A Cosmic Ray Acceleration Mechanism Based on Background Flow Velocity Inhomogeneities Yielding Power-Law Spectra

In this article, momentum transport generated by the combined effects of pitch-angle diffusion and Background Flow Velocity Inhomogeneities (BFVIs) is proposed to obtain a cosmic rays acceleration mechanism, starting from the well-known focusing equation describing particle diffusion and acceleration. The inhomogeneities of background flow velocity is ubiquitous in astrophysical environment. The isotropic distribution function equation of charged energetic particles is derived, and its solution is obtained, demonstrating the form of momentum power laws of cosmic rays. In addition, if it is assumed that cosmic rays penetrate compressible plasma waves or turbulence, for quasi-steady states, the spectral index $\delta$ of the momentum power law spectrum of cosmic rays is found to be in the range $[-5, -3]$, which includes the observed power law indices of galactic cosmic rays. The results obtained in this article demonstrate that the mechanism proposed in this article, along with shock acceleration, may also contribute to the acceleration of galactic cosmic rays. Furthermore, when momentum convection effect and higher-order momentum derivative terms are considered, the indices of power laws should be smaller than $-5$. This may explain the power laws of solar energetic particle events.