Cosmic-ray acceleration by stellar winds. II - The spectrum of accelerated particles
Consideration is given to the spectrum of particles accelerated at a stellar wind terminal shock and, at the same time, modulated by convection and diffusion in the stellar wind and decelerated by adiabatic expansion of the unshocked wind. It is noted that a proper study of this problem combines the whole of the cosmic-ray modulation problem with first-order Fermi acceleration at the shock due to repeated diffusive scattering across the shock. The modulation aspect of the model described here is much simpler than modern models of modulation alone; this makes it possible to derive illustrative analytical expressions for the accelerated spectra. Particular cases of monoenergetic and truncated power laws are computed. Energy losses and convective modulation compete with acceleration at the shock so effectively in this model that enhancement of the background cosmic-ray flux by a factor of more than 2 is extremely unlikely.