Description of solar cosmic ray propagation in the interplanetary medium on the base of kinetic equation
1Fedorov, YI, 1Shakhov, BA 1Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Kyiv, Ukraine |
Kinemat. fiz. nebesnyh tel (Online) 2018, 34(3):3-24 |
https://doi.org/10.15407/kfnt2018.03.003 |
Start Page: Space Physics |
Language: Russian |
Abstract: The propagation of solar cosmic rays in the interplanetary space is considered based on the kinetic equation. The expression for cosmic ray density under instantaneous particle injection by point-like source is obtained. The set of differential equations for harmonics of cosmic ray distribution function is obtained starting from kinetic equation. The cosmic ray transport equation, taking into account the presence of the second harmonic of particle angular distribution, is derived and the solution of this equation is obtained. |
Keywords: cosmic rays, diffusion, kinetic equation |
1.M. Abramowitz and I. Stigun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Government Printing Office, Washington, DC, 1964; Nauka, Moscow, 1979).
2.B. A. Gal’perin, I. N. Toptygin, and A. A. Fradkin, “Strong particle scattering in a random inhomogeneous magnetic field,” J. Exp. Theor. Phys. 60, 915–918 (1971).
3.A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Nauka, Moscow, 1981; Gordon and Breach, New York, 1986).
4.Yu. V. Sidorov, M. V. Fedoryuk, and M. I. Shabunin, Lectures on the Theory of Functions of a Complex Variable (Nauka, Moscow, 1976; Mir, Moscow, 1985).
5.I. N. Toptygin, Cosmic Rays in Interplanetary Magnetic Fields (Nauka, Moscow, 1983; Reidel, Dordrecht, 1985).
https://doi.org/10.1007/978-94-009-5257-7
6.B. A. Shakhov, Yu. I. Fedorov, Yu. V. Kyz’yurov, and S. F. Nosov, “Description of solar cosmic ray propagation on the basis of the analytical solution of the kinetic equation,” Izv. Ross. Akad. Nauk, Ser. Fiz. 59 (4), 48 (1995).
7.V. I. Shishov, “On high-energy solar proton propagation in interplanetary magnetic field,” Geomagn. Aeron. 6, 223 (1966).MathSciNet
8.W. I. Axford, “Anisotropic diffusion of solar cosmic rays,” Planet. Space Sci. 13, 1301–1309 (1965).
https://doi.org/10.1016/0032-0633(65)90063-2
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).
https://doi.org/10.1088/0004-637X/771/2/92
10.L. I. Dorman and M. E. Katz, “Cosmic ray kinetics in space,” Space Sci. Rev. 20, 529–575 (1977).
https://doi.org/10.1007/BF02186896
11.S. P. Duggal, “Relativistic solar cosmic rays,” Rev. Geophys. Space Phys. 17, 1021–1058 (1979).
https://doi.org/10.1029/RG017i005p01021
12.J. A. Earl, “Diffusion of charged particles in a random magnetic field,” Astrophys. J. 180, 227–238 (1973).
https://doi.org/10.1086/151957
13.J. A. Earl, “New description of charged particle propagation in random magnetic field,” Astrophys. J. 425, 331–342 (1994).
https://doi.org/10.1086/173988
14.F. Effenberger and Y. Litvinenko, “The diffusion approximation versus the telegraph equation for modeling solar energetic particle transport with adiabatic focusing. I. Isotropic pitch-angle scattering,” Astrophys. J. 783, 15 (2014).
https://doi.org/10.1088/0004-637X/783/1/15
15.Yu. I. Fedorov, Yu. V. Kyzyurov, S. F. Nosov, and B. A. Shakhov, “Solution of the Boltzmann equation for nondiffusive solar cosmic ray propagation,” Ann. Geophys. 14, 1016–1018 (1996).
16.Yu. I. Fedorov and B. A. Shakhov, “Solar cosmic rays in homogeneous magnetic field,” in Proc. 23rd Int. Cosmic Ray Conf., Calgary, Canada, July 19–30, 1993 (World Sci., Singapore, 1993), Vol. 3, pp. 215–218.
17.Yu. I. Fedorov and B. A. Shakhov, “Description of non-diffusive solar cosmic ray propagation in a homogeneous regular magnetic field,” Astron. Astrophys. 402, 805–817 (2003).
https://doi.org/10.1051/0004-6361:20030169
18.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).
19.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).
https://doi.org/10.1023/A:1020581705981
20.L. A. Fisk and W. I. Axford, “Anisotropies of solar cosmic rays,” Sol. Phys. 7, 486–498 (1969).
https://doi.org/10.1007/BF00146151
21.T. J. Gombosi, J. R. Jokipii, J. Kota, et al., “The telegraph equation in charged particle transport,” Astrophys. J. 403, 377 (1993).
https://doi.org/10.1086/172209
22.N. Gopolswamy, H. Xie, S. Yashiro, et al., “Properties of ground level enhancement events and the associated solar eruptions during solar cycle 23,” Space Sci. Rev. 171, 23–60 (2012).
https://doi.org/10.1007/s11214-012-9890-4
23.E. Kh. Kagashvili, 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).
https://doi.org/10.1017/S0022377803002745
24.J. Kota, “Coherent pulses in the diffusive transport of charged particles,” Astrophys. J. 427, 1035–1041 (1994).
https://doi.org/10.1086/174209
25.M. A. Malkov and R. Z. Sagdeev, “Cosmic ray transport with magnetic focusing and the “telegraph” model,” Astrophys. J. 808, 157 (2015).
https://doi.org/10.1088/0004-637X/808/2/157
26.L. I. Miroshnichenko, Solar Cosmic Rays (Kluwer, Dordrecht, 2001).
https://doi.org/10.1007/978-94-015-9646-6
27.L. I. Miroshnichenko and J. A. Perez-Peraza, “Astrophysical aspects in the studies of solar cosmic rays,” Int. J. Modern Phys. A 23, 1 (2008).
https://doi.org/10.1142/S0217751X08037312
28.A. Sáiz, D. Ruffolo, J. W. Bieber, and P. Evenson, “Modeling relativistic solar particles in the inner solar system during an extreme event,” in Proc. 32-nd Int. Cosmic Ray Conf., Beijing, Aug. 11–18, 2011 (Inst. of High Energy Physics, Beijing, 2011), Vol. 10, p. 210.
29.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).
https://doi.org/10.1016/S0022-4073(02)00175-9
30.M. A. Shea and D. F. Smart, “Space weather and the ground-level solar proton events of the 23rd solar cycle,” Space Sci. Rev. 71, 161–188 (2012).
https://doi.org/10.1007/s11214-012-9923-z
31.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).MathSciNet
https://doi.org/10.1088/0305-4470/33/16/307
32.G. P. Zank, J. Y. Lu, W. K. M. Rice, and G. M. Webb, “Transport of energetic charged particles in a radial magnetic field. Part 1. Large-angle scattering,” J. Plasma Phys. 64, 507–541 (2000).
https://doi.org/10.1017/S0022377800008709