 Solution

An atmospheric thermal inversion is an altitude interval with $dT/dz>0$. Absorption of incoming stellar radiation above the usual thermal-emitting layers can heat the upper gas faster than it cools, producing an inversion. In a semi-grey irradiated atmosphere, a large shortwave-to-infrared opacity ratio favors such high-altitude energy deposition; local infrared emitters and the intrinsic flux also matter.

In the Solar system, Earth and all four giant planets have well-known stratospheric inversions. The ozone layer absorbs ultraviolet sunlight on Earth; methane and photochemical hydrocarbons absorb solar radiation in the giant planets, with aerosols contributing. The giant planets are Jupiter, Saturn, Uranus and Neptune. This refers to their stratospheric temperature rise, not to the gradient at every atmospheric level.

For hot Jupiters, influential factors include the stellar flux and spectrum; the abundances of high-altitude absorbers such as titanium monoxide and vanadium monoxide; atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio; thermal dissociation, atmospheric photochemistry and condensation; a atmospheric cold trap or atmospheric condensate rainout that removes absorbers; replenishment by vertical mixing; exoplanet clouds and atmospheric hazes; and heat redistribution by circulation. The ratio of visible heating to infrared cooling, rather than a single chemical species in isolation, determines whether an inversion persists.