Computer Science – Numerical Analysis
Scientific paper
Jul 1983
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983apj...270..319w&link_type=abstract
Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 270, July 1, 1983, p. 319-338.
Computer Science
Numerical Analysis
17
Charged Particles, Cosmic Rays, Energetic Particles, Particle Acceleration, Shock Waves, Eigenvalues, Numerical Analysis, Perturbation Theory, Steady State, Transport Properties
Scientific paper
Steady state solutions are derived for both the cosmic ray transport equation describing the first-order Fermi acceleration of energetic charged particles at a plane shock, and the second-order Fermi acceleration in the downstream region of the shock. Being superpositions of series of power law momentum spectra, with the power law indices being the roots of an eigenvalue equation, the solutions for the isotropic part of the phase space distribution function may be expressed as eigenfunction expansions. While the exact analytic solutions are for the case of a spatial diffusion coefficient that is independent of momentum, the solutions in general depend on shock compression ratio, modulation parameters in the upstream and downstream regions of the shock, and a dimensionless parameter that characterizes second-order Fermi acceleration.
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