Spectrum of Galactic Cosmic Rays Accelerated in Supernova Remnants

The spectra of high-energy protons and nuclei accelerated by supernova remnant shocks are calculated taking into account magnetic field amplification and Alfvenic drift both upstream and downstream of the shock for different types of supernova remnants during their evolution. The maximum energy of accelerated particles may reach $5\cdot10^{18}$ eV for Fe ions in Type IIb SNRs. The calculated energy spectrum of cosmic rays after propagation through the Galaxy is in good agreement with the spectrum measured at the Earth.Comment: 9 pages, 3 figures, accepted to Ap

Two-Dimensional particle-in-cell simulations of the nonresonant, cosmic-ray driven instability in SNR shocks

In supernova remnants, the nonlinear amplification of magnetic fields upstream of collisionless shocks is essential for the acceleration of cosmic rays to the energy of the "knee" at 10^{15.5}eV. A nonresonant instability driven by the cosmic ray current is thought to be responsible for this effect. We perform two-dimensional, particle-in-cell simulations of this instability. We observe an initial growth of circularly polarized non-propagating magnetic waves as predicted in linear theory. It is demonstrated that in some cases the magnetic energy density in the growing waves, can grow to at least 10 times its initial value. We find no evidence of competing modes, nor of significant modification by thermal effects. At late times we observe saturation of the instability in the simulation, but the mechanism responsible is an artefact of the periodic boundary conditions and has no counterpart in the supernova-shock scenario.Comment: 18 pages, 6 figures, accepted for publication in Ap

2D Elastostatic Problems in Parabolic Coordinates

In the present chapter, the boundary value problems are considered in a parabolic coordinate system. In terms of parabolic coordinates, the equilibrium equation system and Hooke’s law are written, and analytical (exact) solutions of 2D problems of elasticity are constructed in the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system. Analytical solutions are obtained using the method of separation of variables. The solution is constructed using its general representation by two harmonic functions. Using the MATLAB software, numerical results and constructed graphs of the some boundary value problems are obtained