Self-similar solutions to stellar winds with cosmic rays

A two-fluid model is used to study the time evolution of stellar winds including the dynamical effect of cosmic rays. Neglecting the diffusion of cosmic rays, we seek self-similar solutions to spherically symmetric winds with a termination shock. The velocity upstream of the shock is taken to be zero. Physical solutions are those that can connect the shock to the star with the velocity approaching zero at the star. Two parameters govern the behaviour of the solutions, namely, the ratio of the upstream sound speed to the shock speed (in an inertial frame) and the gravitation potential of the central star. In some parameter regimes, no physical solution is possible.