offering a natural explanation for both the spectral break and the shift toward heavier composition. The multi-messenger predictions would also be valuable for future observations.

major comments (3)
  1. [Abstract] Abstract: the preferred model is stated to reproduce the LHAASO spectrum and composition, yet the obliquity angles and relative supernova-versus-wind contributions function as free parameters whose values are selected to match the very data being explained, creating a circularity that undermines the claim of explanatory power.
  2. [Model description] Model description (as summarized in abstract and skeptic note): the distribution of shock obliquity angles is parameterized rather than derived from the three-dimensional MHD structure of cluster winds and supernova remnants, so the model does not demonstrate that quasi-oblique shocks capable of multi-PeV acceleration are sufficiently common in realistic geometries.
  3. [Results section] Results section: no quantitative comparison to alternative knee explanations (standard SNR shocks, galactic wind models, etc.) or presentation of fit uncertainties, chi-squared values, or sensitivity tests is provided, leaving open whether the reproduction of the LHAASO data is unique or merely achievable through tuning.
minor comments (2)
  1. [Abstract] Abstract: the sentence 'We illustrate, within our model that oblique shocks' contains a grammatical error and should be rephrased for clarity.
  2. [Throughout] Throughout: all free parameters (obliquity angle, relative contributions) should be explicitly tabulated with their adopted ranges and any external constraints from prior hydrodynamic studies.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each major comment below, indicating where revisions will strengthen the manuscript while maintaining the core claims supported by our calculations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the preferred model is stated to reproduce the LHAASO spectrum and composition, yet the obliquity angles and relative supernova-versus-wind contributions function as free parameters whose values are selected to match the very data being explained, creating a circularity that undermines the claim of explanatory power.

    Authors: We acknowledge that the preferred parameterization involves choices for the obliquity distribution and supernova-to-wind ratio that are tuned to match LHAASO observations. These choices are, however, constrained to ranges motivated by DSA theory for oblique shocks and by existing hydrodynamic models of cluster winds. The explanatory value lies in showing that a single rigidity-dependent cutoff mechanism, enabled by obliquity, simultaneously accounts for the spectral shape and the progressive heavy-element dominance. In revision we will modify the abstract and introduction to state the physical priors on the parameters explicitly and to frame the preferred model as a viable realization rather than an over-claimed unique solution. revision: partial

  2. Referee: [Model description] Model description (as summarized in abstract and skeptic note): the distribution of shock obliquity angles is parameterized rather than derived from the three-dimensional MHD structure of cluster winds and supernova remnants, so the model does not demonstrate that quasi-oblique shocks capable of multi-PeV acceleration are sufficiently common in realistic geometries.

    Authors: The referee is correct that the obliquity-angle distribution is introduced parametrically rather than extracted from a self-consistent 3D MHD simulation of an entire cluster. Performing such a simulation that includes multiple core-collapse events, continuous wind injection, and realistic magnetic-field evolution lies beyond the scope and computational resources of the present study. The adopted distribution is instead informed by published MHD results on individual oblique shocks and wind-blown bubbles. We will expand the model-description section to include additional references and a clearer statement of this limitation, while preserving the parametric approach as a first-step exploration of the mechanism. revision: partial

  3. Referee: [Results section] Results section: no quantitative comparison to alternative knee explanations (standard SNR shocks, galactic wind models, etc.) or presentation of fit uncertainties, chi-squared values, or sensitivity tests is provided, leaving open whether the reproduction of the LHAASO data is unique or merely achievable through tuning.

    Authors: We agree that the absence of direct comparisons and statistical diagnostics weakens the presentation. In the revised manuscript we will add a dedicated subsection that contrasts our all-particle spectrum and composition with predictions from standard SNR shock models and galactic-wind scenarios. We will also include sensitivity plots varying the key parameters and, where the data permit, report chi-squared values together with a brief discussion of fit uncertainties. revision: yes

Circularity Check

1 steps flagged

Preferred model selected by tuning obliquity parameters to reproduce LHAASO spectrum and composition

specific steps
  1. fitted input called prediction [Abstract]
    "Our preferred model, which incorporates oblique shocks, reproduces the all-particle spectrum and composition observed by The Large High Altitude Air Shower Observatory (LHAASO), interpreting the knee as arising from a sequence of rigidity-dependent cutoffs."

    The model is labeled 'preferred' precisely because its parameters (obliquity angles and SN+wind contributions) are chosen to match the LHAASO spectrum and composition. The claimed explanation of the knee therefore reduces to a parameterization tuned to the very observations it is said to predict, rather than an a priori derivation of shock geometry or obliquity distribution.

full rationale

The paper claims oblique shocks enable multi-PeV acceleration and explain the knee via rigidity-dependent cutoffs. However, the central result is obtained by designating a 'preferred model' that incorporates oblique shocks and matches the LHAASO all-particle spectrum and composition. This selection implies that the obliquity angles, the relative weighting of supernova versus collective wind shocks, and the resulting cutoff rigidities function as adjustable inputs constrained by the target data rather than outputs derived from cluster hydrodynamics or first-principles shock geometry. The reproduction is therefore a fit by construction, not an independent prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard diffusive shock acceleration applied to oblique geometries plus several adjustable parameters for shock angles and relative contributions that are tuned to LHAASO observations.

free parameters (2)
  • shock obliquity angle
    Determines maximum particle energy and acceleration efficiency; values are chosen to allow multi-PeV reach and to fit the observed cutoffs.
  • relative supernova versus wind shock contribution
    Combined in the preferred model to reproduce the all-particle spectrum and composition; weights are not independently derived.
axioms (2)
  • domain assumption Diffusive shock acceleration theory remains valid for oblique shocks at supernova remnants and stellar winds
    Invoked as the foundation for the acceleration efficiency calculation.
  • domain assumption Massive star clusters contain the necessary population of supernova remnants and collective winds to produce the required oblique shocks
    Required for the model to be applicable to the observed knee.

pith-pipeline@v0.9.0 · 5467 in / 1544 out tokens · 71655 ms · 2026-05-10T17:03:35.583391+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

130 extracted references · 124 canonical work pages · 1 internal anchor

  1. [1]

    G., Ackermann, M., Adams, J., et al

    Aartsen, M. G., Ackermann, M., Adams, J., et al. 2017, ApJ, 849, 67, doi: 10.3847/1538-4357/aa8dfb

    work page doi:10.3847/1538-4357/aa8dfb 2017

  2. [2]

    G., Ackermann, M., Adams, J., et al

    Aartsen, M. G., Ackermann, M., Adams, J., et al. 2019, PhRvD, 100, 082002, doi: 10.1103/PhysRevD.100.082002

    work page doi:10.1103/physrevd.100.082002 2019

  3. [3]

    G., Abbasi, R., Ackermann, M., et al

    Aartsen, M. G., Abbasi, R., Ackermann, M., et al. 2020, PhRvD, 102, 122001, doi: 10.1103/PhysRevD.102.122001

    work page doi:10.1103/physrevd.102.122001 2020

  4. [4]

    U., Abe, M., Abu-Zayyad, T., et al

    Abbasi, R. U., Abe, M., Abu-Zayyad, T., et al. 2021, ApJ, 909, 178, doi: 10.3847/1538-4357/abdd30

    work page doi:10.3847/1538-4357/abdd30 2021

  5. [5]

    Nature Astronomy , keywords =

    Abeysekara, A. U., Albert, A., Alfaro, R., et al. 2021, Nature Astronomy, 5, 465, doi: 10.1038/s41550-021-01318-y

    work page doi:10.1038/s41550-021-01318-y 2021

  6. [6]

    M., et al

    Abreu, P., Aglietta, M., Albury, J. M., et al. 2021, European Physical Journal C, 81, 966, doi: 10.1140/epjc/s10052-021-09700-w

    work page doi:10.1140/epjc/s10052-021-09700-w 2021

  7. [7]

    2013, Science, 339, 807, doi: 10.1126/science.1231160

    Ackermann, M., Ajello, M., Allafort, A., et al. 2013, Science, 339, 807, doi: 10.1126/science.1231160

    work page doi:10.1126/science.1231160 2013

  8. [8]

    2015, ApJ, 799, 86, doi: 10.1088/0004-637X/799/1/86

    Ackermann, M., Ajello, M., Albert, A., et al. 2015, ApJ, 799, 86, doi: 10.1088/0004-637X/799/1/86

    work page doi:10.1088/0004-637x/799/1/86 2015

  9. [9]

    Nature Astronomy , keywords =

    Aharonian, F., Yang, R., & de O˜ na Wilhelmi, E. 2019, Nature Astronomy, 3, 561, doi: 10.1038/s41550-019-0724-0

    work page doi:10.1038/s41550-019-0724-0 2019

  10. [10]

    G., Bazer-Bachi, A

    Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006, A&A, 449, 223, doi: 10.1051/0004-6361:20054279

    work page doi:10.1051/0004-6361:20054279 2006

  11. [11]

    , keywords =

    Aharonian, F., Ashkar, H., Backes, M., et al. 2022, A&A, 666, A124, doi: 10.1051/0004-6361/202244323

    work page doi:10.1051/0004-6361/202244323 2022

  12. [12]

    2014, PhRvD, 90, 023010, doi: 10.1103/PhysRevD.90.023010

    Ahlers, M., & Murase, K. 2014, PhRvD, 90, 023010, doi: 10.1103/PhysRevD.90.023010

    work page doi:10.1103/physrevd.90.023010 2014

  13. [13]

    K., Lipari, P., & Stanev, T

    Ahn, E.-J., Engel, R., Gaisser, T. K., Lipari, P., & Stanev, T. 2009, PhRvD, 80, 094003, doi: 10.1103/PhysRevD.80.094003

    work page doi:10.1103/physrevd.80.094003 2009

  14. [14]

    D., et al

    Alfaro, R., Alvarez, C., ´Alvarez, J. D., et al. 2017, PhRvD, 96, 122001, doi: 10.1103/PhysRevD.96.122001

    work page doi:10.1103/physrevd.96.122001 2017

  15. [15]

    2022, Reviews of Modern Plasma Physics, 6, 29, doi: 10.1007/s41614-022-00093-1

    Amano, T., Matsumoto, Y., Bohdan, A., et al. 2022, Reviews of Modern Plasma Physics, 6, 29, doi: 10.1007/s41614-022-00093-1

    work page doi:10.1007/s41614-022-00093-1 2022

  16. [16]

    2019, A&A, 623, A86, doi: 10.1051/0004-6361/201833985

    Ambrogi, L., Zanin, R., Casanova, S., et al. 2019, A&A, 623, A86, doi: 10.1051/0004-6361/201833985

    work page doi:10.1051/0004-6361/201833985 2019

  17. [17]

    Z., et al

    Amenomori, M., Cao, Z., Dai, B. Z., et al. 1996, ApJ, 461, 408, doi: 10.1086/177069

    work page doi:10.1086/177069 1996

  18. [18]

    J., Chen, D., et al

    Amenomori, M., Bi, X. J., Chen, D., et al. 2009, ApJ, 692, 61, doi: 10.1088/0004-637X/692/1/61 20

    work page doi:10.1088/0004-637x/692/1/61 2009

  19. [19]

    W., Bi, X

    Amenomori, M., Bao, Y. W., Bi, X. J., et al. 2021, PhRvL, 126, 141101, doi: 10.1103/PhysRevLett.126.141101

    work page doi:10.1103/physrevlett.126.141101 2021

  20. [20]

    D., Badea, A

    Antoni, T., Apel, W. D., Badea, A. F., et al. 2005, Astroparticle Physics, 24, 1, doi: 10.1016/j.astropartphys.2005.04.001

    work page doi:10.1016/j.astropartphys.2005.04.001 2005

  21. [21]

    I., Leer, E., & Skadron, G

    Axford, W. I., Leer, E., & Skadron, G. 1977, in International Cosmic Ray Conference, Vol. 11, 132

    1977

  22. [22]

    V., Bykov, A

    Badmaev, D. V., Bykov, A. M., & Kalyashova, M. E. 2022, MNRAS, 517, 2818, doi: 10.1093/mnras/stac2738

    work page doi:10.1093/mnras/stac2738 2022

  23. [23]

    V., Bykov, A

    Badmaev, D. V., Bykov, A. M., & Kalyashova, M. E. 2024, MNRAS, 527, 3749, doi: 10.1093/mnras/stad3389

    work page doi:10.1093/mnras/stad3389 2024

  24. [24]

    J., et al

    Bartoli, B., Bernardini, P., Bi, X. J., et al. 2015, ApJ, 806, 20, doi: 10.1088/0004-637X/806/1/20

    work page doi:10.1088/0004-637x/806/1/20 2015

  25. [25]

    1991, MNRAS, 249, 76, doi: 10.1093/mnras/249.1.76

    Battinelli, P., & Capuzzo-Dolcetta, R. 1991, MNRAS, 249, 76, doi: 10.1093/mnras/249.1.76

    work page doi:10.1093/mnras/249.1.76 1991

  26. [26]

    Bell, A. R. 2004, MNRAS, 353, 550, doi: 10.1111/j.1365-2966.2004.08097.x

    work page doi:10.1111/j.1365-2966.2004.08097.x 2004

  27. [27]

    , archivePrefix = "arXiv", eprint =

    Bell, A. R., Schure, K. M., & Reville, B. 2011, MNRAS, 418, 1208, doi: 10.1111/j.1365-2966.2011.19571.x

    work page doi:10.1111/j.1365-2966.2011.19571.x 2011

  28. [28]

    R., Schure, K

    Bell, A. R., Schure, K. M., Reville, B., & Giacinti, G. 2013, MNRAS, 431, 415, doi: 10.1093/mnras/stt179

    work page doi:10.1093/mnras/stt179 2013

  29. [29]

    D., Arteaga-Vel´ azquez, J

    Bertaina, M., Apel, W. D., Arteaga-Vel´ azquez, J. C., et al. 2015, in International Cosmic Ray Conference, Vol. 34, 34th International Cosmic Ray Conference (ICRC2015), 359, doi: 10.22323/1.236.0359

    work page doi:10.22323/1.236.0359 2015

  30. [30]

    A., Budnev, N

    Bezyazeekov, P. A., Budnev, N. M., Chernykh, D., et al. 2018, PhRvD, 97, 122004, doi: 10.1103/PhysRevD.97.122004

    work page doi:10.1103/physrevd.97.122004 2018

  31. [31]

    M., & Barbuy, B

    Bica, E., Dutra, C. M., & Barbuy, B. 2003a, A&A, 397, 177, doi: 10.1051/0004-6361:20021479

    work page doi:10.1051/0004-6361:20021479

  32. [32]

    M., Soares, J., & Barbuy, B

    Bica, E., Dutra, C. M., Soares, J., & Barbuy, B. 2003b, A&A, 404, 223, doi: 10.1051/0004-6361:20030486

    work page doi:10.1051/0004-6361:20030486

  33. [33]

    , keywords =

    Blasi, P. 2025, A&A, 694, A244, doi: 10.1051/0004-6361/202453017

    work page doi:10.1051/0004-6361/202453017 2025

  34. [34]

    2024, MNRAS, 533, 561, doi: 10.1093/mnras/stae1782 Bl¨ umer, J., Engel, R., & H¨ orandel, J

    Blasi, P., & Morlino, G. 2024, MNRAS, 533, 561, doi: 10.1093/mnras/stae1782 Bl¨ umer, J., Engel, R., & H¨ orandel, J. R. 2009, Progress in Particle and Nuclear Physics, 63, 293, doi: 10.1016/j.ppnp.2009.05.002

    work page doi:10.1093/mnras/stae1782 2024

  35. [35]

    2023, Plasma Physics and Controlled Fusion, 65, 014002, doi: 10.1088/1361-6587/aca5b2

    Bohdan, A. 2023, Plasma Physics and Controlled Fusion, 65, 014002, doi: 10.1088/1361-6587/aca5b2

    work page doi:10.1088/1361-6587/aca5b2 2023

  36. [36]

    A., Chantell, M

    Borione, A., Catanese, M. A., Chantell, M. C., et al. 1998, ApJ, 493, 175, doi: 10.1086/305096

    work page doi:10.1086/305096 1998

  37. [37]

    M., Chiavassa, A., Gress, O

    Budnev, N. M., Chiavassa, A., Gress, O. A., et al. 2020, Astroparticle Physics, 117, 102406, doi: 10.1016/j.astropartphys.2019.102406

    work page doi:10.1016/j.astropartphys.2019.102406 2020

  38. [38]

    M., Kuzmichev, L., Mirzoyan, R., et al

    Budnev, N. M., Kuzmichev, L., Mirzoyan, R., et al. 2022, in 37th International Cosmic Ray Conference, 731, doi: 10.22323/1.395.0731

    work page doi:10.22323/1.395.0731 2022

  39. [39]

    2011, AcA, 61, 231, doi: 10.48550/arXiv.1107.5119

    Bukowiecki, L., Maciejewski, G., Konorski, P., & Strobel, A. 2011, AcA, 61, 231, doi: 10.48550/arXiv.1107.5119

    work page doi:10.48550/arxiv.1107.5119 2011

  40. [40]

    M., Ellison, D

    Bykov, A. M., Ellison, D. C., Gladilin, P. E., & Osipov, S. M. 2018, Nuclear and Particle Physics Proceedings, 297-299, 183, doi: 10.1016/j.nuclphysbps.2018.07.028

    work page doi:10.1016/j.nuclphysbps.2018.07.028 2018

  41. [41]

    M., & Toptygin, I

    Bykov, A. M., & Toptygin, I. N. 2001, Astronomy Letters, 27, 625, doi: 10.1134/1.1404456

    work page doi:10.1134/1.1404456 2001

  42. [42]

    Astronomy & Astrophysics , author =

    Cantat-Gaudin, T., & Anders, F. 2020, A&A, 633, A99, doi: 10.1051/0004-6361/201936691

    work page doi:10.1051/0004-6361/201936691 2020

  43. [43]

    2018, , 618, A93, 10.1051/0004-6361/201833476

    Cantat-Gaudin, T., Jordi, C., Vallenari, A., et al. 2018, A&A, 618, A93, doi: 10.1051/0004-6361/201833476

    work page doi:10.1051/0004-6361/201833476 2018

  44. [44]

    Astronomy & Astrophysics , author =

    Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al. 2020, A&A, 640, A1, doi: 10.1051/0004-6361/202038192

    work page doi:10.1051/0004-6361/202038192 2020

  45. [45]

    Addaziet al.(LHAASO), The Large High Altitude Air Shower Observatory (LHAASO) Science Book (2021 Edi- tion), Chin

    Cao, Z., della Volpe, D., Liu, S., et al. 2019, arXiv e-prints, arXiv:1905.02773, doi: 10.48550/arXiv.1905.02773

    work page doi:10.48550/arxiv.1905.02773 2019

  46. [46]

    2023, PhRvL, 131, 151001, doi: 10.1103/PhysRevLett.131.151001

    Cao, Z., Aharonian, F., An, Q., et al. 2023, PhRvL, 131, 151001, doi: 10.1103/PhysRevLett.131.151001

    work page doi:10.1103/physrevlett.131.151001 2023

  47. [47]

    Measurements of All-Particle Energy Spectrum and Mean Logarithmic Mass of Cosmic Rays from 0.3 to 30 PeV with LHAASO-KM2A.PRL2024, 132, 131002, [arXiv:astro-ph.HE/2403.10010]

    Cao, Z., Aharonian, F., Axikegu, et al. 2024, PhRvL, 132, 131002, doi: 10.1103/PhysRevLett.132.131002

    work page doi:10.1103/physrevlett.132.131002 2024

  48. [48]

    and Wilkinson, M

    Caprioli, D., Blasi, P., & Amato, E. 2009, MNRAS, 396, 2065, doi: 10.1111/j.1365-2966.2008.14298.x

    work page doi:10.1111/j.1365-2966.2008.14298.x 2009

  49. [49]

    R., & Spitkovsky, A

    Caprioli, D., Pop, A. R., & Spitkovsky, A. 2015, ApJL, 798, L28, doi: 10.1088/2041-8205/798/2/L28

    work page doi:10.1088/2041-8205/798/2/l28 2015

  50. [50]

    , keywords =

    Morlino, G. 2024, A&A, 686, A118, doi: 10.1051/0004-6361/202348541

    work page doi:10.1051/0004-6361/202348541 2024

  51. [51]

    Galactic

    Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392

    work page internal anchor Pith review doi:10.1086/376392 2003

  52. [52]

    Cranfill, C. W. 1978, ApJ, 226, 650, doi: 10.1086/156649

    work page doi:10.1086/156649 1978

  53. [53]

    2020, Astroparticle Physics, 123, 102492, doi: 10.1016/j.astropartphys.2020.102492

    Cristofari, P., Blasi, P., & Amato, E. 2020, Astroparticle Physics, 123, 102492, doi: 10.1016/j.astropartphys.2020.102492

    work page doi:10.1016/j.astropartphys.2020.102492 2020

  54. [54]

    2021, A&A, 650, A62, doi: 10.1051/0004-6361/202140448 De La Torre Luque, P., Gaggero, D., Grasso, D., et al

    Cristofari, P., Blasi, P., & Caprioli, D. 2021, A&A, 650, A62, doi: 10.1051/0004-6361/202140448 De La Torre Luque, P., Gaggero, D., Grasso, D., et al. 2023, A&A, 672, A58, doi: 10.1051/0004-6361/202243714

    work page doi:10.1051/0004-6361/202140448 2021

  55. [55]

    A., Ramsay, G., Andronov, I., et al

    Dehnen, W., & Binney, J. 1998, MNRAS, 294, 429, doi: 10.1046/j.1365-8711.1998.01282.x10.1111/j.1365- 8711.1998.01282.x

    work page doi:10.1046/j.1365-8711.1998.01282.x10.1111/j.1365- 1998

  56. [56]

    M., Bica, E., Soares, J., & Barbuy, B

    Dutra, C. M., Bica, E., Soares, J., & Barbuy, B. 2003, A&A, 400, 533, doi: 10.1051/0004-6361:20030005

    work page doi:10.1051/0004-6361:20030005 2003

  57. [57]

    K., Lipari, P., & Stanev, T

    Engel, J., Gaisser, T. K., Lipari, P., & Stanev, T. 1992, PhRvD, 46, 5013, doi: 10.1103/PhysRevD.46.5013

    work page doi:10.1103/physrevd.46.5013 1992

  58. [58]

    D., & Wolfendale, A

    Erlykin, A. D., & Wolfendale, A. W. 2001, Advances in Space Research, 27, 803, doi: 10.1016/S0273-1177(01)00125-9

    work page doi:10.1016/s0273-1177(01)00125-9 2001

  59. [59]

    2020, The Cosmic-Ray Energy Spectrum, Zenodo, doi: 10.5281/zenodo.4396125

    Evoli, C. 2020, The Cosmic-Ray Energy Spectrum, Zenodo, doi: 10.5281/zenodo.4396125

    work page doi:10.5281/zenodo.4396125 2020

  60. [60]

    1949, Physical Review, 75, 1169, doi: 10.1103/PhysRev.75.1169 21

    Fermi, E. 1949, Physical Review, 75, 1169, doi: 10.1103/PhysRev.75.1169 21

    work page doi:10.1103/physrev.75.1169 1949

  61. [61]

    2010, A&A, 510, A101, doi: 10.1051/0004-6361/200913520

    Ferrand, G., & Marcowith, A. 2010, A&A, 510, A101, doi: 10.1051/0004-6361/200913520

    work page doi:10.1051/0004-6361/200913520 2010

  62. [62]

    Friel, E. D. 1995, ARA&A, 33, 381, doi: 10.1146/annurev.aa.33.090195.002121

    work page doi:10.1146/annurev.aa.33.090195.002121 1995

  63. [63]

    Froebrich, D., Schmeja, S., Samuel, D., & Lucas, P. W. 2010, MNRAS, 409, 1281, doi: 10.1111/j.1365-2966.2010.17390.x

    work page doi:10.1111/j.1365-2966.2010.17390.x 2010

  64. [64]

    Froebrich, D., Scholz, A., & Raftery, C. L. 2007, MNRAS, 374, 399, doi: 10.1111/j.1365-2966.2006.11148.x

    work page doi:10.1111/j.1365-2966.2006.11148.x 2007

  65. [65]

    2024, in 7th Heidelberg International Symposium on High-Energy Gamma-Ray Astronomy, 16, doi: 10.48550/arXiv.2301.06505

    Gabici, S. 2024, in 7th Heidelberg International Symposium on High-Energy Gamma-Ray Astronomy, 16, doi: 10.48550/arXiv.2301.06505

    work page doi:10.48550/arxiv.2301.06505 2024

  66. [66]

    International Journal of Modern Physics D , keywords =

    Gabici, S., Evoli, C., Gaggero, D., et al. 2019, International Journal of Modern Physics D, 28, 1930022, doi: 10.1142/S0218271819300222

    work page doi:10.1142/s0218271819300222 2019

  67. [67]

    K., Protheroe, R

    Gaisser, T. K., Protheroe, R. J., & Stanev, T. 1998, ApJ, 492, 219, doi: 10.1086/305011

    work page doi:10.1086/305011 1998

  68. [68]

    Glasmacher, M. A. K., Catanese, M. A., Chantell, M. C., et al. 1999, Astroparticle Physics, 10, 291, doi: 10.1016/S0927-6505(98)00070-X

    work page doi:10.1016/s0927-6505(98)00070-x 1999

  69. [69]

    2023, in European Physical Journal Web of Conferences, Vol

    Globus, N., & Blandford, R. 2023, in European Physical Journal Web of Conferences, Vol. 283, European Physical Journal Web of Conferences (EDP), 04001, doi: 10.1051/epjconf/202328304001

    work page doi:10.1051/epjconf/202328304001 2023

  70. [70]

    2017, Nuclear and Particle Physics Proceedings, 291-293, 9, doi: 10.1016/j.nuclphysbps.2017.06.003 H¨ arer, L., Vieu, T., & Reville, B

    Grasso, D., Gaggero, D., Marinelli, A., Urbano, A., & Valli, M. 2017, Nuclear and Particle Physics Proceedings, 291-293, 9, doi: 10.1016/j.nuclphysbps.2017.06.003 H¨ arer, L., Vieu, T., & Reville, B. 2025, A&A, 698, A6, doi: 10.1051/0004-6361/202554057

    work page doi:10.1016/j.nuclphysbps.2017.06.003 2017

  71. [71]

    2024, arXiv e-prints, arXiv:2411.13793, doi: 10.48550/arXiv.2411.13793

    He, H., Zhang, H., Cheng, Q., Ma, L., & Feng, C. 2024, arXiv e-prints, arXiv:2411.13793, doi: 10.48550/arXiv.2411.13793

    work page doi:10.48550/arxiv.2411.13793 2024

  72. [72]

    Hillas, A. M. 2005, Journal of Physics G Nuclear Physics, 31, R95, doi: 10.1088/0954-3899/31/5/R02 H¨ orandel, J. R. 2003, Astroparticle Physics, 19, 193, doi: 10.1016/S0927-6505(02)00198-6 H¨ orandel, J. R. 2004, Astroparticle Physics, 21, 241, doi: 10.1016/j.astropartphys.2004.01.004 IceCube Collaboration, Abbasi, R., Abdou, Y., et al. 2013, Astropartic...

    work page doi:10.1088/0954-3899/31/5/r02 2005

  73. [73]

    2020, Chinese Physics C, 44, 065002, doi: 10.1088/1674-1137/44/6/065002 J´ ohannesson, G., Ruiz de Austri, R., Vincent, A

    Jin, C., Chen, S.-Z., He, H.-H., & Collaboration), L. 2020, Chinese Physics C, 44, 065002, doi: 10.1088/1674-1137/44/6/065002 J´ ohannesson, G., Ruiz de Austri, R., Vincent, A. C., et al. 2016, ApJ, 824, 16, doi: 10.3847/0004-637X/824/1/16

    work page doi:10.1088/1674-1137/44/6/065002 2020

  74. [74]

    Jokipii, J. R. 1987, ApJ, 313, 842, doi: 10.1086/165022

    work page doi:10.1086/165022 1987

  75. [75]

    M., & Vila, G

    Kafexhiu, E., Aharonian, F., Taylor, A. M., & Vila, G. S. 2014, PhRvD, 90, 123014, doi: 10.1103/PhysRevD.90.123014

    work page doi:10.1103/physrevd.90.123014 2014

  76. [76]

    2012, Astroparticle Physics, 35, 660, doi: 10.1016/j.astropartphys.2012.02.004

    Kampert, K.-H., & Unger, M. 2012, Astroparticle Physics, 35, 660, doi: 10.1016/j.astropartphys.2012.02.004

    work page doi:10.1016/j.astropartphys.2012.02.004 2012

  77. [77]

    V., Piskunov, A

    Kharchenko, N. V., Piskunov, A. E., Schilbach, E., R¨ oser, S., & Scholz, R.-D. 2013, A&A, 558, A53, doi: 10.1051/0004-6361/201322302

    work page doi:10.1051/0004-6361/201322302 2013

  78. [78]

    The Astronomical Journal , author =

    King, I. 1962, AJ, 67, 471, doi: 10.1086/108756

    work page doi:10.1086/108756 1962

  79. [79]

    S., & Glover, S

    Klessen, R. S., & Glover, S. C. O. 2016, Saas-Fee Advanced Course, 43, 85, doi: 10.1007/978-3-662-47890-5 2

    work page doi:10.1007/978-3-662-47890-5 2016

  80. [80]

    S., & Samura, T

    Kobayakawa, K., Honda, Y. S., & Samura, T. 2002, PhRvD, 66, 083004, doi: 10.1103/PhysRevD.66.083004

    work page doi:10.1103/physrevd.66.083004 2002

Showing first 80 references.