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In local thermodynamic equilibrium, thermal intensity samples the Planck function near an optical depth of order unity. The Eddington-Barbier relation makes this explicit: $I_{\lambda }(0,\mu )\simeq B_{\lambda }[T({\tau }_{\lambda }=\mu )]$. A molecular band has greater opacity than its adjacent continuum and therefore samples a higher layer. A band in emission relative to the continuum implies that this higher layer is hotter: the line-forming region has an atmospheric thermal inversion under the assumed LTE, thermal interpretation.

The continuum is thermal radiation from an optically thick, deeper photosphere, with comparatively smooth opacity. In an H/He hot Jupiter, collision-induced absorption by H2-H2 and H2-He collisions supplies an important continuum; weak overlapping molecular lines and opaque exoplanet clouds can contribute too. It is not a separate blackbody emitter floating above the gas. A strongly isothermal layer would erase LTE molecular contrast rather than generate emission peaks.

In the emitting inversion, $dT/dz>0$, whereas the dry adiabatic lapse rate has $dT/dz=-g/C_p<0$. Thereforewhich lies on the stable side of the Schwarzschild criterion. An upward-displaced parcel cools and becomes denser than the ambient hot upper gas. The region can thus carry and redistribute thermal energy by radiative transfer, not by unstable thermal convection. Winds may transport energy horizontally; stability rules out the specified buoyant vertical convection, not every possible motion.

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.

Cloud particles can scatter or absorb, so not every atmospheric haze makes a high-albedo planet. Weak molecular bands alone also admit low abundance or high mean molecular weight; a consistent spectral combination is stronger evidence than a single signature.

The inferred dayside water mixing ratio is about $500$ times below the solar-equilibrium reference. Several explanations are possible. A high atmospheric carbon-to-oxygen ratio, especially near or above unity in hot carbon monoxide-dominated gas, locks much of the oxygen in CO and leaves little water. A genuinely oxygen-poor or low-metallicity envelope also reduces water, although bulk elemental abundances must apply to both hemispheres. Atmospheric photochemistry can destroy water at low pressures. In portions of the dayside that are substantially hotter than the stated minimum, thermal dissociation can also reduce water, especially at low pressure; $1800\mathrm{K}$ alone does not establish strong dissociation throughout the emitting region. Temperature/opacity degeneracies or incomplete treatment of exoplanet clouds and horizontal structure in an atmospheric retrieval can bias the inferred abundance.

At the cooler terminator, $10^{-4}$ is only a factor of five below the reference. Moderately reduced oxygen abundance, an enhanced C/O ratio, and exoplanet cloud/atmospheric haze dilution of spectral features are plausible explanations. Water is comparatively stable at $1000\mathrm{K}$, so strong thermal dissociation is not the natural explanation there. A exoplanet transmission spectrum samples a slant path through the limb, and the degeneracy among exoplanet cloud height, reference pressure and gas abundance can mimic a lower mixing ratio.

The two measurements need not describe the same pressure range or longitude. Local dissociation/photochemistry on the hotter dayside and reformation on the cooler limb can produce a real spatial difference. Even at one elemental C/O ratio, hot carbon monoxide-rich chemistry can leave less oxygen for water than cooler methane-rich chemistry, an example of carbon partition and atmospheric water abundance. Transport can modify or homogenize those tendencies, depending on the reaction and advection timescales. Alternatively, inconsistent assumptions in the emission and transmission retrievals can create an apparent discrepancy. A reconciliation should use one bulk elemental inventory, separate dayside/limb temperature profiles and contribution pressures, and consistent exoplanet cloud and transport physics; it should not assign independent planetary metallicities to the two hemispheres.

The two relevant categories are strongly irradiated, often transiting hot Jupiters, and young directly imaged exoplanets with large residual-entropy radii. The young objects are inflated relative to an old cold mass-radius sequence; a large young radius is not automatically anomalous relative to an age-appropriate formation model.

For a transit, the depth gives $(R_p/R_{\star }{)}^2$ after geometric and limb darkening corrections; an independently estimated stellar radius sets $R_p$. A measured planet mass then tests its location on the planetary mass-radius relation. For a directly imaged object, distance and integrated spectral flux give luminosity; a fitted effective temperature suppliesThe imaged radius is model-dependent because temperature, gravity and exoplanet clouds are inferred from its spectrum; its mass may also depend on age and evolutionary models unless dynamically measured.

Two broad explanatory classes are slower loss of existing heat and addition of heat to the deep interior. Specific delayed-cooling proposals are enhanced atmospheric opacity, which restricts radiative loss, and composition gradients producing layered convection in a giant planet. Specific heating proposals are tidal heating and Ohmic heating from currents induced by magnetized atmospheric winds, the proposed Ohmic heating of a giant planet. Heating must reach or influence sufficiently deep layers to maintain the interior specific entropy; merely heating the optically thin upper atmosphere is not equivalent. For young distant objects, retention of formation heat, described by hot and cold starts of a giant planet, itself explains much of the large radius without requiring the hot-Jupiter heating mechanisms.

The spherical stellar structure equations apply to a planet as well, with its appropriate equation of state:The last equation is the local first law, including gravitational contraction through the specific entropy change. For an isolated non-burning planet, ${ϵ}_{\mathrm{nuc}}\simeq 0$, soIf the convective interior has nearly uniform specific entropy $s(t)$, positive outgoing luminosity gives $\dot{s}=-L_{\mathrm{int}}/\int Tdm<0$. This entropy loss of an isolated convective planet describes its secular cooling.

During assembly, the gravitational energy scale $G{M}^2/R$ provides a source of heat. Integrating hydrostatic equilibrium gives the virial theorem, $\Omega +3\int PdV=0$, neglecting the surface-pressure term. For a monatomic ideal gas this becomes $2U+\Omega =0$, hence $E=U+\Omega =\Omega /2$. Contraction makes $\Omega$ more negative, releases radiation, and heats part of the gas. A characteristic formation temperature has scale $k_{B}T\sim \mu m_{H}GM/R$ up to structure factors. The retained fraction depends on the accretion shock, producing hot and cold starts of a giant planet.

Formation releases heat, and an isolated planet subsequently loses specific entropy and intrinsic luminosity as it approaches a cooler, more degenerate state. The structure equations alone do not fix one initial temperature, and the early contraction of an ideal gas can increase its central temperature while its total energy decreases. “Cools” therefore need not mean that every depth becomes colder at every instant. Once degeneracy limits contraction, the thermal reservoir declines more directly; the emergent effective temperature usually falls along a fixed-mass cooling track.

This makes young giant planets attractive for exoplanet direct imaging: their thermal near/mid-infrared emission can be far brighter than that of old planets of the same mass. Imaging and spectroscopy measure projected separation, astrometry, luminosity, an exoplanet emission spectrum, temperatures and atmospheric molecular and exoplanet cloud signatures. Radius follows from luminosity and fitted temperature; mass and initial specific entropy generally require age/evolutionary information, or an independent dynamical mass measurement. A detected luminosity alone is not a model-independent planet mass.

Transmission spectroscopy, eclipse/emission spectroscopy and direct-imaging spectroscopy probe atmospheres in complementary geometries.

A exoplanet transmission spectrum uses the wavelength dependence of primary-transit depth to measure slant-path absorption at the limb. Its long path length gives sensitivity to trace absorbers, but only transiting systems qualify; exoplanet clouds, stellar heterogeneity and the reference-radius/abundance degeneracy complicate interpretation, and the result samples the terminator rather than the dayside.

An exoplanet secondary eclipse isolates the dayside thermal/reflected spectrum by removing the planet behind the star. It constrains brightness temperatures, molecules and thermal inversions, but the planet/star contrast is small, a suitable orbital geometry is needed, and temperature-abundance degeneracies remain. A exoplanet thermal phase curve extends this method to longitudinal information.

Exoplanet direct imaging spatially separates a planet from its star and permits repeated spectroscopy without requiring a transit. It is effective for young, luminous, widely separated giant planets, but demands high contrast and angular separation; old small planets close to their stars are difficult. Inferred gravities, radii and masses can depend strongly on atmosphere and evolutionary models.

Three useful approaches are self-consistent one-dimensional models, atmospheric retrievals and three-dimensional circulation models.

A one-dimensional radiative-convective equilibrium model solves energy balance and radiative transfer, typically with a prescribed gravity, irradiation and composition or a chemical-equilibrium/kinetic calculation. It predicts physically coupled temperature profiles and spectra economically, but averages away longitude/latitude structure and inherits opacity, exoplanet cloud and chemistry assumptions.

An atmospheric retrieval parametrizes temperature, gas abundances, exoplanet clouds and other nuisance quantities and fits them to observations through a forward radiative-transfer model. It quantifies parameter constraints and degeneracies without enforcing every theoretical relation, but limited data can leave nonunique answers and priors/model structure can dominate.

A general circulation model solves rotating fluid dynamics with radiative heating/cooling and, when feasible, chemistry and exoplanet cloud transport. It predicts winds, phase curves and horizontal contrasts, but is computationally costly and still needs approximations for unresolved turbulence, exoplanet cloud microphysics and chemical networks. The approaches answer different questions and can be used jointly.

Three signatures are a displaced exoplanet thermal phase curve maximum, the day-night temperature contrast, and wind-induced Doppler shifts.

The maximum of an exoplanet thermal phase curve need not coincide with secondary eclipse. An eastward hot spot and a maximum before eclipse indicate eastward advection/atmospheric superrotation, provided reflected light and exoplanet clouds are accounted for.

The amplitude and shape of an exoplanet thermal phase curve measure day-night heat redistribution: a warm nightside and reduced day-night contrast imply effective transport relative to radiative relaxation time in a planetary atmosphere. Wavelength-dependent contrasts can reveal different circulation efficiencies at different contribution pressures.

High-resolution transmission or emission spectra can show line-centroid shifts/broadening relative to the orbital velocity. A limb blueshift is consistent with a dayside-to-nightside flow toward the observer; rotation and limb asymmetry affect the pattern. Such Doppler spectroscopy probes velocity directly, while phase curves probe the transported temperature pattern.

Three observable signatures are a blue optical transmission slope, muted molecular transmission features, and enhanced reflected optical light.

Small-particle atmospheric haze can increase the apparent transit radius toward short wavelengths, roughly $0.3$--$1\mu \mathrm{m}$. For Rayleigh scattering-like extinction, $\sigma \propto {\lambda }^{-4}$ and an isothermal atmospheric scale height model gives $d{R}_p/d\ln \lambda \simeq -4H$. Larger particles instead produce a flatter, approximately grey continuum.

An optically thick exoplanet cloud deck can truncate the slant path through gas and flatten molecular bands in a exoplanet transmission spectrum, for example over $0.4$--$2.5\mu \mathrm{m}$ and the water band around $1.4\mu \mathrm{m}$. The observed quantity is reduced band-to-continuum radius contrast, not necessarily a low molecular abundance.

exoplanet cloud scattering can produce a high geometric albedo and distinctive optical secondary eclipses/phase curves around $0.4$--$0.9\mu \mathrm{m}$. Longitudinal exoplanet cloud variation can shift a reflected-light phase maximum. A spectral slope or flat spectrum alone is not a unique exoplanet cloud identification: particle size, gas scattering, metallicity and stellar contamination must be considered together.

Three complementary methods available in the setting of this 2017 paper are:

Transmission probes the limb, eclipses probe the dayside, and direct imaging isolates spatially resolved planetary light. Radial velocities alone measure orbital motion rather than atmospheric composition; high-resolution planetary Doppler spectroscopy can instead detect atmospheric spectral lines if planetary light is isolated statistically.

The three parameter groups are local temperature, local pressure, and elemental composition. The elemental composition includes both atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio; these are independent compositional controls, so a specification by $T$ and $P$ alone is incomplete. Given these inputs, thermochemical equilibrium minimizes Gibbs free energy with element conservation.

Three useful chemical observables in a hydrogen-rich hot Jupiter are:

CO/CH4, H2O, and CO2 bands provide complementary temperature and elemental-abundance diagnostics, but their observed strengths also depend on the atmospheric pressure-temperature profile, exoplanet clouds and observing geometry. A spectral feature is not by itself a direct equilibrium abundance.