Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 1 / e

A thermal inversion means temperature rises outward, so $dT/d\tau <0$. From part (c), near the observable atmosphere this requiresAt the top, the condition isIt 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.