• [1]

    Elena Orlando and Andrew Strong, Astrophys. Space Sci. 309, 359 (2007), arXiv:astro-ph/0607563

  • [2]

    Igor V. Moskalenko, Troy A. Porter, and Seth W. Digel, Astrophys. J. 652 , L65-L68 (2006) [Erratum: Astrophys. J. 664 , L143 (2007)], arXiv: astro-ph/0607521

  • [3]

    Elena Orlando and Andrew Strong, Journal of Cosmology and Astroparticle Physics 2021, 004 (2021)

  • [4]

    Elena Orlando, Astrophys. J. 943, 173 (2023), arXiv:2212.01364[astro-ph.HE]

  • [5]

    I. V. Moskalenko, S. Karakula, and W. Tkaczyk, Astron. Astrophys. 248, L5 (1991)

  • [6]

    D. Seckel, Todor Stanev, and T. K. Gaisser, Astrophys. J. 382, 652 (1991)

  • [7]

    G. Ingelman and M. Thunman, Phys. Rev. D 54, 4385 (1996), arXiv:hep-ph/9604288[hep-ph]

  • [8]

    D. J. Thompson, D. L. Bertsch, D. J. Morrisv et al., J. Geophys. Res. 102, 14735 (1997)

  • [9]

    Elena Orlando and Andrew W. Strong, Astron. Astrophys. 480, 847 (2008), arXiv:0801.2178[astro-ph]

  • [10]

    A. A. Abdo et al. (Fermi-LAT), Astrophys. J. 734, 116 (2011), arXiv:1104.2093[astro-ph.HE]

  • [11]

    Kenny C. Y. Ng, John F. Beacom, Annika H. G. Peter et al., Phys. Rev. D 94, 023004 (2016), arXiv:1508.06276[astroph.HE]

  • [12]

    Tim Linden, Bei Zhou, John F. Beacom et al., Phys. Rev. Lett. 121, 131103 (2018), arXiv:1803.05436[astro-ph.HE]

  • [13]

    Qing-Wen Tang, Kenny C. Y. Ng, Tim Linden et al., Phys. Rev. D 98, 063019 (2018), arXiv:1804.06846[astro-ph.HE]

  • [14]

    Mehr Nisa, John Beacom, Annika Peter et al., BAAS 51, 194 (2019), arXiv:1903.06349[astroph.HE]

  • [15]

    Tim Linden, John F. Beacom, Annika H. G. Peter et al., First Observations of Solar Disk Gamma Rays over a Full Solar Cycle, (2020), arXiv: 2012.04654[astro-ph.HE]

  • [16]

    B. Bartoli et al. (ARGO-YBJ), Astrophys. J. 872, 143 (2019), arXiv:1901.04201[astro-ph.HE]

  • [17]

    A. Albert et al. (HAWC), Phys. Rev. D 98, 123011 (2018), arXiv:1808.05620[astroph.HE]

  • [18]

    A. Albert et al. (HAWC), Phys. Rev. Lett. 131, 051201 (2023), arXiv:2212.00815[astro-ph.HE]

  • [19]

    M.G. Aartsen et al. (IceCube), Searches for neutrinos from cosmic-ray interactions in the Sun using seven years of IceCube data, (2019), arXiv: 1912.13135[astro-ph.HE]

  • [20]

    C. A. Argüelles, G. de Wasseige, A. Fedynitch et al., JCAP 1707, 024 (2017), arXiv:1703.07798[astro-ph.HE]

  • [21]

    Joakim Edsjo, Jessica Elevant, Rikard Enberg et al., JCAP 1706, 033 (2017), arXiv:1704.02892[astroph.HE]

  • [22]

    Bei Zhou, Kenny C. Y. Ng, John F. Beacom et al., Phys. Rev. D 96, 023015 (2017), arXiv:1612.02420[astro-ph.HE]

  • [23]

    Baosheng Gao, Songzhan Chen, Zhe Li et al., PoS ICRC 2017, 878 (2018)

  • [24]

    M.N. Mazziotta, F. Loparco, D. Serini et al., Phys. Rev. D 102, 022003 (2020), arXiv:2006.04114[astro-ph.HE]

  • [25]

    Hugh S. Hudson, Alec MacKinnon, Mikolaj Szydlarski et al., MNRAS 491, 4852 (2020), arXiv:1910.01186[astro-ph.SR]

  • [26]

    J. Allison et al., Nucl. Instrum. Meth. A 835, 186 (2016)

  • [27]

    https://geant4-userdoc.web.cern.ch/UsersGuides/PhysicsListGuide/html/reference_PL/FTFP_BERT.html.

  • [28]

    Aatos Heikkinen, Nikita Stepanov, and Johannes Peter Wellisch, Bertini intranuclear cascade implementation in GEANT4, Proceedings, 13th International Conference on Computing in High-Enery and Nuclear Physics (CHEP 2003): La Jolla, California, March 24-28, 2003, eConf C0303241 , MOMT008 (2003), arXiv: nuclth/0306008

  • [29]

    On behalf of Geant4 Hadronic Group Julia Yarba, Recent Development and Validation of Geant4 Hadronic Physics, CHEP 2012, NEW YORK CITY, NY

  • [30]

    Kristoffer K. Andersen and Spencer R. Klein, Phys. Rev. D 83, 103519 (2011), arXiv:1103.5090[astro-ph.HE]

  • [31]

    Thomas Wiegelmann, Julia K. Thalmann, and Sami K. Solanki, A&A Rev. 22, 78 (2014), arXiv:1410.4214[astro-ph.SR]

  • [32]

    Norman H. Baker and Stefan Temesváry, Tables of Convective Stellar Envelope Models (1966).

  • [33]

    White, Marvin L. and Kim, Koo Sun, Temperature models for the outer solar atmosphere, Icarus 5 , 124129 (1966)

  • [34]

    Kenneth H. Schatten, John M. Wilcox, and Norman F. Ness, Sol Phys 6, 442 (1969)

  • [35]

    Martin D. Altschuler and Gordon Newkirk, Sol. Phys. 9, 131 (1969)

  • [36]

    J. T. Hoeksema, Structure and Evolution of the Large Scale Solar and Heliospheric Magnetic Fields., Ph.D. thesis, Stanford Univ., CA. (1984).

  • [37]

    Y. M. Wang and Jr. Sheeley, N. R., ApJ 392 , 310 (1992)

  • [38]

    https://gong.nso.edu.

  • [39]

    https://sdo.gsfc.nasa.gov/.

  • [40]

    Pete Riley, J. A. Linker, Z. Mikić et al., ApJ 653, 1510 (2006)

  • [41]

    http://www.mssl.ucl.ac.uk/surf/sswdoc/solarsoft/ssw_install_howto.html.

  • [42]

    We thank the anonymous referee for pointing this out

  • [43]

    L.J. Gleeson and W.I. Axford, Astrophys. J. 154, 1011 (1968)

  • [44]

    Marius Potgieter, Living Rev. Solar Phys. 10, 3 (2013), arXiv:1306.4421[physics.space-ph]

  • [45]

    O. Adriani et al., Astrophys. J. 765, 91 (2013), arXiv:1301.4108[astro-ph.HE]

  • [46]

    M. Aguilar et al.(AMS), Phys. Rev. Lett. 114, 171103 (2015)

  • [47]

    Y. S. Yoon et al., Astrophys. J. 728, 122 (2011), arXiv:1102.2575[astro-ph.HE]

  • [48]

    Niklas Karlsson and Tuneyoshi Kamae, Astrophys. J. 674, 278 (2008), arXiv:0709.0233[astroph]

  • [49]

    Hylke B. J. Koers, Asaf Pe’er, and Ralph A. M. J. Wijers, Parameterization of the energy and rapidity dis-tributions of secondary pions and kaons produced in energetic proton-proton collisions, (2006), arXiv: hepph/0611219

  • [50]

    M. S. Potgieter, E. E. Vos, M. Boezio et al., Sol. Phys. 289, 391 (2014), arXiv:1302.1284[astro-ph.SR]

  • [51]

    Z Fujii and FB McDonald, Advances in Space Research 35, 611616 (2005)

  • [52]

    Zhe Li, Songzhan Chen, Yuncheng Nan et al., PoS ICRC 2019, 1182 (2019)

  • [53]

    T.T. Böhlen, F. Cerutti, M.P.W. Chin et al., Nucl. Data Sheets 120, 211 (2014)

  • [54]

    F. W. Stecker, Nature Physical Science 242, 59 (1973)

  • [55]

    R. Ramaty, B. Kozlovsky, and R. E. Lingenfelter, Space Sci. Rev. 18, 341 (1975)

  • [56]

    Michael Kachelriess, Igor V. Moskalenko, and Sergey S. Ostapchenko, Astrophys. J. 789, 136 (2014), arXiv:1406.0035[astro-ph.HE]

  • [57]

    N. Grevesse and A. J. Sauval, Advances in Space Research 30, 3 (2002)

  • [58]

    M. Amenomori et al. (Tibet ASgamma), Phys. Rev. Lett. 111, 011101 (2013), arXiv:1306.3009

  • [59]

    M. Amenomori, X. J. Bi, D. Chen et al, Phys. Rev. Lett. 120, 031101 (2018), arXiv:1801.06942[astro-ph.SR]

  • [60]

    M.G. Aartsen et al. (IceCube), Astrophys. J. 872, 133 (2019), arXiv:1811.02015[astro-ph.HE]

  • [61]

    Julia Becker Tjus, Paolo Desiati, Niklas Döpper et al., Astron. Astrophys. 633, A83 (2020), arXiv:1903.12638[astro-ph.HE]

  • [62]

    M. G. Aartsen et al. (IceCube Collaboration), Phys. Rev. D 103, 042005 (2021), arXiv:2006.16298

  • [63]

    Miguel Gutiérrez and Manuel Masip, Astropart. Phys. 119, 102440 (2020), arXiv:1911.07530[hep-ph]

  • [64]

    Prabir Banik, Arunava Bhadra, and Sanjay K. Ghosh, Sun is a cosmic ray TeVatron, (2023), arXiv: 2305.17086[astro-ph.HE]

  • [65]

    I. V. Moskalenko and S. Karakula, J. Phys. G 19, 1399 (1993)

  • [66]

    M. Masip, Astropart. Phys. 97, 63 (2018), arXiv:1706.01290[hepph]

  • [67]

    William H. Press and David N. Spergel, Astrophys. J. 296, 679 (1985)

  • [68]

    Lawrence M. Krauss, K. Freese, W. Press et al., Astrophys. J. 299, 1001 (1985)

  • [69]

    Joseph Silk, Keith A. Olive, and Mark Srednicki, Phys. Rev. Lett. 55, 257 (1985)

  • [70]

    Andrew Gould, Astrophys. J. 321, 571 (1987)

  • [71]

    Andrew Gould, Astrophys. J. 321, 560 (1987)

  • [72]

    Annika H.G. Peter, Phys. Rev. D 79, 103532 (2009), arXiv:0902.1347[astro-ph.HE]

  • [73]

    Nicole F. Bell and Kalliopi Petraki, JCAP 04, 003 (2011), arXiv:1102.2958

  • [74]

    Carl Niblaeus, Ankit Beniwal, and Joakim Edsjo, JCAP 11, 011 (2019), arXiv:1903.11363

  • [75]

    Brian Batell, Maxim Pospelov, Adam Ritz et al., Phys. Rev. D 81, 075004 (2010), arXiv:0910.1567[hep-ph]

  • [76]

    Rebecca K. Leane, Kenny C. Y. Ng, and John F. Beacom, Phys. Rev. D 95, 123016 (2017), arXiv:1703.04629[astro-ph.HE]

  • [77]

    Chiara Arina, Mihailo Backović, Jan Heisig et al., Phys. Rev. D 96, 063010 (2017), arXiv:1703.08087[astro-ph.HE]

  • [78]

    M.G. Aartsen et al. (IceCube), Eur. Phys. J. C 77, 146 (2017), arXiv:1612.05949[astro-ph.HE]

  • [79]

    S. Adrian-Martinez et al. (ANTARES), Phys. Lett. B 759, 69 (2016), arXiv:1603.02228[astro-ph.HE]

  • [80]

    S. Adrián-Martínez et al. (ANTARES), JCAP 05, 016 (2016), arXiv:1602.07000

  • [81]

    A. Albert et al., Phys. Rev. D 98, 123012 (2018), arXiv:1808.05624[hepph]

  • [82]

    A. Cuoco, P. De La Torre Luque, F. Gargano et al., Phys. Rev. D 101, 022002 (2020), arXiv:1912.09373[astro-ph.HE]

  • [83]

    Kenny C. Y. Ng, John F. Beacom, Annika H. G. Peter et al., Phys. Rev. D 96, 103006 (2017), arXiv:1703.10280[astro-ph.HE]

  • [84]

    R. D. Strauss, M. S. Potgieter, and S. E. S. Ferreira, Adv. Space Res. 49, 392 (2012)

  • [85]

    Mathew J. Owens and Robert J. Forsyth, Living Reviews in Solar Physics 10, 5 (2013)

  • [86]

    Sami K. Solanki, Bernd Inhester, and Manfred Schussler, Rept. Prog. Phys. 69, 563 (2006), arXiv:1008.0771[astro-ph.SR]

  • [87]

    Thomas Wiegelmann and Takashi Sakurai, Living Rev. Sol. Phys. 9, 5 (2012), arXiv:1208.4693

  • [88]

    D. H. Mackay and A. R. Yeates, Living Rev. Sol. Phys. 9, 6 (2012), arXiv:1211.6545

  • [89]

    Zihao Yang, Christian Bethge, Hui Tian et al., Science 369(6504), 694 (2020)