Non-linear diffusion of cosmic rays escaping from supernova remnants – II. Hot ionized media

We study the problem of the escape and transport of Cosmic-Rays (CR) from a source embedded in a fully ionised, hot phase of the interstellar medium (HIM). In particular, we model the CR escape and their propagation in the source vicinity taking into account excitation of Alfv\'enic turbulence by CR streaming and mechanisms damping the self-excited turbulence itself. Our estimates of escape radii and times result in large values (100 pc, $2\times10^5$ yr) for particle energies $\lesssim20$ GeV and smaller values for particles with increasing energies (35 pc and 14 kyr at 1 TeV). These escape times and radii, when used as initial conditions for the CR propagation outside the source, result in relevant suppression of the diffusion coefficient (by a factor 5-10) on time-scales comparable with their (energy dependent) escape time-scale. The damping mechanisms are fast enough that even on shorter time scales the Alfv\'enic turbulence is efficiently damped, and the ratio between random and ordered component of the magnetic field is $\delta B/B_0\ll 1$, justifying the use of quasi-linear theory. In spite of the suppressed diffusion coefficient, and then the increased residence time in the vicinity (<200 pc) of their source, the grammage accumulated by CRs after their escape is found to be negligible (at all energies) as compared to the one accumulated while diffusing in the whole Galaxy, due to the low density of the HIM.