Photon number emission in synchrotron radiation: Systematics for high-energy particles

A recent derivation of an explicit elementary expression for the mean number $<N>$ of photons emitted per revolution in synchrotron radiation allows a systematic high-energy analysis leading to the result $<N>\simeq 5\pi\alpha/\sqrt{3(1-\beta^2)}+a_0\alpha+\pi\alpha\sqrt{1-\beta^2}/10\sqrt{3}$ where $a_0$ is a constant, with relative errors of 2.2%, 0.64%, 0.017%, in comparison to the well-known formula tabulated in the literature of 160%, 82%, 17% for $\beta=0.8$, 0.9, 0.99, respectively.