 Solution

Relativistic radiation is concentrated into an angle of order $1/\gamma$ around the instantaneous velocity. The emission direction sweeps past a tangent observer over an orbital interval of order $\rho /(\gamma c)$. Since the charge moves toward the observer, arrival times are compressed by an additional factor of order $1/{\gamma }^2$, giving the spectral scale $\omega \sim c{\gamma }^3/\rho$.

The conventional numerical factor follows from the radiation Fourier phase. Near the tangent direction, circular motion givesWith $u=c\gamma t/\rho$, this becomes $[\omega \rho /(2c{\gamma }^3)](u+u^3/3)$. The resulting Airy-type integrals are expressible through modified Bessel functions with tangent-plane argument $\xi =\omega \rho /(3c{\gamma }^3)$. The standard angle-integrated synchrotron radiation spectrum has shapeIt therefore identifiesThe beaming argument supplies the physical scaling; the spectral integral fixes the conventional factor $3/2$. Critical energy is a broad spectral scale, not a hard photon-energy cutoff.^[ref]