Cosmic-Ray Modulation Equations


The temporal variation of the cosmic-ray intensity in the heliosphere is called cosmic-ray modulation. The main periodicity is the response to the 11-year solar activity cycle. Other variations include a 27-day solar rotation variation, a diurnal variation, and irregular variations such as Forbush decreases. General awareness of the importance of this cosmic-ray modulation has greatly increased in the last two decades, mainly in communities studying cosmogenic nuclides, upper atmospheric physics and climate, helio-climatology, and space weather, where corrections need to be made for these modulation effects. Parameterized descriptions of the modulation are even used in archeology and in planning the flight paths of commercial passenger jets.

The qualitative, physical part of the modulation is generally well-understood in these communities. The mathematical formalism that is most often used to quantify it is the so-called Force-Field approach, but the origins of this approach are somewhat obscure and it is not always used correct. This is mainly because the theory was developed over more than 40 years, and all its aspects are not collated in a single document.

This paper contains a formal mathematical description intended for these wider communities. It consists of four parts: (1) a description of the relations between four indicators of “energy”, namely energy, speed, momentum and rigidity, (2) the various ways of how to count particles, (3) the description of particle motion with transport equations, and (4) the solution of such equations, and what these solutions mean. Part (4) was previously described in Caballero-Lopez and Moraal (J. Geophys. Res, 109: A05105, doi:10.1029/2003JA010358 2004). Therefore, the details are not all repeated here.

The style of this paper is not to be rigorous. It rather tries to capture the relevant tools to do modulation studies, to show how seemingly unrelated results are, in fact, related to one another, and to point out the historical context of some of the results. The paper adds no new knowledge. The summary contains advice on how to use the theory most effectively.

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  1. R.A. Burger, D.J. Visser, Astrophys. J. 725, 1366–1372 (2010)

    ADS  Article  Google Scholar 

  2. R.A. Burger, T.P.J. Krüger, M. Hitge, N.E. Engelbrecht, Astrophys. J. 674, 511 (2008)

    ADS  Article  Google Scholar 

  3. R.A. Caballero-Lopez, H. Moraal, J. Geophys. Res. 109, 01101 (2004). doi:10.1029/3002JA010098

    Article  Google Scholar 

  4. R.A. Caballero-Lopez, H. Moraal, F.B. McDonald, J. Geophys. Res. 109, A05105 (2004). doi:10.1029/2003JA010358

    ADS  Article  Google Scholar 

  5. W. Dröge, Space Sci. Rev. 93, 121 (2000)

    ADS  Article  Google Scholar 

  6. L.A. Fisk, J. Geophys. Res. 76, 221 (1971)

    ADS  Article  Google Scholar 

  7. M.A. Forman, Planet. Space Sci. 18, 25 (1970)

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  9. L.J. Gleeson, I.A. Urch, Astrophys. Space Sci. 11, 288 (1971)

    ADS  Article  Google Scholar 

  10. L.J. Gleeson, I.A. Urch, Astrophys. Space Sci. 25, 387 (1973)

    ADS  Article  Google Scholar 

  11. L.J. Gleeson, G.M. Webb, Astrophys. Space Sci. 58, 21 (1978)

    ADS  Article  Google Scholar 

  12. P.A. Isenberg, J.R. Jokipii, Astrophys. J. 234, 746 (1979)

    ADS  Article  Google Scholar 

  13. J.R. Jokipii, Astrophys. J. 146, 480 (1966)

    ADS  Article  Google Scholar 

  14. J.R. Jokipii, J. Geophys. Res. 91, 2929 (1986)

    ADS  Article  Google Scholar 

  15. J.R. Jokipii, H. Levy, W.B. Hubbard, Astrophys. J. 213, 861 (1977)

    ADS  Article  Google Scholar 

  16. J. Kóta, J.R. Jokipii, Astrophys. J. 265, 573 (1983)

    ADS  Article  Google Scholar 

  17. J. Kóta, J.R. Jokipii, Geophys. Res. Lett. 18, 1797 (1991)

    ADS  Article  Google Scholar 

  18. W.H. Matthaeus, G. Qin, J.W. Bieber, G.P. Zank, Astrophys. J. 590, L53 (2003)

    ADS  Article  Google Scholar 

  19. H. Moraal, Proc. Suppl. Nucl. Phys. B 33 B, 161 (1991)

    ADS  Google Scholar 

  20. H. Moraal, in Cospar Colloquia Series, ed. by K. Scherer et al.. vol. 11 (Pergamon, Elmsford, 2001), p. 147

    Google Scholar 

  21. H. Moraal, L.J. Gleeson, in Proc. 14th Int. Cosmic Ray Conf. vol. 12, (1975) p. 4189

    Google Scholar 

  22. E.N. Parker, Planet. Space Sci. 13, 9 (1965)

    ADS  Article  Google Scholar 

  23. M.S. Potgieter, H. Moraal, Astrophys. J. 294, 425 (1985)

    ADS  Article  Google Scholar 

  24. C. Pei, J.W. Bieber, R.A. Burger, J. Clem, W.H. Matthaeus, in Proc. 31st Int. Cosmic Ray Conf., Paper 731. (2009).

  25. A. Salchi, Astrophys. and Space Sci. Library, (2009). doi:10.1007/798-3-642-00309-7-1

  26. R. Steenkamp, Ph.D. thesis, Potchefstroom University, South Africa, (1995)

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Moraal, H. Cosmic-Ray Modulation Equations. Space Sci Rev 176, 299–319 (2013).

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  • Cosmic rays
  • Modulation
  • Force field
  • Transport equation