Spectra for B = 2 μG, near the shock and at several positions downstream of the shock (up to 100 kpc), normalized by the spectrum at the shock. In terms of time, the red curves are “spaced out” by ≈13 Myr. Both the radiative losses and adiabatic expansion are taken into account. The magenta dashed line shows the spectrum at 100 kpc from the shock if changes in the particles’ Lorentz factor and magnetic fields associated with the re-expansion of the gas are ignored. The blue dashed line shows the integrated spectrum over the regions up to 100 kpc from the shock. The thin black line shows the power law with the index large by 0.4 than the initial index at the shock front (see Fig. 19 in Bonafede et al. 2022).