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Hot-Jupiter radius inflation is the unexpectedly large radius of some strongly irradiated giant exoplanets compared with ordinary cooling and contraction models at the inferred age. A giant planet loses internal energy through its exoplanet atmosphere; slowing that loss or depositing additional energy sufficiently deep can keep its interior hotter and more extended.

Atmospheric opacity and day-night heat redistribution affect the boundary condition seen by the interior. Analytic irradiated-atmosphere work shows why deep temperatures depend on global transport as well as incident power, while also illustrating the difficulty of explaining observed radii by simple opacity changes alone.^[ref] An expanded radius is thus a constraint on the thermal history, rather than direct evidence for one unique heating mechanism.

An atmospheric thermal inversion has outward-increasing temperature. In a semi-grey irradiated atmosphere, define downward thermal optical depth $\tau$, constant opacity ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}>0$, irradiation temperature $T_{\mathrm{irr}}$ and internal effective temperature of a planet $T_{\mathrm{int}}$. In the isotropic-irradiation closure, write the incident mean irradiance as $f_r{\sigma }_{\mathrm{SB}}{T}_{\mathrm{irr}}^4$, with $f_r>0$ explicitly specifying the averaging convention. Common choices are $1/4$ for a whole-sphere average and $1/2$ for a dayside average.

The analytic radiative equilibrium profile in this closure isDifferentiation givesSince $\tau$ increases inward, an atmospheric thermal inversion occurs when this derivative is negative. Thus $\gamma >1$ is necessary in this closure; a nonzero intrinsic radiative flux also requires sufficiently strong irradiation. For $T_{\mathrm{int}}>0$, any inversion is restricted toprovided the logarithm is positive. The $T_{\mathrm{int}}=0$ limit gives the threshold $\gamma >1$. These coefficients depend on the angular closure; a collimated beam has an incidence-angle dependence.^[ref]

An irradiated planetary atmosphere receives stellar radiation from above while intrinsic planetary heat flows outward from below. Stellar heating is therefore deposited where incident light is absorbed, rather than being supplied entirely at the bottom. Different opacities for stellar and planetary radiation can produce a deep, nearly isothermal region or an atmospheric thermal inversion.^[ref]

The absorbed power sets a global energy constraint, summarized by the planetary equilibrium temperature. The vertical atmospheric pressure-temperature profile additionally requires radiative transfer, convection and the heating depth. On a exoplanet under tidal locking, day-night heat redistribution also matters because only one hemisphere receives direct starlight.

A semi-grey irradiated atmosphere uses separate averaged opacities for incoming stellar light and outgoing planetary thermal radiation. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ compares where the two energy streams interact with matter. Large visible opacity relative to infrared opacity deposits stellar heat at a relatively small thermal optical depth; sufficiently strong high-altitude heating can produce an atmospheric thermal inversion.^[ref]

The model connects irradiation temperature, internal effective temperature of a planet and heating depth to an atmospheric pressure-temperature profile. It is useful for interpreting broad thermal structure, but averaging each band suppresses spectral windows and molecular lines that matter in detailed atmospheric retrieval.

Tristan Guillot developed analytic radiative-equilibrium models for irradiated planetary atmospheres. They connect incoming stellar radiation and outgoing thermal radiation with a temperature-pressure structure, making assumptions about opacity and irradiation explicit.^[ref]