Radiative muon capture rates and the maximum photon energy

The differential photon spectrum for radiative muon capture is expanded about the average maximum photon energy $k_m$ and the correction terms evaluated using for one a modified Thomas-Reiche-Kuhn sum rule, thus extending previous work for ordinary capture. The resulting rate is much less dependent on $k_m$ than the usual closure result. The ratio $\frac{k_m}{\nu }$ appropriate for closure calculations, with $\nu$ the average neutrino energy, is determined and found to be approximately constant and, when correction terms are included, somewhat higher than values previously used. By similar techniques a consistency relation is derived which can be solved to explicitly estimate "physical" values of $k_m$ and $\nu$.