Inversion criterion with intrinsic planetary flux

For the usual semi-grey irradiated atmosphere with downward thermal optical depth $\tau$, an inversion occurs where $f_r{T}_{\mathrm{irr}}^4({\gamma }^2-1)e^{-\sqrt{3}\gamma \tau }>T_{\mathrm{int}}^4$, with $\gamma$ the visible-to-thermal opacity ratio. At negligible intrinsic flux the threshold reduces to $\gamma >1$.