Appendix B: The synchrotron cooling break

We argue that the conventional synchrotron spectral break occurs at a very non-relativistic electron energy. The corresponding break in the radio spectrum is unobservable.

In writing the electron energy-loss rate Eq. (10), we have assumed that synchrotron radiation, which is quadratic in energy, dominates (inverse Compton scattering has the same energy dependence, but it is negligible, since the magnetic energy density within a CB is much higher than the radiation energy density). The general result for the energy loss of high-energy electrons is of the form:

$$-{{\rm d}E\over {\rm d}t}\simeq A_{\rm C}\,({\rm ln} {E\over m_{e}\,c^2}+a)+A_{\rm B}\,E+A_{\rm S}\, E^2.$$ (42)

The term proportional to $A_{\rm C}$ describes Coulomb scattering and ionization losses in the CB, which are negligible at high energies. The second term represents bremsstrahlung and its coefficient is (e.g. Shu 1991)

$$A_{\rm B}={34.35\,\alpha\, \sigma_{\rm T}\, c\,\bar{n}_{\rm b} \over 2\, \pi} ,$$ (43)

where $\alpha=1/137$ and $\rm\bar n_b$ is the baryon number density in the CB. Adiabatic losses and electron escape would have the same energy dependence as bremsstrahlung, but in the AG regime we are discussing the CBs are no longer expanding and the electron's Larmor radii are so small - relative to the CB's radius - that escape losses should also be negligible.

The spectral index of high energy electrons injected with a power-law spectrum steepens by one unit at a ``cooling break'' energy $[\beta(E_{\rm c})]^2\,E_{\rm c}=A_{\rm B}/A_{\rm S}$. For $\gamma \simeq 10^3$ and the reference values of $\bar n_{\rm b}=\bar n_{e}$ and n_e, the synchrotron cooling break is at a subrelativistic energy ( $\beta\sim 0.8$). This is in contrast with the injection bend at the highly relativistic energy $E_{\rm b}\simeq \gamma\, m_{e}\, c^2\,.$The synchrotron radiation of electrons below the cooling break is, for the current data, at unobservably low observer's radio frequencies.