Critical adiabatic exponent for spherical power-law flow (${\gamma }_{\mathrm{crit}}=(4+\beta )/(4-\beta )(0<\beta <4)$)

For $0<\beta <4$, a nondegenerate transonic spherical flow in a power-law potential connected to infinity requires $\gamma <(4+\beta )/(4-\beta )$. For $\beta \ge 4$, every finite $\gamma >1$ satisfies the local crossing condition; there is no finite upper bound. The uniformly valid inequality is $2\beta -(4-\beta )(\gamma -1)>0$.