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Intensity of Cosmic Rays at the Initial Stage of a Solar Flare

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Abstract

The propagation of solar cosmic rays in the interplanetary space is analyzed by solving the Fokker–Planck equation in the small-angle approximation. The particle source is assumed to be instantaneous and point-like. The spatiotemporal distribution of density of energetic particles in the anisotropic phase of a solar cosmic-ray enhancement is examined. Prolonged particle injection into the interplanetary medium is also discussed.

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References

  1. M. Abramovitz and I. Stigun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964; Nauka, Moscow, 1979).

    Google Scholar 

  2. G. A. Bazilevskaya and R. M. Golynskaya, “On distribution of solar cosmic rays in interstellar medium with consideration of adiabatic focusing,” Geomagn. Aeron. 29, 204–209 (1989).

    ADS  Google Scholar 

  3. B. A. Gal’perin, I. N. Toptygin, and A. A. Fradkin, “Scattering of particles by magnetic inhomogeneities in a strong magnetic field,” J. Exp. Theor. Phys. 33, 526–531 (1971).

    ADS  Google Scholar 

  4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic, New York, 1965).

    MATH  Google Scholar 

  5. L. I. Dorman and M. E. Katz, “On intensity fluctuations of solar cosmic rays,” in Proc. Solar Cosmic Rays and Their Penetration into the Earth’s Magnetosphere: 5th Leningrad Int. Semin., Leningrad, July 26–29, 1973 (Leningrad Fiz.-Tekh. Inst. im. A. F. Ioffe, Leningrad, 1973), pp. 311–321.

    Google Scholar 

  6. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics (Nauka, Moscow, 1978; Birkhäuser, Basel, 1985).

    MATH  Google Scholar 

  7. I. N. Toptygin, “On the time dependence of intensity of cosmic rays at the anisotropic stage of solar flares,” Geomagn. Aeron. 12, 989–995 (1972).

    Google Scholar 

  8. I. N. Toptygin, Cosmic Rays in Interplanetary Magnetic Fields (Nauka, Moscow, 1983; Reidel, Dordrecht, 1985).

    Google Scholar 

  9. J. W. Bieber, J. Clem, P. Evenson, et al., “Giant ground level enhancement of relativistic solar protons on 2005 January 20. I. Spaceship Earth observations,” Astrophys. J. 771, 92 (2013).

    Article  ADS  Google Scholar 

  10. J. W. Bieber, J. A. Earl, G. Green, et al., “Interplanetary pitch angle scattering and coronal transport of solar energetic particles: New information from Helios,” J. Geophys. Res.: Space Phys. 85, 2313–2323 (1980).

    Article  ADS  Google Scholar 

  11. J. W. Bieber, P. A. Evenson, and M. A. Pomerantz, “Focusing anisotropy of solar cosmic rays,” J. Geophys. Res.: Space Phys. 91, 8713–8724 (1986).

    Article  ADS  Google Scholar 

  12. D. J. Bombardieri, M. L. Duldig, J. E. Humble, and K. J. Michael, “An improved model for relativistic solar proton acceleration applied to the 2005 January 20 and earlier events,” Astrophys. J. 682, 1315–1327 (2008).

    Article  ADS  Google Scholar 

  13. J. L. Cramp, M. L. Duldig, E. O. Flückiger, J. E. Humble, M. A. Shea, and D. F. Smart, “The October 22, 1989, solar cosmic ray enhancement: An analysis of the anisotropy and spectral characteristics,” J. Geophys. Res.: Space Phys. 102, 24237–24248 (1997).

    Article  ADS  Google Scholar 

  14. R. J. Danos, J. D. Fiege, and A. Shalchi, “Numerical analysis of the Fokker–Planck equation with adiabatic focusing: Isotropic pitch-angle scattering,” Astrophys. J. 772, 35 (2013).

    Article  ADS  Google Scholar 

  15. H. Debrunner, J. A. Lockwood, and J. M. Ryan, “The solar flare event on 1990 May 24 — Evidence for two separate particle accelerations,” Astrophys. J., Part 2 — Lett. 387, L51–L54 (1992).

    Article  ADS  Google Scholar 

  16. L. I. Dorman and M. E. Katz, “Cosmic ray kinetics in space,” Space Sci. Rev. 70, 529–575 (1977).

    Article  ADS  Google Scholar 

  17. W. Dröge, Y. Y. Kartavych, B. Klecker, and G. A. Kovaltsov, “Anisotropic three-dimensional focused transport of solar energetic particles in the inner heliosphere,” Astrophys. J. 709, 912–919 (2010).

    Article  ADS  Google Scholar 

  18. S. P. Duggal, “Relativistic solar cosmic rays,” Rev. Geophys. Space Phys. 17, 1021–1058 (1979).

    Article  ADS  Google Scholar 

  19. Yu. I. Fedorov and B. A. Shakhov, “Solar cosmic rays in homogeneous magnetic field,” in Proc. 23rd Int. Cosmic Ray Conf., Calgary, July 19–30, 1993 (World Sci., Singapore, 1993), Vol. 3, pp. 215–218.

    ADS  Google Scholar 

  20. Yu. I. Fedorov, B. A. Shakhov, and M. Stehlik, “Non-diffusive transport of cosmic rays in homogeneous regular magnetic fields,” Astron. Astrophys. 302, 623–634 (1995).

    ADS  Google Scholar 

  21. Yu. I. Fedorov, M. Stehlik, K. Kudela, and J. Kassavicova, “Non-diffusive particle pulse transport — Application to an anisotropic solar GLE,” Sol. Phys. 208, 325–334 (2002).

    Article  ADS  Google Scholar 

  22. L. J. Gleeson and G. M. Webb, “Energy changes of cosmic rays in the interplanetary region,” Astrophys. Space Sci. 58, 21–39 (1978).

    Article  ADS  Google Scholar 

  23. E. Kh. Kaghashvili, G. P. Zank, J. Y. Lu, and W. Dröge, “Transport of energetic charged particles. Part 2. Smallangle scattering,” J. Plasma Phys. 70, 505–532 (2004).

    Article  ADS  Google Scholar 

  24. J. Kota, “Coherent pulses in the diffusive transport of charged particles,” Astrophys. J. 427, 1035–1040 (1994).

    Article  ADS  Google Scholar 

  25. J. Y. Lu, J. P. Zank, R. Rankin, and R. Marchand, “The transport of charged particles in a flowing medium,” Astrophys. J. 576, 574–586 (2002).

    Article  ADS  Google Scholar 

  26. J. Y. Lu, G. P. Zank, and G. M. Webb, “Numerical solution of the time-dependent kinetic equation for anisotropic pitch-angle scattering,” Astrophys. J. 550, 34–51 (2001).

    Article  ADS  Google Scholar 

  27. K. G. McCracken, H. Moraal, and P. H. Stoker, “Investigation of the multiple-component structure of the 20 January 2005 cosmic ray ground level enhancement,” J. Geophys. Res.: Space Phys. 113, A12101 (2008).

    Article  ADS  Google Scholar 

  28. L. I. Miroshnichenko, Solar Cosmic Rays (Kluwer, Dordrecht, 2001).

    Book  Google Scholar 

  29. C. Plainaki, A. Belov, H. Mavromichalaki, and V. Yanke, “Modeling ground level enhancements: Event of 20 January 2005,” J. Geophys. Res.: Space Phys. 112, A04102 (2007).

    Article  ADS  Google Scholar 

  30. D. Ruffolo, “Effect of adiabatic deceleration on the focused transport of solar cosmic rays,” Astrophys. J. 442, 861–874 (1995).

    Article  ADS  Google Scholar 

  31. B. A. Shakhov and M. Stehlik, “The Fokker–Planck equation in the second-order pitch angle approximation and its exact solution,” J. Quant. Spectrosc. Radiat. Transfer 78, 31–39 (2003).

    Article  ADS  Google Scholar 

  32. G. M. Simnett, “The timing of relativistic proton acceleration in the 20 January 2005 flare,” Astron. Astrophys. 445, 715–724 (2006).

    Article  ADS  Google Scholar 

  33. G. M. Webb, M. Pantazopolou, and G. P. Zank, “Multiple scattering and the BGK Boltzmann equation,” J. Phys. A: Math. Gen. 33, 3137–3160 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. G. P. Zank, J. Y. Lu, W. K. M. Rise, and G. M. Webb, “Transport of energetic charged particles in a radial magnetic field. Part 1. Large-angle scattering,” J. Plasma Phys. 64, 507 (2000).

    Article  ADS  Google Scholar 

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Correspondence to Yu. I. Fedorov.

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Original Russian Text © Yu.I. Fedorov, 2018, published in Kinematika i Fizika Nebesnykh Tel, 2018, Vol. 34, No. 1, pp. 3–20.

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Fedorov, Y.I. Intensity of Cosmic Rays at the Initial Stage of a Solar Flare. Kinemat. Phys. Celest. Bodies 34, 1–12 (2018). https://doi.org/10.3103/S0884591318010038

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