radiation.

The particle acceleration is combined with a propagation model relevant to . As described in Section \[sec:CSF\], high-energy particles will obtain a hard spectrum when their mean free path is large enough so they scatter back and forth between the two converging shocks. Lower energy particles however, will be confined to, and accelerated by a single shock and obtain a softer spectrum. This phenomenon is illustrated in Fig. 3 of @MNRAS_BGO13.

It is important to note that even though the CSF acceleration is efficient and the CR population modifies the structure of the plasma flows, the spectrum of high-energy CRs scattering between the two shocks remains hard until they gain enough energy to escape from the system. The shock modification can cause ${t_\mathrm{acc}}$ to increase, and the energy where the exponential turnover in Eq. (\[SolveDC\]) starts to dominate drop, but below the turnover, the spectrum remains close to $N(\gamma) \propto \gamma^{-1}$.

Thus, the CSF system will have two spectral regimes for trapped CRs: a low-energy region ($\lsimX 1$ TeV) from particles accelerated in a single SNR shock (produced both before and after the start of the two-shock acceleration period), and a high-energy, hard spectrum region ($\gsimX 1$ TeV) from 