Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-24T21:18:33.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical forms of the first 14 moments of the cosmic ray Fokker–Planck equation

Part of: Focus on Plasma Astrophysics

Published online by Cambridge University Press:  15 November 2017

A. Shalchi Open the ORCID record for A. Shalchi [Opens in a new window]
A. Shalchi*
Affiliation:
Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
*
†Email address for correspondence: andreasm4@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The Fokker–Planck transport equation describing the motion of energetic particles through a plasma is explored analytically. The latter equation provides a pitch-angle and position-dependent distribution function of the charged particles. In the current paper the first 14 moments of this equation are computed exactly for an arbitrary initial pitch angle. Such analytical forms are required in nonlinear treatments of perpendicular transport and other scenarios in plasma physics and astrophysics.

Keywords

energetic particlesmagnetized plasmaturbulence
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 6 , December 2017 , 905830603
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Earl, J. A. 1974 The diffusive idealization of charged-particle transport in random magnetic fields. Astrophys. J. 193, 231.CrossRefGoogle Scholar
Engelbrecht, N. E. & Burger, R. A. 2015 Sensitivity of cosmic-ray proton spectra to the low-wavenumber behavior of the 2D turbulence power spectrum. Astrophys. J. 814, 152.CrossRefGoogle Scholar
Ferrand, G., Danos, R. J., Shalchi, A., Safi-Harb, S., Edmon, P. & Mendygral, P. 2014 Cosmic ray acceleration at perpendicular shocks in supernova remnants. Astrophys. J. 792, 133.Google Scholar
Green, M. S. 1951 Brownian motion in a gas of noninteracting molecules. J. Chem. Phys. 19, 1036.Google Scholar
Kubo, R. 1957 Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12, 570.Google Scholar
Li, G., Shalchi, A., Ao, X., Zank, G. & Verkhoglyadova, O. P. 2012 Particle acceleration and transport at an oblique CME-driven shock. Adv. Space Res. 49, 1067–1075.Google Scholar
Litvinenko, Y. E., Effenberger, F. & Schlickeiser, R. 2015 The telegraph approximation for focused cosmic-ray transport in the presence of boundaries. Astrophys. J. 806, 217.Google Scholar
Litvinenko, Y. E. & Schlickeiser, R. 2013 The telegraph equation for cosmic-ray transport with weak adiabatic focusing. Astron. Astrophys. 554, A59.Google Scholar
Malkov, M. A. 2017 Exact solution of the Fokker–Planck equation for isotropic scattering. Phys. Rev. D 95, 023007.Google Scholar
Matthaeus, W. H., Qin, G., Bieber, J. W. & Zank, G. P. 2003 Nonlinear collisionless perpendicular diffusion of charged particles. Astrophys. J. 590, L53–L56.Google Scholar
Miyake, S., Muraishi, H. & Yanagita, S. 2015 A stochastic simulation of the propagation of Galactic cosmic rays reflecting the discreteness of cosmic ray sources Age and path length distribution. Astron. Astrophys. 573, A134.Google Scholar
Mulcahy, D. D., Fletcher, A., Beck, R., Mitra, D. & Scaife, A. M. M. 2016 Modelling the cosmic ray electron propagation in M. Astron. Astrophys. 592, A123.Google Scholar
Owens, A. J. 1974 The effects of nonlinear terms in cosmic-ray diffusion theory. Astrophys. J. 191, 235–244.Google Scholar
Porth, O., Vorster, M. J., Lyutikov, M. & Engelbrecht, N. E. 2016 Diffusion in pulsar wind nebulae: an investigation using magnetohydrodynamic and particle transport models. Mon. Not. R. Astron. Soc. 460, 4135–4149.Google Scholar
Schlickeiser, R. 2002 Cosmic Ray Astrophysics. Springer.Google Scholar
Shalchi, A. 2006 Analytical investigation of the two-dimensional cosmic ray Fokker–Planck equation. Astron. Astrophys. 448, 809–816.CrossRefGoogle Scholar
Shalchi, A. 2009 Nonlinear Cosmic Ray Diffusion Theories, Astrophysics and Space Science Library, vol. 362. Springer.Google Scholar
Shalchi, A. 2010 A unified particle diffusion theory for cross-field scattering: subdiffusion, recovery of diffusion, and diffusion in three-dimensional turbulence. Astrophys. J. 720, L127–L130.Google Scholar
Shalchi, A. 2011a Velocity correlation functions of charged particles derived from the Fokker–Planck equation. Adv. Space Res. 47, 1147–1164.Google Scholar
Shalchi, A. 2011b Applicability of the Taylor–Green–Kubo formula in particle diffusion theory. Phys. Rev. E 83, 046402.Google Scholar
Shalchi, A. 2017 Time-dependent perpendicular transport of energetic particles in magnetic turbulence with transverse complexity. Phys. Plasmas 24, 050702.Google Scholar
Shalchi, A. & Schlickeiser, R. 2005 Evidence for the nonlinear transport of galactic cosmic rays. Astrophys. J. 626, L97–L99.Google Scholar
Shalchi, A., Škoda, T., Tautz, R. C. & Schlickeiser, R. 2009 Analytical description of nonlinear cosmic ray scattering: isotropic and quasilinear regimes of pitch-angle diffusion. Astron. Astrophys. 507, 589–597.Google Scholar
Skilling, J. 1975 Cosmic ray streaming. I - Effect of Alfvén waves on particles. Mon. Not. R. Astron. Soc. 172, 557–566.Google Scholar
Strauss, R. D. T., Dresing, N. & Engelbrecht, N. E. 2017 Perpendicular diffusion of solar energetic particles: model results and implications for electrons. Astrophys. J. 837, 43.Google Scholar
Swordy, S. P., Mueller, D., Meyer, P., L’Heureux, J. & Grunsfeld, J. M. 1990 Relative abundances of secondary and primary cosmic rays at high energies. Astrophys. J. 349, 625–633.Google Scholar
Tautz, R. C. & Lerche, I. 2016 Application of the three-dimensional telegraph equation to cosmic-ray transport. Res. Astron. Astrophys. 16, 162.CrossRefGoogle Scholar
Taylor, G. I. 1922 Diffusion by continuous movement. Proc. Lond. Math. Soc. 20, 196.Google Scholar
Zank, G. P. 2014 Transport Processes in Space Physics and Astrophysics, Lecture Notes in Physics, vol. 877. Springer.Google Scholar