Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 4 / d

At a molecular line, the large opacity moves the optical-depth-one surface to lower pressure and higher altitude than the neighboring continuum. The Eddington-Barbier relation makes the emergent intensity approximately the Planck function at that layer. In an ordinary outward-cooling atmosphere, the line-forming layer is cooler and the feature is in absorption. In an atmospheric thermal inversion, it is hotter, so the spectral-line emission from an atmospheric thermal inversion exceeds the continuum brightness and the feature appears in emission.

Factors that can create or suppress inversions include the abundance of high-altitude optical absorbers such as TiO, VO, or atomic metals; the host star's irradiation level and spectral energy distribution; and clouds, hazes, composition, and day-night circulation, all of which alter where stellar and thermal radiation are absorbed.