= Solution
The <formal solution of the radiative transfer equation> and the <Eddington-Barbier relation> imply that each wavelength samples the temperature near its own photosphere. High opacity moves the emitting level upward. If temperature decreases upward, a molecular band samples cooler gas than the continuum and appears in absorption. If an <atmospheric thermal inversion> makes temperature increase upward, the same band can appear in emission.
In hot Jupiters, strong optical and ultraviolet absorption by TiO, VO, atomic metals, metal hydrides, or hydrogen-minus opacity can heat low-pressure gas; low infrared cooling efficiency and stellar activity also matter. Solar-System inversions arise from selective absorption, including ozone ultraviolet heating on Earth, methane and hydrocarbons on giant planets, and aerosols or dust. Around M dwarfs, water, methane, carbon dioxide, and hazes absorb the star's near-infrared radiation high in a sub-Neptune atmosphere.
On a temperate planet, a stratospheric inversion can provide a cold trap that limits water loss and can alter photochemistry and circulation. Excess upper-atmosphere heating can instead weaken the cold trap, promote escape, or create hostile surface conditions. Earth's ozone-heated stratosphere is the standard example of a habitable planet with an inversion.
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