Cosmic ray distribution function under anisotropic particle scattering on magnetic field fluctuations

Heading: 
1Fedorov, YI
1Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
Kinemat. fiz. nebesnyh tel (Online) 2019, 35(1):3-26
https://doi.org/10.15407/kfnt2019.01.003
Start Page: Space Physics
Language: Russian
Abstract: 

The acceleration of energetic particles and their propagation in magnetic fields of the solar wind and the Galactic is one of the actual astrophysical problems. Cosmic rays affect communication, spacecraft electronics, disturb magnetosphere and ionosphere of the Earth. The particle scattering on the magnetic field irregularities appears as the basic mechanism, which control cosmic ray propagation in the interplanetary medium. If energetic particle scattering in the interplanetary medium is relatively weak, thus the particle mean free path is comparable to the heliocentric distance; it is necessary to use kinetic equation description of cosmic ray propagation. The energetic charged particle propagation in magnetic field, which is a superposition of the mean homogeneous magnetic field and magnetic inhomogeneities of various scales, is considered based on the kinetic equation. Fokker-Planck kinetic equation corresponds to the multiple small angle scattering and the scattering integral of this equation describes particle diffusion in the momentum space. Based on kinetic equation the set of differential equations for the spherical harmonics of cosmic ray distribution function is obtained. The cosmic ray transport equations are derived and solutions of these equations are obtained. The evolution of cosmic ray distribution function under anisotropic particle scattering on magnetic field fluctuations is studied. It is shown that the particle angular distribution depends sufficiently on the level of their scattering anisotropy. The temporal dependence of the cosmic ray distribution function is analyzed and the estimate of the parameter, which characterized the particle scattering anisotropy, is obtained.

Keywords: cosmic rays, diffusion, kinetic equation, telegraph equation
References: 

1. Galperin B. A., Toptygin I. N., Fradkin A. A. (1971) Rasseyaniye chastits magnitnymi neodnorodnostyami v sil’nom magnitnom pole. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 60(3), 972 (in Russian).

2. Prudnikov A. P., Brychkov Yu. A., Marichev O. I. (1981) Integraly i ryady. M.: Nauka. 800 p. (in Russian).

3. Toptygin I. N. (1972) O vremennoy zavisimosti intensivnosti kosmicheskikh luchey na anizotropnoy stadii solnechnykh vspyshek. Geomagnetizm i Aeronomiia, 12, 989  (in Russian).

4. Toptygin I. N. (1983) Kosmicheskiye luchi v mezhplanetnykh magnitnykh polyakh. M.: Nauka. 304 p. (in Russian).

5. Fedorov Yu. I. (2018) Intensity of Cosmic Rays at the Initial Stage of a Solar Flare. Kinematics Phys. Celestial Bodies.  34 (1), 1-12  (in Russian).
https://doi.org/10.3103/S0884591318010038

6. Shishov V. I. (1966) O rasprostranenii vysokoenergichnykh solnechnykh protonov v mezhplanetnom magnitnom pole. Geomagnetizm i aeronomiya,  6, 223 (in Russian).

7. Axford W. I. (1965) Anisotropic diffusion of solar cosmic rays. Planet. Space Sci., 13(12), 1301.
https://doi.org/10.1016/0032-0633(65)90063-2

8. Beeck J., Wibberenz G. (1986) Pitch angle distributions of solar energetic particles and the local scattering properties of the interplanetary medium. Astrophys. J., 311, 437.
https://doi.org/10.1086/164784

9. Bieber J. W., Earl J. A., Green G., et al. (1980)  Interplanetary pitch-angle scattering and coronal transport of solar energetic particles: New information from Helios.  J. Geophys. Res., 85(A5), 213.
https://doi.org/10.1029/JA085iA05p02313

10. Bieber J. W., Evenson P. A., Pomerantz M. A. (1986)  Focusing anisotropy of solar cosmic rays. J. Geophys. Res., 91(A8),  8713.
https://doi.org/10.1029/JA091iA08p08713

11. Dorman L. I., Katz M. E. (1977) Cosmic ray kinetics in space. Space Sci. Rev., 70, 529—575.
https://doi.org/10.1007/BF02186896

12. Earl J. A. (1973) Diffusion of charged particles in a random magnetic field. Astrophys. J., 180, 227.
https://doi.org/10.1086/151957

13. Earl J. A. (1994) New description of charged particle propagation in random magnetic field. Astrophys. J., 425, 331.
https://doi.org/10.1086/173988

14. Effenberger F., Litvinenko Y. ( 2014) The diffusion approximation versus the telegraph equation for modeling solar energetic particle transport with adiabatic focusing. 1. Isotropic pitch angle scattering.  Astrophys. J., 783, 15.
https://doi.org/10.1088/0004-637X/783/1/15

15. Fedorov Yu. I., Shakhov B. A., Stehlik M. (1995) Non-diffusive transport of cosmic rays in homogeneous regular magnetic fields. Astron. and Astrophys., 302 (2), 623—634.

16. Fedorov Yu. I., Stehlik M., Kudela K., Kassavicova J. (2002)  Non-diffusive particle pulse transport: Application to an anisotropic solar GLE. Solar Phys.,  208 (2),  325—334.
https://doi.org/10.1023/A:1020581705981

17. Fedorov Yu. I., Shakhov B. A. (2003) Description of non-diffusive cosmic ray propagation in a homogeneous regular magnetic field. Astron. and Astrophys., 402, 805.
https://doi.org/10.1051/0004-6361:20030169

18. Fisk L. A., Axford W. I. (1969) Anisotropies of solar cosmic rays. Solar Phys., 7, 486.
https://doi.org/10.1007/BF00146151

19. Gombosi T. J., Jokipii J. R., Kota J., et al. (1993) The telegraph equation in charged particle transport.  Astrophys. J., 403, 377.
https://doi.org/10.1086/172209

20. Hasselmann K., Wibberenz G. (1968) Scattering charged particles by random electromagnetic fields. Z. Geophys., 34,  353.

21. Hasselmann K., Wibberenz G. (1970) A note of the parallel diffusion coefficient. Astrophys. J., 162, 1049.
https://doi.org/10.1086/150736

22. Jokipii J. R. (1966) Cosmic ray propagation. 1. Charged particle in a random magnetic field. Astrophys. J., 146, 480.
https://doi.org/10.1086/148912

23. Kagashvili E. Kh., Zank G. P., Lu J. Y., Droge W. (2004) Transport of energetic charged particles. 2. Small-angle scattering. J. Plasma Phys., 70(part 5), 505—532.
https://doi.org/10.1017/S0022377803002745

24. Kota J. (1994) Coherent pulses in the diffusive transport of charged particles. Astrophys. J., 427(2), 1035—1080.
https://doi.org/10.1086/174209

25. Li G., Moore R., Mewaldt R. A., et al. (2012) A twin-CME scenario for ground level enhancement events. Space Sci. Rev., 171, 141.
https://doi.org/10.1007/s11214-011-9823-7

26. Litvinenko Yu. E., Noble P. L. (2016) Comparison of the telegraph and hyperdiffusion approximations in cosmic ray transport. Phys. Plasmas,  23. 062901 (8 p.).
https://doi.org/10.1063/1.4953564

27. Malkov M. A., Sagdeev R. Z. (2015) Cosmic ray transport with magnetic focusing and the “telegraph” model. Astrophys. J., 808, 157.
https://doi.org/10.1088/0004-637X/808/2/157

28. Miroshnichenko L. I., Perez-Peraza J. A. (2008) Astrophysical aspects in the studies of solar cosmic rays. Int . J. Modern Phys. A, 23(1), 1.
https://doi.org/10.1142/S0217751X08037312

29. Schwadron N. A., Gombosi T. I. (1994) A unifying comparison of nearly scatter free transport models.  J. Geophys. Res.,  99( NA10), 19301.
https://doi.org/10.1029/94JA01737

30. Shakhov B. A., Stehlik M. (2003) The Fokker-Planck equation in the second order pitch angle approximation and its exact solution. J. Quant. Spectrosc. Radiat. Transfer, 78, 31—39.
https://doi.org/10.1016/S0022-4073(02)00175-9

31. Shea M. A., Smart D. F. ( 2012) Space weather and the ground-level solar proton events of the 23rd solar cycle. Space Sci. Rev., 71, 161.
https://doi.org/10.1007/s11214-012-9923-z

32. Webb G. M., Pantazopolou M., Zank G. P. (2000) Multiple scattering and the BGK Boltzmann equation. J. Phys. A Math. Gen., 33, 3137—3160.
https://doi.org/10.1088/0305-4470/33/16/307

33. Wibberenz G., Green G. (1988) New methods and results in the field of interplanetary propagation. Proc. 11-th Europ. Cosmic Ray Symp. Balatonfured: Invited Talks.