Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 2 / c / i / Solution

Optical depth increases inward, so an atmospheric thermal inversion occurs where temperature decreases with $\tau$. Differentiation givesTherefore the inversion condition in the observable atmosphere isIn particular, an inversion reaches the top when $\beta C>B$. A hot Jupiter can satisfy this when visible absorbers such as atomic metals, metal oxides, or negative hydrogen absorb incident starlight above the infrared photosphere. This makes the shortwave opacity large relative to the thermal opacity and deposits heat at low pressure.