Cherenkov escape condition at a flat dielectric interface ($n_1\sin {\theta }_C\le n_2$)

A photon emitted at angle ${\theta }_C$ to a normally incident particle can cross a planar boundary from index $n_1$ to $n_2$ only if Snell's law permits a real transmitted angle. Combining $\cos {\theta }_C=1/(n_1\beta )$ with that condition gives $n_1^2-{\beta }^{-2}\le n_2^2$. Otherwise the photon undergoes total internal reflection. For a relativistic particle in glass of index $1.55$ followed by air, the condition fails despite emission inside the glass.