Cosmic-ray acceleration at stellar wind terminal shocks

Steady-state, spherically symmetric, analytic solutions of the cosmic-ray transport equations, applicable to the problem of acceleration of cosmic rays at the terminal shock to a stellar wind, are studied. The spectra, gradients, and flow patterns of particles modulated and accelerated by the stellar wind and shock are investigated by means of monoenergetic-source solutions at finite radius, as well as solutions with monoenergetic and power-law galactic spectra. The solutions obtained apply in the test particle limit in which the cosmic rays do not modify the background flow. The model is used to estimate the kinetic energy flux of cosmic rays from typical stellar winds and to investigate whether stellar winds can provide a significant portion of the power required to maintain the galactic cosmic-ray energy density.