Some general properties of radiation processes as related to "secondary" cosmic rays

The individual processes of synchrotron radiation and inverse Compton scattering produce quite different photon spectra for electrons of a given energy. However, an inverse power-law spectrum of electrons, j(γ) = k′γ−m, will produce a photon spectrum N(α) = Kα−(m+1)/2 for either process. The explanation for this lies in the fact that both individual radiation processes produce spectra of the form N(α) = F(αc/αc)(αc)−n where the characteristic energy αc is proportional to some power of the electron energy and contains all of the dependence on the electron energy that the spectrum exhibits. This form of the spectrum coupled with an inverse power-law electron spectrum produces the desired result. An investigation reveals that other radiation and secondary particle production processes possess this form of spectrum. This gives a very general explanation for the well-known fact that secondary particles, even those several generations removed from the primary cosmic rays, exhibit inverse power-law spectra.