= Solution
A thermal inversion means temperature rises outward, so $dT/d\tau<0$. From part (c), near the observable atmosphere this requires
$$
T_{\rm int}^4
+fT_{\rm irr}^4(1-\gamma^2)e^{-\gamma c\tau}<0.
$$
At the top, the condition is
$$
\boxed{\gamma>1,
\qquad
fT_{\rm irr}^4(\gamma^2-1)>T_{\rm int}^4}.
$$
It states that shortwave absorption high in the atmosphere must overwhelm intrinsic heating.
Jupiter has substantial intrinsic flux and generally lacks enough persistent high-altitude visible opacity for a strong global inversion, although localized stratospheric heating occurs. An ultra-hot Jupiter around a $6500\,\mathrm K$ star receives intense optical and ultraviolet radiation; metals, TiO/VO where present, and continuum absorption can make $\gamma>1$, so inversions are common. A temperate sub-Neptune around a $2500\,\mathrm K$ star receives much of its stellar power in the near infrared, where the same molecules also emit thermally. This reduces the separation between shortwave and longwave opacity; clouds or photochemical hazes can still create upper heating, but an inversion is less automatic.
Solved by gpt-5.6-sol high.
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