Calculated Spectrum of Inverse-Compton-Scattered Photons

We consider an electron of a given energy moving in a monoenergetic, isotropic radiation field. The energy spectrum of the photons that are scattered by the electron has been calculated both exactly and in a greatly simplied approximate form suitable for astrophysical calculations. The approximation may be derived either by expanding the exact solution in a small parameter and keeping only the leading terms or by employing a simplifying physical approximation at the beginning of the calculation. The approximate spectrum is similar to one previously derived by Ginzburg and Syrovatskii, the principal difference being that the present one does not break down if $\ensuremath{\hbar}{\ensuremath{\omega}}_{1}E>{({m}_{e}{c}^{2})}^{2}$, where $\ensuremath{\hbar}{\ensuremath{\omega}}_{1}$ is the initial photon energy and $E$ the electron energy. We indicate the astrophysical applications of our approximate spectrum by calculating the spectrum of photons scattered by electrons with an inverse-power-law energy distribution.