Cosmic-ray acceleration at stellar wind terminal shocks

Abstract

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 solutions show a characteristic power-law momentum spectrum for accelerated particles and a more complex spectrum of particles that are decelerated in the stellar wind. The power-law spectral index depends on the compression ratio of the shock and on the modulation parameters characterizing propagation conditions in the upstream and downstream regions of the shock. Solutions of the transport equations for the total density N (integrated over all energies), pressure P c , and energy flux F c of Galactic cosmic rays interacting with a stellar wind and shock are also studied. The density N(r) increases with radius r, and for strong shocks with large enough modulation parameters, there may be a significant enhancement of the pressure of weakly relativistic particles near the shock compared to the cosmic-ray background pressure P^. The emergent energy flux at infinity is of the order of 4nR 2 V 1 P ao (1^ is wind velocity upstream of the shock, R is shock radius). This is also of the order of the mechanical power ^MV 2 (M is mass loss rate) in the wind. On this basis, early-type stars could supply a significant fraction of the 3 x 10 40 ergs s _1 required by Galactic cosmic rays.