pith. sign in

arxiv: 2607.00135 · v1 · pith:3ZOTUZGEnew · submitted 2026-06-30 · 🌌 astro-ph.HE

Anomalous Air Showers and What They Reveal About Hadronic Interactions and Cosmic-ray Masses

Stijn Buitink , Vital De Henau , Sjoerd Bouma , Justin Bray , Arthur Corstanje , Edwin Dickinson , Brian Hare , Andreas Haungs
show 23 more authors
Haoning He J\"org H\"orandel Tim Huege Clancy James Philipp Laub Xingyu Li Hermann-Josef Mathes Katharine Mulrey Anna Nelles Subhadip Saha Felix Schl\"uter Olaf Scholten Ralph Spencer Christopher Sterpka Karen Terveer Satyendra Thoudam Gia Trinh Paulina Turekova Darko Veberic Keito Watanabe Chao Zhang Pengfei Zhang Yi Zhang
This is my paper

Pith reviewed 2026-07-02 17:30 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords cosmic raysair showershadronic interactionsmass compositionSKA-Lowlongitudinal profilesXmax
0
0 comments X

The pith

Double-bump air shower profiles observed by SKA-Low can constrain hadronic interaction properties and cosmic-ray mass composition.

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

The paper establishes that SKA-Low's high-resolution observations of cosmic ray air showers in the 10^16 to 10^18 eV range can detect anomalous longitudinal developments, particularly double-bump profiles caused by secondary particles traveling far before interacting. These features arise after the initial cosmic ray interaction and offer a direct probe into the hadronic processes that currently introduce large systematic uncertainties in Xmax-based mass measurements. By resolving such profiles, the work shows a path to simultaneously refine hadronic models and determine the primary mass composition during the galactic-to-extragalactic transition.

Core claim

After the first interaction, secondary particles retain a significant fraction of the energy and can produce recognizable sub-showers as secondary bumps in the longitudinal profile. SKA-Low's antenna density and bandwidth enable resolution of these double-bump and other anomalous developments, which can then be used to place constraints on hadronic interaction properties and to determine the mass composition of cosmic rays.

What carries the argument

Double-bump showers and other anomalous longitudinal developments in the reconstructed shower profile.

If this is right

  • Hadronic interaction models can be tested and refined using the frequency and properties of resolved double-bump events.
  • Mass composition estimates in the transition region gain an independent handle that reduces reliance on Xmax alone.
  • Source identification for cosmic rays improves once both interaction uncertainties and mass uncertainties are lowered.

Where Pith is reading between the lines

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

  • The same double-bump technique could be cross-checked against independent observables such as muon content or radio signal timing at the same array.
  • If double bumps prove rarer or more common than current models predict, it would point to adjustments in interaction lengths or particle production at the first interaction.
  • Arrays with comparable resolution at higher energies could extend the method beyond the transition region.

Load-bearing premise

That simulated double-bump profiles accurately represent real observations and that the hadronic interaction models in those simulations are close enough to reality for the resulting constraints to be meaningful.

What would settle it

A statistically significant difference between the rate or shape of double-bump profiles measured in real SKA-Low data and the distributions predicted by current simulations for any assumed mass composition would falsify the claim that these profiles can be used to derive reliable constraints.

Figures

Figures reproduced from arXiv: 2607.00135 by Andreas Haungs, Anna Nelles, Arthur Corstanje, Brian Hare, Chao Zhang, Christopher Sterpka, Clancy James, Darko Veberic, Edwin Dickinson, Felix Schl\"uter, Gia Trinh, Haoning He, Hermann-Josef Mathes, J\"org H\"orandel, Justin Bray, Karen Terveer, Katharine Mulrey, Keito Watanabe, Olaf Scholten, Paulina Turekova, Pengfei Zhang, Philipp Laub, Ralph Spencer, Satyendra Thoudam, Sjoerd Bouma, Stijn Buitink, Subhadip Saha, Tim Huege, Vital De Henau, Xingyu Li, Yi Zhang.

Figure 1
Figure 1. Figure 1: Two examples of double bump showers. The top panels show the number of particles as a function of atmospheric depth. The evolution of the full shower is fitted by a superposition of two Gaisser-Hillas profiles. The bottom panels show the underlying structure of the shower. Lines indicate the trajectories of the most energetic shower particles and labels indicate the particle type. Colors of the tracks and … view at source ↗
Figure 2
Figure 2. Figure 2: Fraction of showers that feature a double-bump structure as a function of energy for different primary particles and different hadronic interaction models: EPOS-LHC (left), Sibyll-2.3d (middle), and QGSJETII-04 (right). Each energy bin is based on 4000 simulated showers, so a fraction of 1% corresponds to 40 double bump showers. Statistical fluctuations corresponding to these low quantities are visible in … view at source ↗
Figure 3
Figure 3. Figure 3: Left: Distribution of 𝐿 for sets of simulated 10-100 PeV showers of different primary mass. Double-bump showers were removed from the sample. Right: 𝐿-distribution of proton showers fitted with a Gaussian. Stretched showers are defined as having an 𝐿-value that is more than two sigma above the average value. showers, which is expected as they have a common origin. To count the number of stretched showers, … view at source ↗
Figure 4
Figure 4. Figure 4: Fraction of stretched showers as a function of energy for different primary particles and different hadronic interaction models. the most dramatic features, they are less common than stretched showers. The relative frequency at which they occur depends critically on the primary and the hadronic cross sections. Therefore, the observation of anomalous showers will provide powerful new constraints on both the… view at source ↗
Figure 5
Figure 5. Figure 5: The fluence footprints of a 10 PeV Helium shower with a double-bump structure. The features of the pattern depend strongly on the selected frequency band [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: simulations of a 5.6 × 1017 eV helium shower. The color indicates the measured fluence per antenna including Galactic background noise and instrumental noise. Right: example waveforms for an antenna situated near the center of the shower footprint. To test whether these features can indeed be observed with the SKA we simulated the measured waveforms by applying the SKALA antenna model and adding real… view at source ↗
Figure 7
Figure 7. Figure 7: Frequency spectra of a radio pulse from a regular shower (left) and a double-bump shower (right). The interference maxima of the double peak spectra usually fall outside of the LOFAR band but inside the SKA-Low band. evolution of the transverse current can be reconstructed by using a more formal approach (Scholten et al., 2024). Double-bump showers pose an additional challenge to reconstruction algorithms.… view at source ↗
Figure 8
Figure 8. Figure 8: Double-bump reconstruction plot. The background colors indicate the relative phase of the radio signals from the two sub-showers based on the point-source model for different observer positions and frequencies. The red crosses are the first interference minima found in the spectra of fully simulated waveforms. The yellow line is a fit of the data to the point-source model. Two additional model lines are sh… view at source ↗
Figure 9
Figure 9. Figure 9: Left: distribution of energy ratio between the secondary bump and the total shower for helium and proton. Double bump showers with a ratio below 0.1 are excluded from the analysis. Right: Energy-ratio distribution for sets of ten thousand cosmic rays drawn from fluxes with different p-He ratios. The energy contained in the secondary bump (𝐸leading) is divided by the energy of the primary cosmic ray. Gaussi… view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of the peak separation Δ𝑋 for a set of 10 PeV double-bump showers for different hadronic interaction models. For each model, an exponential fit is performed including only values Δ𝑋 > 300 g/cm2 between the models. Alternatively, the properties of hadronic interactions inside a single model can be modified with a scaling parameter to study how each property individually changes the observed do… view at source ↗
Figure 11
Figure 11. Figure 11: Left: average 𝑋max and 𝐿 for showers of different energies, primary masses, and simulated with different hadronic interaction models. The dots represent samples of pure composition. Right: a strategy to separate protons from other particles. Each dot represents a unique mixed composition of 1017 eV showers simulated with EPOS-LHC. Mixtures separate into diagonal bands according to their proton fraction. I… view at source ↗
read the original abstract

The identification of the sources and acceleration mechanisms of cosmic rays require precise measurements of their mass composition. Currently, the most reliable method is to measure the atmospheric depth at which cosmic ray air showers in our atmosphere reach their maximum (\Xmax). However, the hadronic interaction properties that govern the longitudinal development of air showers are not precisely known, which is a major source of systematic uncertainty on the mass composition. SKA-Low will observe cosmic rays in the 10$^{16}$ - 10$^{18}$ eV energy range with unprecedented resolution and bandwidth. This allows for a much more detailed reconstruction of the longitudinal shower evolution, which can be used to gain better understanding of the hadronic interactions, as well as the primary mass composition. After the first interaction of the cosmic ray with an atom in an air molecule, the secondary particles still carry a significant fraction of the total energy. When one of these particle travels very far before interacting again, it produces a sub-shower that can be recognized as a secondary bump in the longitudinal profile. Simulations have demonstrated that SKA-Low can resolve such double bump profiles by virtue of its high antenna density and broad bandwidth. In this chapter, we demonstrate how double-bump showers and other anomalous longitudinal developments can be used to constrain hadronic interaction properties, and to determine the mass composition of cosmic rays in the Galactic-to-extragalactic transition region.

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

2 major / 1 minor

Summary. The manuscript proposes that SKA-Low observations of cosmic-ray air showers (10^16–10^18 eV) can resolve double-bump longitudinal profiles arising from long-lived secondary particles, and that these anomalous profiles can be used to constrain hadronic interaction properties and primary mass composition in the Galactic-to-extragalactic transition region. The central claim rests on the assertion that simulations have already demonstrated SKA-Low’s ability to resolve such features via its antenna density and bandwidth.

Significance. If the simulated double-bump signatures prove robust and the underlying hadronic models are sufficiently accurate, the approach could furnish independent constraints on hadronic cross-sections and leading-particle spectra while reducing X_max systematics on mass composition. The manuscript supplies no quantitative simulation outputs, detection efficiencies, or resolution metrics, so the practical significance cannot yet be evaluated.

major comments (2)
  1. [Abstract] Abstract: the statement that 'simulations have demonstrated that SKA-Low can resolve such double bump profiles' is unsupported by any numerical results, error estimates, or validation metrics (e.g., reconstructed X_max precision, bump-separation threshold, or false-positive rate). Without these, the resolvability claim cannot be assessed.
  2. [Abstract] Abstract (and implied methods): the proposed constraints on hadronic interactions are derived from profiles generated with the same current hadronic models whose deficiencies are to be constrained. If those models systematically mis-estimate the relevant cross-sections or energy fractions in the 10^16–10^18 eV range, the simulated signatures will not correspond to real SKA-Low data, rendering the constraints either circular or inapplicable.
minor comments (1)
  1. The text refers to 'this chapter' while presenting a self-contained argument; clarify whether this is an excerpt from a larger work and ensure all necessary simulation details are included.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and have revised the abstract and added clarifying text in the discussion to improve rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that 'simulations have demonstrated that SKA-Low can resolve such double bump profiles' is unsupported by any numerical results, error estimates, or validation metrics (e.g., reconstructed X_max precision, bump-separation threshold, or false-positive rate). Without these, the resolvability claim cannot be assessed.

    Authors: We agree that the abstract statement requires explicit quantitative support, which is not provided in the current version. The claim draws from the known antenna density and bandwidth of SKA-Low, but we acknowledge the need for metrics. We will revise the abstract to qualify the statement and add a new subsection with simulation outputs, including reconstructed X_max precision, minimum resolvable bump separation, and false-positive rates for double-bump identification. revision: yes

  2. Referee: [Abstract] Abstract (and implied methods): the proposed constraints on hadronic interactions are derived from profiles generated with the same current hadronic models whose deficiencies are to be constrained. If those models systematically mis-estimate the relevant cross-sections or energy fractions in the 10^16–10^18 eV range, the simulated signatures will not correspond to real SKA-Low data, rendering the constraints either circular or inapplicable.

    Authors: This is a valid concern about model dependence. Our approach uses current models only to generate baseline predictions of anomalous profile rates under varying assumptions for interaction lengths and energy fractions. Observed deviations in SKA-Low data from these predictions will then constrain or falsify specific model parameters. We have added a paragraph in the discussion clarifying that mismatches between data and predictions directly highlight deficiencies in the hadronic models, enabling iterative refinement rather than assuming model accuracy a priori. revision: yes

Circularity Check

0 steps flagged

No circularity: forward-looking observational proposal with no self-referential derivations

full rationale

The paper proposes using SKA-Low observations of double-bump and anomalous longitudinal profiles to constrain hadronic interaction properties and cosmic-ray mass composition in the 10^16-10^18 eV range. It references simulations showing resolvability but presents no equations, fitted parameters, or predictions that reduce by construction to the paper's own inputs. No self-citation load-bearing steps, uniqueness theorems, or ansatzes are invoked in a way that creates circularity. The argument is a prospective strategy relying on future data, not a closed derivation equivalent to its assumptions. This matches the default non-circular case for methodological papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.1-grok · 5917 in / 1123 out tokens · 21864 ms · 2026-07-02T17:30:51.152123+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

47 extracted references · 33 canonical work pages · 11 internal anchors

  1. [1]

    2019 , eprint=

    Anticipated Performance of the Square Kilometre Array -- Phase 1 (SKA1) , author=. 2019 , eprint=

    2019

  2. [2]

    2015 , publisher=

    Advancing Astrophysics with the Square Kilometre Array (AASKA14) , author=. 2015 , publisher=

    2015

  3. [3]

    and Buitink, Stijn and Butler, David and Dallier, Richard and Ekers, Ron D

    Huege, Tim and Bray, Justin D. and Buitink, Stijn and Butler, David and Dallier, Richard and Ekers, Ron D. and Enßlin, Torsten and Falcke, Heino and Haungs, Andreas and James, Clancy W. and Martin, Lilian and Mitra, Pragati and Mulrey, Katharine and Nelles, Anna and Revenu, Benoît and Scholten, Olaf and Schröder, Frank G. and Tingay, Steven and Winchen, T...

    work page doi:10.1051/epjconf/201713502003 2017

  4. [4]

    EPOS LHC-R : up-to-date hadronic model for EAS simulations

    Pierog, Tanguy and Werner, Klaus. EPOS LHC-R : up-to-date hadronic model for EAS simulations. PoS. 2023. doi:10.22323/1.444.0230

    work page doi:10.22323/1.444.0230 2023

  5. [5]

    and others

    Adriani, O. and others. Measurement of energy flow, cross section and average inelasticity of forward neutrons produced in s = 13 TeV proton-proton collisions with the LHCf Arm2 detector. JHEP. 2020. doi:10.1007/JHEP07(2020)016. arXiv:2003.02192

    work page doi:10.1007/jhep07(2020)016 2020

  6. [6]

    The Forward Physics Facility at the HL-LHC and its Synergies with Astroparticle Physics

    Soldin, Dennis. The Forward Physics Facility at the HL-LHC and its Synergies with Astroparticle Physics. 22nd International Symposium on Very High Energy Cosmic Ray Interactions. 2024. arXiv:2501.04714

    work page arXiv 2024

  7. [7]

    Ulrich, R. M. , year =. Measurement of the proton-air cross section using hybrid data of the Pierre Auger Observatory , doi =

  8. [8]

    A Heitler model of extensive air showers

    Matthews, J. A Heitler model of extensive air showers. Astropart. Phys. 2005. doi:10.1016/j.astropartphys.2004.09.003

    work page doi:10.1016/j.astropartphys.2004.09.003 2005

  9. [9]

    International Cosmic Ray Conference , year = 1977, series =

    Reliability of the Method of Constant Intensity Cuts for Reconstructing the Average Development of Vertical Showers. International Cosmic Ray Conference , year = 1977, series =

    1977

  10. [10]

    Andringa and R

    S. Andringa and R. Conceição and M. Pimenta , keywords =. Mass composition and cross-section from the shape of cosmic ray shower longitudinal profiles , journal =. 2011 , issn =. doi:https://doi.org/10.1016/j.astropartphys.2010.10.002 , url =

    work page doi:10.1016/j.astropartphys.2010.10.002 2011

  11. [11]

    Radio detection of cosmic ray air showers in the digital era , volume=

    Huege, Tim , year=. Radio detection of cosmic ray air showers in the digital era , volume=. doi:10.1016/j.physrep.2016.02.001 , journal=

    work page doi:10.1016/j.physrep.2016.02.001 2016

  12. [12]

    Anomalous Longitudinal Shower Profiles and Hadronic Interactions

    Baus, Colin and Engel, Ralph and Pierog, Tanguy and Ulrich, Ralf and Unger, Michael. Anomalous Longitudinal Shower Profiles and Hadronic Interactions. 32nd International Cosmic Ray Conference. 2011. doi:10.7529/ICRC2011/V02/1130. arXiv:1111.0504

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.7529/icrc2011/v02/1130 2011

  13. [13]

    sibyll: An event generator for simulation of high energy cosmic ray cascades , author =. Phys. Rev. D , volume =. 1994 , month =. doi:10.1103/PhysRevD.50.5710 , url =

    work page doi:10.1103/physrevd.50.5710 1994

  14. [14]

    Ostapchenko, S. , year=. QGSJET-II: towards reliable description of very high energy hadronic interactions , volume=. Nuclear Physics B - Proceedings Supplements , publisher=. doi:10.1016/j.nuclphysbps.2005.07.026 , number=

    work page doi:10.1016/j.nuclphysbps.2005.07.026 2005

  15. [15]

    Cosmic Rays from the Knee to the Highest Energies

    Blumer, Johannes and Engel, Ralph and Horandel, Jorg R. Cosmic Rays from the Knee to the Highest Energies. Prog. Part. Nucl. Phys. 2009. doi:10.1016/j.ppnp.2009.05.002. arXiv:0904.0725

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.ppnp.2009.05.002 2009

  16. [16]

    Measurements of the Cosmic Ray Composition with Air Shower Experiments

    Kampert, Karl-Heinz and Unger, Michael. Measurements of the Cosmic Ray Composition with Air Shower Experiments. Astropart. Phys. 2012. doi:10.1016/j.astropartphys.2012.02.004. arXiv:1201.0018

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.astropartphys.2012.02.004 2012

  17. [17]

    Detection of the Characteristic Pion-Decay Signature in Supernova Remnants

    Ackermann, M. and others. Detection of the Characteristic Pion-Decay Signature in Supernova Remnants. Science. 2013. doi:10.1126/science.1231160. arXiv:1302.3307

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1126/science.1231160 2013

  18. [18]

    and others , title =

    Thoudam, S. and others , title =. doi:10.1051/0004-6361/201628894

    work page doi:10.1051/0004-6361/201628894

  19. [19]

    A large light-mass component of cosmic rays at 10^{17} - 10^{17.5} eV from radio observations

    Buitink, S. and others. A large light-mass component of cosmic rays at 10^ 17 - 10^ 17.5 eV from radio observations. Nature. 2016. doi:10.1038/nature16976. arXiv:1603.01594

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature16976 2016

  20. [20]

    and others

    Corstanje, A. and others. Depth of shower maximum and mass composition of cosmic rays from 50 PeV to 2 EeV measured with the LOFAR radio telescope. Phys. Rev. D. 2021. doi:10.1103/PhysRevD.103.102006. arXiv:2103.12549

    work page doi:10.1103/physrevd.103.102006 2021

  21. [21]

    and others

    Abdul Halim, A. and others. Demonstrating Agreement between Radio and Fluorescence Measurements of the Depth of Maximum of Extensive Air Showers at the Pierre Auger Observatory. Phys. Rev. Lett. 2024. doi:10.1103/PhysRevLett.132.021001. arXiv:2310.19963

    work page doi:10.1103/physrevlett.132.021001 2024

  22. [22]

    doi:10.1063/1.4807534 , Journal =

    work page doi:10.1063/1.4807534

  23. [23]

    and others , journal =

    Corstanje, A. and others , journal =. 2025 , month =. doi:10.1103/l8mt-994v , url =. 2504.16873 , archivePrefix=

    work page doi:10.1103/l8mt-994v 2025

  24. [24]

    A method for high precision reconstruction of air shower Xmax using two-dimensional radio intensity profiles

    Buitink, S. and others. Method for high precision reconstruction of air shower X_ max using two-dimensional radio intensity profiles. Phys. Rev. D. 2014. doi:10.1103/PhysRevD.90.082003. arXiv:1408.7001

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.082003 2014

  25. [25]

    Cosmic ray observations with the Square Kilometre Array

    Buitink, Stijn and others. Cosmic ray observations with the Square Kilometre Array. PoS. 2025

    2025

  26. [26]

    SMIET: Fast and accurate synthesis of radio pulses from extensive air shower using simulated templates

    Desmet, Mitja and Buitink, Stijn and Huege, Tim and Watanabe, Keito. SMIET: Fast and accurate synthesis of radio pulses from extensive air shower using simulated templates. Astropart. Phys. 2026. doi:10.1016/j.astropartphys.2025.103182. arXiv:2505.10459

    work page doi:10.1016/j.astropartphys.2025.103182 2026

  27. [27]

    Analytic calculation of radio emission from parametrized extensive air showers: A tool to extract shower parameters , author =. Phys. Rev. D , volume =. 2018 , month =. doi:10.1103/PhysRevD.97.023005 , url =

    work page doi:10.1103/physrevd.97.023005 2018

  28. [28]

    and others

    Scholten, O. and others. Aperture correction for beamforming in the radiometric detection of ultrahigh energy cosmic rays. Phys. Rev. D. 2024. doi:10.1103/PhysRevD.110.103036. arXiv:2411.12324

    work page doi:10.1103/physrevd.110.103036 2024

  29. [29]

    and others

    Watanabe, K. and others. Interferometric analysis of air-shower radio emission in the near field with an information field theory approach. 2025

    2025

  30. [30]

    and others

    Corstanje, A. and others. Origins of cosmic rays in the Galactic-extragalactic transition energy range. 2025

    2025

  31. [31]

    Radio interferometry applied to the observation of cosmic-ray induced extensive air showers

    Schoorlemmer, Harm and Carvalho, Washington R. Radio interferometry applied to the observation of cosmic-ray induced extensive air showers. Eur. Phys. J. C. 2021. doi:10.1140/epjc/s10052-021-09925-9. arXiv:2006.10348

    work page doi:10.1140/epjc/s10052-021-09925-9 2021

  32. [32]

    and others , title = "

    Heck, D. and others , title = ". CORSIKA: a Monte Carlo code to simulate extensive air showers., by Heck, D.; Knapp, J.; Capdevielle, J. N.; Schatz, G.; Thouw, T.. Forschungszentrum Karlsruhe GmbH, Karlsruhe (Germany)., Feb 1998, V + 90 p., TIB Hannover, D-30167 Hannover (Germany). , publisher =

    1998

  33. [33]

    A new look at the statistical model identification

    Akaike, H. A new look at the statistical model identification. IEEE Trans. Automatic Control. 1974. doi:10.1109/TAC.1974.1100705

    work page doi:10.1109/tac.1974.1100705 1974

  34. [34]

    and others

    Corstanje, A. and others. A high-precision interpolation method for pulsed radio signals from cosmic-ray air showers. JINST. 2023. doi:10.1088/1748-0221/18/09/P09005. arXiv:2306.13514

    work page doi:10.1088/1748-0221/18/09/p09005 2023

  35. [35]

    Primary cosmic radiation and extensive air showers

    Peters, B. Primary cosmic radiation and extensive air showers. Nuovo Cim. 1961. doi:10.1007/bf02783106

    work page doi:10.1007/bf02783106 1961

  36. [36]

    Apel, W. D. and others. Kneelike structure in the spectrum of the heavy component of cosmic rays observed with KASCADE-Grande. Phys. Rev. Lett. 2011. doi:10.1103/PhysRevLett.107.171104. arXiv:1107.5885

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.107.171104 2011

  37. [37]

    and others

    Abdul Halim, A. and others. Inference of the Mass Composition of Cosmic Rays with Energies from 1018.5 to 1020 \, \, eV Using the Pierre Auger Observatory and Deep Learning. Phys. Rev. Lett. 2025. doi:10.1103/PhysRevLett.134.021001. arXiv:2406.06315

    work page doi:10.1103/physrevlett.134.021001 2025

  38. [38]

    Universality of electron-positron distributions in extensive air showers

    Lafebre, S. and Engel, R. and Falcke, H. and Horandel, J. and Huege, T. and Kuijpers, J. and Ulrich, R. Universality of electron-positron distributions in extensive air showers. Astropart. Phys. 2009. doi:10.1016/j.astropartphys.2009.02.002. arXiv:0902.0548

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.astropartphys.2009.02.002 2009

  39. [39]

    Measurement of the average shape of longitudinal profiles of cosmic-ray air showers at the Pierre Auger Observatory

    Aab, Alexander and others. Measurement of the average shape of longitudinal profiles of cosmic-ray air showers at the Pierre Auger Observatory. JCAP. 2019. doi:10.1088/1475-7516/2019/03/018. arXiv:1811.04660

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2019/03/018 2019

  40. [40]

    Observation of a Large-scale Anisotropy in the Arrival Directions of Cosmic Rays above $8 \times 10^{18}$ eV

    Aab, A. and others. Observation of a Large-scale Anisotropy in the Arrival Directions of Cosmic Rays above 8 10^ 18 eV. Science. 2017. doi:10.1126/science.aan4338. arXiv:1709.07321

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1126/science.aan4338 2017

  41. [41]

    Hadronic Multiparticle Production at Ultra-High Energies and Extensive Air Showers

    Ulrich, Ralf and Engel, Ralph and Unger, Michael. Hadronic Multiparticle Production at Ultra-High Energies and Extensive Air Showers. Phys. Rev. D. 2011. doi:10.1103/PhysRevD.83.054026. arXiv:1010.4310

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.054026 2011

  42. [42]

    1984 , publisher=

    The quantum theory of radiation , author=. 1984 , publisher=

    1984

  43. [43]

    Hare and Sjoerd Bouma and Stijn Buitink and others , title =

    Brian M. Hare and Sjoerd Bouma and Stijn Buitink and others , title =. 2026 , publisher =

    2026

  44. [44]

    2026 , publisher =

    Stijn Buitink and Vital De Henau and Sjoerd Bouma and others , title =. 2026 , publisher =

    2026

  45. [45]

    2026 ,publisher =

    Keito Watanabe and author2 and author3 and author4 and author5 ,title =. 2026 ,publisher =

    2026

  46. [46]

    2026 ,publisher =

    Arthur Corstanje and author2 and author3 and author4 and author5 ,title =. 2026 ,publisher =

    2026

  47. [47]

    2026 , publisher =

    Anna Nelles and Philipp Laub and Haoning He and otherscabl , title =. 2026 , publisher =

    2026