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Fokker-Planck equation for geomagnetically trapped electron distribution as function of longitude, time, energy and mirror-point field intensity
Fokker-Planck equation describing distribution of geomagnetically trapped electrons as function of longitude, time, energy, and mirror point field intensity
General Fokker-Planck equation for geomagnetically trapped electron distribution as function of longitude, time, energy, and mirror point field intensity
Solution of Fokker-Planck diffusion equation in intensity prediction of antiproton component of primary cosmic ray flux
Primary cosmic ray origin and acceleration and antiproton energy spectrum, using Fokker-Planck diffusion equation
Trapped particles pitch angle diffusion, deriving Fokker-Planck equation in spherical coordinates
The power spectrum of the interplanetary magnetic field were tested, as well as the power spectrum-cosmic diffusion ray coefficient relation at low energies. A first order perturbation solution of the Fokker-Planck equation governing the diffusion, convection, and adiabatic deceleration of galactic cosmic rays in the solar medium was found to relate intensity fluctuations of low energy cosmic rays to local changes in the propagation parameters. Diffusion coefficients and their day to day variations were calculated from interplanetary magnetic field data obtained by Pioneer 6 in 1965/1966. These are compared to simultaneous observations by IMP 3 of the proton flux in three energy channels (20 to 40, 40 to 60, 60 to 80 MeV).
Pitch angle diffusion of trapped electrons in terrestrial radiation zones, discussing diffusion theory based on Fokker-Planck equation, atmospheric Coulomb scattering mechanism, etc
Rays associated Fokker-Planck relation for isotropic turbulence diffusion equation, discussing logarithmic refractive index, moments and path length
Fokker-Planck equation for geomagnetically trapped electron distribution as function of longitude, time, energy and mirror-point field intensity
Fokker-Planck equation applied to probability density for finding rays larger than correlation distance during propagation in random medium
The first part of the present theory is devoted to the derivation of a Fokker-Planck equation. The eddies smaller than the hydrodynamic scale of the diffusion cloud form a diffusivity, while the inhomogeneous, bigger eddies give rise to a nonuniform migratory drift. This introduces an eddy-induced shear which reflects on the large-scale diffusion. The eddy-induced shear does not require the presence of a permanent wind shear and is intrinsic to the diffusion. Secondly, a transport theory of diffusivity is developed by the method of repeated-cascade and is based upon a relaxation of a chain of memories with decreasing information. The full range of diffusion consists of inertia, composite, and shear subranges, for which variance and eddy diffusivities are predicted. The coefficients are evaluated. Comparison with experiments in the upper atmosphere and oceans is made.
Stochastic acceleration, diffusion and scattering by weak random electromagnetic fields using Fokker-Planck equation, detailing relativistic particles
An analytic solution was obtained to the complete Fokker-Planck equation including the effects of convection, interplanetary deceleration and acceleration, corotation, and anisotropic diffusion. The solution yields a time to maximum for the particle flux of about 10 h and an exponential decay time of about 5 h. Several solar flare particle events were observed with the solar and galactic cosmic ray experiment on OGO-6. Comparisons of the calculated time dependence of the fluxes with these observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times.
For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximation yield the same integral for the Fokker-Planck coefficient for the pitch angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For large pitch angles these errors become significant, and for pitch angles of 90 deg the actual Fokker-Planck coefficient contains a delta function. The implications for these corrections relating cosmic ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.
An analytic solution was obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion. It is assumed that the particles are injected impulsively at a single point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events were observed with solar and galactic cosmic ray experiment aboard OGO 6. Detailed comparisons of the predictions of the solution with observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times. The solution also yields a time evolution for the vector anisotropy which agrees well with reported observations.
For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximations yield the same integral for the Fokker-Planck coefficient for the pitch-angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For small pitch angles, the errors in previous evaluations are fortuitously of minor importance. For large pitch angles, however, these errors become more significant; and for pitch angles of 90 deg, the actual Fokker-Planck coefficient contains a delta function which has been overlooked in the past. The implications of these corrections on the possibility of relating cosmic-ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.
Diffusion of protons in outer radiation belt due to violation of third adiabatic invariant for trapped particles