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arxiv: 1906.11477 · v2 · pith:T4XHQYXRnew · submitted 2019-06-27 · 🌌 astro-ph.HE · astro-ph.SR· hep-ph· physics.space-ph

Numerical modeling of cosmic-ray transport in the heliosphere and interpretation of the proton-to-helium ratio in Solar Cycle 24

Nicola Tomassetti , Fernando Bar\~ao , Bruna Bertucci , Emanuele Fiandrini , Miguel Orcinha This is my paper

Pith reviewed 2026-05-25 14:45 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.SRhep-phphysics.space-ph
keywords cosmic raysheliospheric transportproton-to-helium ratiodiffusion coefficientsolar modulationsolar cycle 24
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The pith

The time variation of the cosmic-ray proton-to-helium ratio below 3 GV arises from mass-to-charge dependent diffusion during heliospheric transport.

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

The paper develops numerical models of cosmic-ray transport through the Milky Way and the heliosphere to connect diffusion properties to the observed evolution of the proton-to-helium ratio over Solar Cycle 24. It shows that the long-term behavior at rigidities below 3 GV follows from a diffusion coefficient that depends on particle mass and charge, producing different modulation for protons and helium nuclei. A sympathetic reader would care because this links measurable ratio changes directly to the physics of particle scattering off solar-wind irregularities rather than to source or galactic effects. The work uses both full numerical solutions and approximate analytical relations to isolate how the rigidity dependence of diffusion shapes the ratio's time profile.

Core claim

Within the transport model the time-dependent proton-to-helium ratio at R < 3 GV is reproduced by a diffusion coefficient whose magnitude scales with particle mass and charge; this scaling causes protons and helium to experience different levels of solar modulation as the heliospheric conditions evolve through the solar cycle, and the resulting ratio pattern matches the AMS monthly data without requiring changes in the interstellar source composition.

What carries the argument

Mass/charge-dependent diffusion coefficient for low-rigidity cosmic rays in the heliosphere, which produces differential modulation between protons and helium nuclei.

If this is right

  • The ratio pattern supplies a direct constraint on the rigidity and mass/charge scaling of the heliospheric diffusion coefficient.
  • Numerical heliospheric models that incorporate this scaling can be used to correct low-rigidity fluxes measured at Earth for solar modulation.
  • Similar ratio measurements for other light nuclei should exhibit the same mass/charge-driven time dependence if the mechanism is correct.

Where Pith is reading between the lines

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

  • The same diffusion scaling may affect the interpretation of other low-rigidity secondary-to-primary ratios during solar maximum and minimum.
  • If the mass/charge dependence is confirmed, it provides a new observable for testing models of cosmic-ray scattering on magnetic turbulence in the inner heliosphere.
  • Extending the model to include charge-sign dependent drifts could predict different ratio behaviors for positive and negative particles in future solar cycles.

Load-bearing premise

The observed time-dependent changes in the proton-to-helium ratio are caused mainly by heliospheric transport rather than by time-varying source composition or galactic propagation.

What would settle it

A data set in which the proton-to-helium ratio at fixed rigidity shows no solar-cycle variation after the source composition is held fixed would falsify the claim.

Figures

Figures reproduced from arXiv: 1906.11477 by Bruna Bertucci, Emanuele Fiandrini, Fernando Bar\~ao, Miguel Orcinha, Nicola Tomassetti.

Figure 1
Figure 1. Figure 1: Calculations of the B/C ratio and CR propagation uncer￾tainties before and after the use of the new data from AMS (Aguilar et al., 2018c). heavy ions data are available at 110-600 MeV/n (Cum￾mings et al., 2016). These data have been collected by Voyager-1 in the period 2012-2016, i.e., while the space￾craft were moving in the interstellar space. Thus they provide direct and valuable constraints to CR propa… view at source ↗
Figure 3
Figure 3. Figure 3: Proton and helium data from AMS and Voyager-1 (Aguilar et al., 2015a,b; Cummings et al., 2016), in comparison with LIS from various works: long-dashed red (Tomassetti et al., 2017), dashed or￾ange (Boschini et al., 2017), dot-dashed black (Corti et al., 2016), dotted blue (Tomassetti, 2017a), dashed pink (Tomassetti et al., 2018), and solid green line (Tomassetti, 2015a, 2017b). propagation in two diffusiv… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Measured and calculated proton and helium LIS’s. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Modulated energy spectra from 16 selected epochs. The calculations show the best-fit proton fluxes (pink), the predicted helium [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Time-series of the best-fit k0-values derived with the AMS proton data at R = 1 − 60 GV (Aguilar et al., 2018a); (b) time profile of the proton flux measured by AMS at R = 2 GV in comparison with calculations, LIS uncertainties and uncertainties from the fit; (c) time profile of the helium flux measured by AMS at R = 2 GV in comparison with best-fit calculations from the proton-driven fits (for K = βR)… view at source ↗
Figure 6
Figure 6. Figure 6: (a) time profiles of the p/He ratio evaluated the at rigidity R = 2.2, 2.5, 2.8, 3.4, 5.1, 7.4, 10.5, 15, and 22 GV (from top to bottom); (b) rigidity dependence of the ratio Γp/He = [p/He]t2 /[p/He]t1 calculated for February 2014 (t1) and May 2017 (t2). absorbed in the fit, i.e., by a proper scaling of the k0-values and the nuisance parameters. The relevant uncertainties in the proton flux are shown by th… view at source ↗
Figure 7
Figure 7. Figure 7: (a) time profiles of the p/He ratio evaluated at rigidity R = 2.2, 2.5, 2.8, 3.4, 5.1, 7.4, 10.5, 15, and 22 GV (from top to bottom); (b) rigidity dependence of the ratio Γp/He = [p/He]t2 /[p/He]t1 calculated for February 2014 (t1) and May 2017 (t2). In both plots, the new data from AMS are compared with convection-diffusion calculations. fluxes Based on such a correlation, one may expect that the p/He rat… view at source ↗
Figure 8
Figure 8. Figure 8: (a) time profiles of the p/He ratio evaluated the rigidity R = 2.2, 2.5, 2.8, 3.4, 5.1, 7.4, 10.5, 15, and 22 GV (from top to bottom); (b) rigidity dependence of the ratio Γp/He = [p/He]t2 /[p/He]t1 calculated for February 2014 (t1) and May 2017 (t2). In both plots, the new data from AMS are compared with the broken line fits. current moves always outward, following the solar wind, the key CDA hypothesis i… view at source ↗
read the original abstract

Thanks to space-borne experiments of cosmic-ray (CR) detection, such as the AMS and PAMELA missions in low-Earth orbit, or the Voyager-1 spacecraft in the interstellar space, a large collection of multi-channel and time-resolved CR data has become available. Recently, the AMS experiment has released new precision data, on the proton and helium fluxes in CRs, measured on monthly basis during its first six years of mission. The AMS data reveal a remarkable long-term behavior in the temporal evolution of the proton-to-helium ratio at rigidity $R = p/Z <$ 3 GV. As we have argued in a recent work, such a behavior may reflect the transport properties of low-rigidity CRs in the inteplanetary space. In particular, it can be caused by mass/charge dependence of the CR diffusion coefficient. In this paper, we present our developments in the numerical modeling of CR transport in the Milky Way and in the heliosphere. Within our model, and with the help of approximated analytical solutions, we describe in details the relations between the properties of CR diffusion and the time-dependent evolution of the proton-to-helium ratio.

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

1 major / 0 minor

Summary. The manuscript develops numerical models of cosmic-ray transport in the heliosphere (and Milky Way) and uses them, together with approximated analytical solutions, to interpret the long-term temporal evolution of the proton-to-helium ratio at R < 3 GV reported by AMS during Solar Cycle 24. The central claim is that this behavior is produced by a mass/charge dependence in the heliospheric diffusion coefficient.

Significance. If the central claim is substantiated, the work would show that time-dependent heliospheric modulation with explicit (A/Z) dependence can reproduce observed low-rigidity ratio variations, thereby providing a transport-based explanation that does not require time-varying source composition or galactic propagation changes. This would strengthen the interpretive power of combined numerical-plus-analytical heliospheric models for precision CR data.

major comments (1)
  1. [Sections describing the analytical approximations and their application to the p/He ratio] The central claim requires that the approximated analytical solutions remain accurate when the diffusion coefficient carries explicit mass/charge dependence and the modulation potential varies over the multi-year timescale of Solar Cycle 24. No explicit validation (e.g., direct comparison of analytical vs. full numerical solutions for p and He fluxes under the same (A/Z)-dependent diffusion) is described for the strongly modulated, time-dependent regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The manuscript combines numerical transport modeling with analytical approximations to interpret the AMS proton-to-helium ratio variations. Below we address the single major comment point by point. We agree that additional validation will strengthen the presentation and will incorporate it in the revision.

read point-by-point responses

  1. Referee: [Sections describing the analytical approximations and their application to the p/He ratio] The central claim requires that the approximated analytical solutions remain accurate when the diffusion coefficient carries explicit mass/charge dependence and the modulation potential varies over the multi-year timescale of Solar Cycle 24. No explicit validation (e.g., direct comparison of analytical vs. full numerical solutions for p and He fluxes under the same (A/Z)-dependent diffusion) is described for the strongly modulated, time-dependent regime.

    Authors: We agree that explicit validation of the analytical approximations under simultaneous (A/Z) dependence and time variation of the modulation potential is important for substantiating the central claim. Our analytical solutions are derived from the time-dependent transport equation with a rigidity- and (A/Z)-dependent diffusion coefficient; they were previously cross-checked against the numerical code in the steady-state limit and for (A/Z)-independent cases. However, a direct side-by-side comparison for the full time-dependent, (A/Z)-dependent heliospheric modulation over Solar Cycle 24 is not shown. In the revised manuscript we will add this comparison: we will run the numerical model with the same (A/Z)-dependent diffusion coefficient and time-varying modulation potential used in the analytical treatment, then overlay the resulting proton and helium fluxes (and their ratio) against the analytical predictions at rigidities below 3 GV. This will quantify the accuracy of the approximations in the strongly modulated regime relevant to the AMS data. revision: yes

Circularity Check

1 steps flagged

Central interpretation of p/He ratio evolution relies on self-cited prior argument for (A/Z)-dependent diffusion

specific steps
  1. self citation load bearing [Abstract]
    "As we have argued in a recent work, such a behavior may reflect the transport properties of low-rigidity CRs in the inteplanetary space. In particular, it can be caused by mass/charge dependence of the CR diffusion coefficient. In this paper, we present our developments in the numerical modeling of CR transport in the Milky Way and in the heliosphere. Within our model, and with the help of approximated analytical solutions, we describe in details the relations between the properties of CR diffusion and the time-dependent evolution of the proton-to-helium ratio."

    The paper states that the observed ratio evolution 'can be caused by mass/charge dependence' and then uses its numerical/analytical framework to 'describe the relations' between that dependence and the ratio. The justification for invoking the dependence at all is the self-citation ('as we have argued in a recent work'), with no independent derivation or external validation supplied here. The central interpretive claim therefore reduces to the prior paper's conclusion rather than emerging from the present model's first-principles transport equations.

full rationale

The paper's core claim attributes the long-term p/He ratio behavior at R<3 GV to mass/charge-dependent heliospheric diffusion. This attribution is introduced via explicit self-citation to the authors' recent work rather than derived from the numerical model or external constraints within this manuscript. The numerical modeling and analytical approximations are then used to relate diffusion properties to the ratio evolution, but the load-bearing premise (that the dependence exists and drives the observation) traces directly to the self-citation. This creates partial circularity because the interpretation reduces to re-describing the same dataset under an assumption justified only by prior work from the same group. No machine-checked theorem, parameter-free external benchmark, or independent falsification is shown for the (A/Z) dependence itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the diffusion coefficient carries an explicit mass-to-charge dependence whose functional form is chosen to reproduce the AMS ratio time series; no independent evidence for that functional form is supplied in the abstract.

free parameters (1)
  • mass/charge scaling exponent in diffusion coefficient
    The dependence is introduced to match the observed ratio evolution and is therefore fitted rather than derived from first principles.
axioms (1)
  • domain assumption Heliospheric transport dominates the time variation of the low-rigidity p/He ratio
    Invoked in the abstract to attribute the AMS pattern to diffusion properties rather than galactic or source effects.

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 2 internal anchors

  1. [1]

    C., Bazilevskaya, G

    Adriani, O., Barbarino, G. C., Bazilevskaya, G. A., et al., 2013,Time dependence of the proton flux measured by PAMELA during the 2006 July - 2009 December solar minimum, Astrophys. J. 765, 91; doi: 10.1088/0004-637X/765/2/91

    work page doi:10.1088/0004-637x/765/2/91 2013

  2. [3]

    Aguilar, M., Ali Cavasonza, L., Alpat, B., et al., 2018 (b), Obser- vation of Complex Time Structures in the Cosmic-Ray Electron and Positron Fluxes with the Alpha Magnetic Spectrometer on the International Space Station, Phys. Rev. Lett. 121, 051102 doi: 10.1103/PhysRevLett.121.051102

    work page doi:10.1103/physrevlett.121.051102 2018

  3. [4]

    Aguilar, M., Ali Cavasonza, L., Ambrosi, G., et al., 2018 (c) Ob- servation of New Properties of Secondary Cosmic Rays Lithium, Beryllium, and Boron by the Alpha Magnetic Spectrometer on the International Space Station, Phys. Rev. Lett. 120, 021101 doi: 10.1103/PhysRevLett.120.021101

    work page doi:10.1103/physrevlett.120.021101 2018

  4. [5]

    Aguilar, M., Aisa, D., Alpat, B., et al., 2015 (a), Precision Measure- ment of the Proton Flux in Primary Cosmic Rays from Rigidity 1 GV to 1.8 TV with the Alpha Magnetic Spectrometer on the International Space Station , Phys. Rev. Lett. 114, 171103; doi: 10.1103/PhysRevLett.114.171103

    work page doi:10.1103/physrevlett.114.171103 2015

  5. [6]

    Aguilar, M., Aisa, D., Alpat, B., et al., 2015 (b), Precision Measure- ment of the Helium Flux in Primary Cosmic Rays of Rigidities 1.9 GV to 3 TV with the Alpha Magnetic Spectrometer on the International Space Station , Phys. Rev. Lett. 115, 211101; doi: 10.1103/PhysRevLett.115.211101

    work page doi:10.1103/physrevlett.115.211101 2015

  6. [7]

    Space Res

    Bindi, V., Corti, C., Consolandi, C., Hoffman, J., Whitman, K., 2017, Overview of galactic cosmic ray solar modula- tion in the AMS-02 era , Adv. Space Res. 60, 865-878 doi: 10.1016/j.asr.2017.05.025

    work page doi:10.1016/j.asr.2017.05.025 2017

  7. [8]

    Biswas, S., Ramadurai, S., Sreenivasan, N., 1967, 3He/(4He +3He) Ratios in Cosmic Rays, Path Lengths in Space, and Energy Spec- trum of Helium Nuclei in Local Interstellar Space, Phys. Rev. 159, 1063-1069 doi: 10.1103/PhysRev.159.1063 Bobik P., Boschini, M. J., Della Torre, S., et al., 2016, On the forward-backward-in-time approach for Monte Carlo solutio...

    work page doi:10.1103/physrev.159.1063 1967

  8. [9]

    The HelMod Model in the Works for Inner and Outer Heliosphere: from AMS to Voyager Probes Observations

    Boschini, M. J., Della Torre, S., Gervasi, M., La Vacca, G., Rancoita, P. G., 2019, The HelMod Model in the Works for Inner and Outer Heliosphere: from AMS to Voyager Probes Observations , Adv. Space Res. doi: 10.1016/j.asr.2019.04.007 [arXiv:1903.07501]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.asr.2019.04.007 2019

  9. [11]

    S., Bindi, V., et al., 2019, Numerical model- ing of galactic cosmic ray proton and helium observed by AMS-02 during the solar maximum of Solar Cycle 24 Astrophys

    Corti, C., Potgieter, M. S., Bindi, V., et al., 2019, Numerical model- ing of galactic cosmic ray proton and helium observed by AMS-02 during the solar maximum of Solar Cycle 24 Astrophys. J. 871, 253; doi: 10.3847/1538-4357/aafac4

    work page doi:10.3847/1538-4357/aafac4 2019

  10. [12]

    Corti, C., Bindi, V., Consolandi, C., and Whitman, K., 2016, So- lar Modulation of the Local Interstellar Spectrum with Voyager 1, AMS-02, PAMELA, and BESS , Astrophys. J. 829, 8; doi: 10.3847/0004-637X/829/1/8

    work page doi:10.3847/0004-637x/829/1/8 2016

  11. [13]

    C, Stone, E

    Cummings, A. C, Stone, E. C., Heikkila, B. C., et al., 2016, Galactic cosmic rays in the local interstellar medium: Voyager 1 observa- tions and model results , Astrophys. J. 831, 18 doi: 10.3847/0004- 637X/831/1/18 Di Felice, V, Pizzolotto, C., D’Urso, D., et al., 2017, Looking for cosmic ray data? The ASI Cosmic Ray Database , Proc. 35 th ICRC - Bexco, ...

    work page doi:10.3847/0004- 2016

  12. [14]

    O’C., & Strong, A

    Drury, L. O’C., & Strong, A. W., 2017, Power requirements for cosmic ray propagation models involving diffusive reaccelera- tion; estimates and implications for the damping of interstellar turbulence, Astron. & Astrophys.597, A117 doi: 10.1051/0004- 6361/201629526

    work page doi:10.1051/0004- 2017

  13. [15]

    Feng, J., Tomassetti, N., Oliva, A., 2016, Bayesian analysis of spatial-dependent cosmic-ray propagation: Astrophysical back- ground of antiprotons and positrons , Phys. Rev. D 94, 123007 doi: 10.1103/PhysRevD.94.123007

    work page doi:10.1103/physrevd.94.123007 2016

  14. [16]

    A., 1971, Solar modulation of galactic cosmic rays, 2 , Space Phys

    Fisk, L. A., 1971, Solar modulation of galactic cosmic rays, 2 , Space Phys. 76, 221-226 doi: 10.1029/JA076i001p00221

    work page doi:10.1029/ja076i001p00221 1971

  15. [17]

    A., Guzik, T

    Garcia-Munoz, M., Simpson, J. A., Guzik, T. G., Wefel, J. P., Mar- golis, S. H., 1987, Cosmic-ray propagation in the Galaxy and in the heliosphere - The path-length distribution at low energy As- trophys. J. SS 64, 269-304; doi: 10.1086/191197

    work page doi:10.1086/191197 1987

  16. [18]

    Gieseler, J., Heber, B., Herbst, K., 2017, An Empirical Modification of the Force Field Approach to Describe the Modulation of Galac- tic Cosmic Rays Close to Earth in a Broad Range of Rigidities , J. Geophys. Res. 122, 10964-10979 doi: 10.1002/2017JA024763

    work page doi:10.1002/2017ja024763 2017

  17. [19]

    J., & Urch, I

    Gleeson, L. J., & Urch, I. H., 1971, Energy losses and modulation of galactic cosmic rays , Astrophys. Space Sci. 11, 288-308 doi: 10.1007/BF00661360

    work page doi:10.1007/bf00661360 1971

  18. [21]

    A., Black, J

    Grenier, I. A., Black, J. H., and Strong, A. W., 2015, The nine lives of cosmic rays in Galaxies , Annu. Rev. Astron. Astrophys. 53, 199-246; doi: 10.1146/annurev-astro-082214-122457

    work page doi:10.1146/annurev-astro-082214-122457 2015

  19. [22]

    S., and Ferreira, S

    Heber, B., Kopp, A., Gieseler, J., M¨ uller-Mellin, R., Fitchner, H., Scherer, K., Potgieter, M. S., and Ferreira, S. E. S., 2009, Modulation of Galactic cosmic ray protons and electrons dur- ing an unusual solar minimum , Astrophys. J. 699, 1956-1963; doi: 10.1088/0004-637X/699/2/1956

    work page doi:10.1088/0004-637x/699/2/1956 2009

  20. [23]

    Herbst, K., Muscheler, R., Heber, B., 2017, The new local inter- stellar spectra and their influence on the production rates of the cosmogenic radionuclides 10Be and 14C J. Geophys. Res. 122, 23-34 doi: 10.1002/2016JA023207

    work page doi:10.1002/2016ja023207 2017

  21. [24]

    A., & Jokipii, J

    Isenberg, P. A., & Jokipii, J. R., 1978, Effects of particle drift on cosmic ray transport. II - Analytical solution to the modulation problem with no latitudinal diffusion , Astrophys. J. 219, 740-749 doi: 10.1086/155832

    work page doi:10.1086/155832 1978

  22. [25]

    R., Levy, E

    Jokipii, J. R., Levy, E. H., Hubbard, W. B., 1977, Effects of par- ticle drift on cosmic-ray transport. I - General properties, ap- plication to solar modulation , Astrophys. J. 213, 861-868 doi: 10.1086/155218

    work page doi:10.1086/155218 1977

  23. [26]

    R., 1967, Cosmic-Ray Propagation

    Jokipii, J. R., 1967, Cosmic-Ray Propagation. Ii. Diffusion in the Interplanetary Magnetic Field , Astrophys. J. 149, 405-415 doi: 10.1086/149265

    work page doi:10.1086/149265 1967

  24. [27]

    F., 1964, Diffusive mechanism of the diurnal cosmic ray variation, Geomagnetism & A´ eronomy 4, 977-987

    Krymsky, G. F., 1964, Diffusive mechanism of the diurnal cosmic ray variation, Geomagnetism & A´ eronomy 4, 977-987

    work page 1964

  25. [28]

    Y., Belov, A., 2000, Cosmic Rays in Relation to Space Weather Space Sci

    Kudela, K., Storini, M., Hofer, M. Y., Belov, A., 2000, Cosmic Rays in Relation to Space Weather Space Sci. Rev. 93, 153-174 doi: 10.1023/A:1026540327564 K¨ uhl, P., G´ omez-Herrero, R., and Heber, B., 2016,Annual Cosmic Ray Spectra from 250 MeV up to 1.6 GeV from 1995 - 2014 Measured With the Electron Proton Helium Instrument onboard

    work page doi:10.1023/a:1026540327564 2000

  26. [29]

    291, 965-974; doi: 10.1007/s11207-016-0879-0

    SOHO, Solar Phys. 291, 965-974; doi: 10.1007/s11207-016-0879-0

    work page doi:10.1007/s11207-016-0879-0

  27. [30]

    S., Bindi, V., Zhang, M., Feng, X., 2019, A Numerical Study of Cosmic Proton Modulation Using AMS-02 Observations , Astrophys

    Luo, X., Potgieter, M. S., Bindi, V., Zhang, M., Feng, X., 2019, A Numerical Study of Cosmic Proton Modulation Using AMS-02 Observations , Astrophys. J. 878, 6; doi: 10.3847/1538- 4357/ab1b2a

    work page doi:10.3847/1538- 2019

  28. [31]

    Manuel, R., Ferreira, S. E. S., and Potgieter, M. S., 2014, Time- dependent modulation of cosmic rays in the Heliosphere , Solar Phys. 289, 2207-2231 doi: 10.1007/s11207-013-0445-y

    work page doi:10.1007/s11207-013-0445-y 2014

  29. [32]

    Martucci, M., Munini, R., Boezio, M., et al., 2018, Proton Fluxes Measured by the PAMELA Experiment from the Minimum to the Maximum Solar Activity for Solar Cycle 24 Astrophys. J. 854, L2 doi: 10.3847/2041-8213/aaa9b2

    work page doi:10.3847/2041-8213/aaa9b2 2018

  30. [33]

    176 299–319 doi: 10.1007/s11214-011-9819-3

    Moraal, H., 2013, Cosmic-Ray Modulation Equations, Space Sci Rev. 176 299–319 doi: 10.1007/s11214-011-9819-3

    work page doi:10.1007/s11214-011-9819-3 2013

  31. [34]

    N., 1965, The passage of energetic charged particles through interplanetary space , 1965, Planet

    Parker, E. N., 1965, The passage of energetic charged particles through interplanetary space , 1965, Planet. Space Sci. 13, 9-49 doi: 10.1016/0032-0633(65)90131-5

    work page doi:10.1016/0032-0633(65)90131-5 1965

  32. [35]

    S., 2014, The charge-sign dependent effect in the solar modulation of cosmic rays, Adv

    Potgieter, M. S., 2014, The charge-sign dependent effect in the solar modulation of cosmic rays, Adv. Space Res. 53-10, 1415-1425; doi: 12 10.1016/j.asr.2013.04.015

    work page doi:10.1016/j.asr.2013.04.015 2014

  33. [36]

    S., 2013, Solar modulation of cosmic rays, Living Rev

    Potgieter, M. S., 2013, Solar modulation of cosmic rays, Living Rev. Solar Phys., 10, 3 doi: 10.12942/lrsp-2013-3

    work page doi:10.12942/lrsp-2013-3 2013

  34. [37]

    N., 2006, Dissipation of Magnetohydrodynamic Waves on Energetic Particles: Impact on Interstellar Turbu- lence and Cosmic Ray Transport, Astrophys

    Zirakashvili, V. N., 2006, Dissipation of Magnetohydrodynamic Waves on Energetic Particles: Impact on Interstellar Turbu- lence and Cosmic Ray Transport, Astrophys. J. 642, 902-916; doi: 10.1086/501117

    work page doi:10.1086/501117 2006

  35. [38]

    D., Potgieter, M

    Strauss, R. D., Potgieter, M. S., Kopp, A., Buesching, I., 2011, On the propagation times and energy losses of cosmic rays in the heliosphere , J. Geophys. Res. 116, A12105 (2011) doi: 10.1029/2011JA016831

    work page doi:10.1029/2011ja016831 2011

  36. [39]

    Teufel, A., & Schlickeiser, R., 2002,Analytic calculation of the paral- lel mean free path of heliospheric cosmic rays. I. Dynamical mag- netic slab turbulence and random sweeping slab turbulence , As- tron. & Astrophys. 393, 703-715 doi: 10.1051/0004-6361:20021046

    work page doi:10.1051/0004-6361:20021046 2002

  37. [40]

    Tomassetti, N., Bertucci, B., Bar˜ ao, F., et al., 2018, Testing Diffu- sion of Cosmic Rays in the Heliosphere with Proton and Helium Data from AMS, Phys. Rev. Lett. 121, 251104; doi: 10.1103/Phys- RevLett.121.251104

    work page doi:10.1103/phys- 2018

  38. [41]

    Tomassetti, N., Orcinha, M., Bar˜ ao, F., Bertucci, B., 2017,Evidence for a Time Lag in Solar Modulation of Galactic Cosmic Rays , Astrophys. J. Lett. 849, 32 doi: 10.3847/2041-8213/aa9373

    work page doi:10.3847/2041-8213/aa9373 2017

  39. [42]

    Tomassetti, N., 2017 (a), Solar and nuclear physics uncertain- ties in cosmic-ray propagation , Phys. Rev. D 96, 103005 doi: 10.1103/PhysRevD.96.103005

    work page doi:10.1103/physrevd.96.103005 2017

  40. [43]

    Space Res

    Tomassetti, N., 2017 (b), Testing universality of cosmic-ray accel- eration with proton/helium data from AMS and Voyager-1 Adv. Space Res. 60, 815-825 doi: 10.1016/j.asr.2016.10.024

    work page doi:10.1016/j.asr.2016.10.024 2017

  41. [44]

    & Feng, J., 2017, The curious case of high-energy deuterons in Galactic cosmic rays , Astrophys

    Tomassetti, N. & Feng, J., 2017, The curious case of high-energy deuterons in Galactic cosmic rays , Astrophys. J. Lett. 835, L26 doi: 10.3847/2041-8213/835/2/L26

    work page doi:10.3847/2041-8213/835/2/l26 2017

  42. [45]

    Tomassetti, N., 2015 (a), Origin of the proton-to-helium ratio anomaly in cosmic rays , Astrophys. J. Lett. 815, L1; doi: 10.1088/2041-8205/815/1/L1

    work page doi:10.1088/2041-8205/815/1/l1 2015

  43. [46]

    Tomassetti, N., 2015 (b), Cosmic-ray protons, nuclei, electrons, and antiparticles under a two-halo scenario of diffusive propagation , Phys. Rev. D 92, 081301(R); doi: 10.1103/PhysRevD.92.081301

    work page doi:10.1103/physrevd.92.081301 2015

  44. [47]

    Tomassetti, N., 2015 (c), Examination of uncertainties in nuclear data for cosmic ray physics with the AMS experiment , Phys. Rev. C 92, 045808; doi: 10.1103/PhysRevC.92.045808

    work page doi:10.1103/physrevc.92.045808 2015

  45. [48]

    Tomassetti, N., 2012 (a),Origin of the cosmic ray spectral hardening, Astrophys. J. Lett. 752, L13; doi: 10.1088/2041-8205/752/1/L13

    work page doi:10.1088/2041-8205/752/1/l13 2012

  46. [49]

    Space Sci

    Tomassetti, N., 2012 (b), Propagation of H and He cosmic ray iso- topes in the Galaxy: astrophysical and nuclear uncertainties , As- trophys. Space Sci. 342, 131-136; doi: 10.1007/s10509-012-1138-y

    work page doi:10.1007/s10509-012-1138-y 2012

  47. [50]

    V., et al., 2011, Constraints on cosmic ray propagation models from a global Bayesian analysis , Astrophys

    Trotta, R., Johannesson, G., Moskalenko, I. V., et al., 2011, Constraints on cosmic ray propagation models from a global Bayesian analysis , Astrophys. J. 729, 106 doi: 10.1088/0004- 637X/729/2/106

    work page doi:10.1088/0004- 2011

  48. [51]

    Usoskin, I. G., G. A. Bazilevskaya, and G. A. Kovaltsov, 2011, Solar modulation parameter for cosmic rays since 1936 reconstructed from ground-based neutron monitors and ionization chambers , J. Geophys. Res. 116, A02104 doi: 10.1029/2010JA016105

    work page doi:10.1029/2010ja016105 2011

  49. [52]

    G., Alanko-Huotari, K., Kovaltsov, G

    Usoskin, I. G., Alanko-Huotari, K., Kovaltsov, G. A., and Mursula, K., 2005, Heliospheric modulation of cosmic rays: Monthly re- construction for 1951-2004 , J. Geophys. Res. 110, A12108 doi: 10.1029/ 2005JA011250

    work page 2005

  50. [53]

    W., Johannesson, G., et al., 2011, GAL- PROP WebRun: An internet-based service for calculating galactic cosmic ray propagation and associated photon emissions , Comp

    Vladimirov, A., Digel, S. W., Johannesson, G., et al., 2011, GAL- PROP WebRun: An internet-based service for calculating galactic cosmic ray propagation and associated photon emissions , Comp. Phys. Commu. 182, 1156-1161 doi: 10.1016/j.cpc.2011.01.017

    work page doi:10.1016/j.cpc.2011.01.017 2011

  51. [54]

    A Fit to the Galactic Cosmic Ray Hydrogen and Helium Spectra at Voyager 1 at Low Energies and Earth Based Measurements at Much Higher Energies with Identical Rigidity Independent Source Spectra for the Hydrogen and Helium Nuclei

    Webber, W. R., & Higbie, P. R., 2015, A Fit to the Galactic Cosmic Ray Hydrogen and Helium Spectra at Voyager 1 at Low Energies and Earth Based Measurements at Much Higher Energies with Identical Rigidity Independent Source Spectra for the Hydrogen and Helium Nuclei , [arXiv:1503.01035]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  52. [55]

    Wiedenbeck, M., E., 2013, Cosmic-ray energy spectra and time vari- ations in the local interstellar medium: constraints and uncertain- ties, Space Sci. Rev. 176, 35-46 doi: 10.1007/s11214-011-9778-8

    work page doi:10.1007/s11214-011-9778-8 2013

  53. [56]

    E., Davis, A

    Wiedenbeck, M. E., Davis, A. J., Cohen, C. M. S., et al., 2009, Time dependence of solar modulation throughout solar cycle 23 as inferred from ACE measurements of cosmic-ray energy spec- tra; Proceedings of the 31 st International Cosmic Ray Conference (ICRC), 7-15 July 2019, L´ od´ z, Poland. Paper N. 0545, Pag. 1-4 [icrc2009.uni.lodz.pl/proc/pdf/icrc054...

    work page 2009