= 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>, \b[<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.
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