= 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 gives
$$
\omega\left(t-\frac{\hat{\mathbf n}\cdot\mathbf r(t)}c\right)\simeq\omega\left(\frac{t}{2\gamma^2}+\frac{c^2t^3}{6\rho^2}\right).
$$
With $u=c\gamma t/\rho$, 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 shape
$$
F(x)=x\int_x^\infty K_{5/3}(z)\,dz,\qquad x=\frac\omega{\omega_c},\qquad\xi=\frac{x}{2}.
$$
It therefore identifies
$$
\boxed{\omega_c=\frac{3c\gamma^3}{2\rho},\qquad\epsilon_c=\hbar\omega_c=\frac32\frac{\hbar c}{\rho}\gamma^3.}
$$
The 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.<Synchrotron Radiation in the Geant4 Physics Reference Manual by Geant4 Collaboration>[Critical energy and critical mode number]{ref}
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