Inverse Compton Scattering of Cosmic-Ray Electrons

It has long been known that an important mode of energy loss for cosmic-ray electrons is inverse Compton scattering with photons of starlight. Previous calculations of ${〈\frac{d\mathcal{E}}{\mathrm{dt}}〉}_{\mathrm{av}}$ due to this process have involved nonsystematic approximations involving the form of the Klein-Nishina formula and the angular distribution of the radiation as seen in the electron's rest frame. The present paper considers an electron of arbitrary energy in an isotropic thermal radiation field of temperature $T$. A formally correct expression for ${〈\frac{d\mathcal{E}}{\mathrm{dt}}〉}_{\mathrm{av}}$ is obtained as an asymptotic expansion in the quantity $\frac{\mathcal{E}\mathrm{kT}}{{\left(m_e{c}^2\right)}^2}$ considered as a small parameter. The often quoted result ${〈\frac{d\mathcal{E}}{\mathrm{dt}}〉}_{\mathrm{av}}\propto {\mathcal{E}}^2$ is seen to be the zero-order term in this expansion. It is also seen that the energy-loss rate changes sign at an energy $\mathcal{E}\approx \frac{3}{2}\mathrm{kT}$ as would be expected from thermodynamics. A derivation of the zero-order term is given from classical radiation theory, and from this it is seen that this term also describes the energy-loss rate due to synchrotron radiation as well as from inverse Compton scattering.