= Inversion criterion with intrinsic planetary flux
An <atmospheric thermal inversion> has outward-increasing <temperature>. In a <semi-grey irradiated atmosphere>, define downward thermal <optical depth> $\tau$, constant <opacity> ratio $\gamma=\kappa_{\rm vis}/\kappa_{\rm IR}>0$, <irradiation temperature> $T_{\rm irr}$ and <internal effective temperature of a planet> $T_{\rm int}$. In the isotropic-irradiation closure, write the incident mean <irradiance> as $f_r\sigma_{\rm SB}T_{\rm irr}^4$, with $f_r>0$ explicitly specifying the averaging convention. Common choices are $1/4$ for a whole-sphere average and $1/2$ for a dayside average.
The analytic <radiative equilibrium> profile in this closure is
$$
T^4=\frac34T_{\rm int}^4\left(\tau+\frac23\right)
+\frac34f_rT_{\rm irr}^4\left[\frac23+\frac1{\sqrt3\gamma}
+\frac{\gamma-\gamma^{-1}}{\sqrt3}e^{-\sqrt3\gamma\tau}\right].
$$
Differentiation gives
$$
\frac{dT^4}{d\tau}=\frac34\left[T_{\rm int}^4-f_rT_{\rm irr}^4(\gamma^2-1)e^{-\sqrt3\gamma\tau}\right].
$$
Since $\tau$ increases inward, an <atmospheric thermal inversion> occurs when this derivative is negative. Thus $\gamma>1$ is necessary in this closure; a nonzero intrinsic <radiative flux> also requires sufficiently strong irradiation. For $T_{\rm int}>0$, any inversion is restricted to
$$
0\leq\tau<\frac1{\sqrt3\gamma}\log\left[\frac{f_rT_{\rm irr}^4(\gamma^2-1)}{T_{\rm int}^4}\right],
$$
provided the logarithm is positive. The $T_{\rm int}=0$ limit gives the threshold $\gamma>1$. These coefficients depend on the angular closure; a collimated beam has an incidence-angle dependence.<On the radiative equilibrium of irradiated planetary atmospheres by Tristan Guillot>[Section 2.4, equations (27) and (29): collimated and isotropic irradiation closures]{ref}
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