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A brightness temperature equates a measured intensity with a Planck function; it does not determine its contribution pressure. In hydrostatic equilibrium, ${\tau }_{\lambda }(P)={\int }_0^P{\kappa }_{\lambda }d{P}^{\prime }/g$. Estimating $P\sim g/{\kappa }_{\lambda }$ therefore needs gravity and opacity, as well as a model of the contribution function. Different pressures can produce the same band brightness.

Assume $R_p/R_{\star }=0.10$, $T_{\star }=5800\mathrm{K}$, uniform disk-averaged thermal emission, a nearly blackbody stellar continuum, negligible reflected light at $20\mu \mathrm{m}$, and a flat response over the narrow band. If the eclipse depth is normalized to the out-of-eclipse star-plus-planet flux, the planet/star flux ratio is $r=0.005/(1-0.005)=0.005025$; using $r=0.005$ changes the estimate by less than one percent. The exoplanet secondary eclipse then givesUsing the Planck law at $\lambda =20\mu \mathrm{m}$ gives $T_1\simeq 3.1\times 10^3\mathrm{K}$, or about $3000\mathrm{K}$ at the precision justified by the radius assumption. The Rayleigh-Jeans law instead gives $T_1\simeq 0.5T_{\star }\simeq 2900\mathrm{K}$, a useful rough check. This is a brightness temperature, not the bolometric planetary equilibrium temperature. With zero Bond albedo and global redistribution, the latter would be only about $1250\mathrm{K}$ at $0.05\mathrm{AU}$; a thermal window can sample hotter, deeper gas.

A continuum photon normally emerges from an optical depth of order unity. In hydrostatic equilibrium,The emission-pressure degeneracy of a brightness temperature means that pressure cannot be uniquely recovered from the measured brightness temperature without $g$ and an opacity model. For an explicit nominal estimate, take $g=20\mathrm{m}{\mathrm{s}}^{-2}$ and an effective continuum mass absorption coefficient ${\kappa }_{20}=2\times 10^{-4}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$. Then $P_{\tau \sim 1}\sim g/{\kappa }_{20}=10^5\mathrm{Pa}$, or approximately one bar. In a hydrogen-rich atmosphere, collision-induced absorption by transient ${\mathrm{H}}_2$ pairs supplies continuum opacity even without ordinary molecular lines; its density dependence makes the constant-opacity calculation an illustrative estimate. With a normal deep atmospheric pressure-temperature profile, temperature increases at larger pressures, eventually following an interior adiabatic temperature gradient.

Two mechanisms for $T_1\ne T_2$ are wavelength-dependent sampling of a vertically varying temperature, and wavelength-dependent weighting of a horizontally nonuniform dayside. In the first, carbon monoxide-rich opacity near $4.5\mu \mathrm{m}$ can sample cooler gas around $0.1$ bar while the $20\mu \mathrm{m}$ continuum samples hotter gas around one bar. In the second, hot and cool areas contribute different proportions of the total flux in different bands because the Planck function depends nonlinearly on temperature. The vertical mechanism alone produces the illustrated pair of temperatures; a single isothermal, horizontally uniform blackbody would have the same brightness temperature in both bands.

If the J-band observation is thermal emission, its low-opacity window normally samples at least as deeply as the continuum at $20\mu \mathrm{m}$. On the illustrated non-inverted profile, $T_3\gtrsim T_1>T_2$. The ordering is conditional: molecular-line absence alone does not specify the relative continuum opacities, and an inversion or reflected-light contamination can change the inference. The exoplanet emission spectrum measures weighted contribution regions rather than three exact thermometers at known pressures.

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 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.

Let an isothermal non-scattering atmospheric layer of temperature $T_a$ lie above an optically thick continuum source with brightness temperature $T_b$. Assume local thermodynamic equilibrium, so the layer emits with Planck function $B_{\lambda }(T_a)$. Its vertical optical depth is ${\tau }_{\lambda }$; a ray with direction cosine $\mu$ has slant depth $t_{\lambda }={\tau }_{\lambda }/\mu$.

The formal solution of the radiative transfer equation givesThe two terms are attenuated background and layer emission. Compare a line and nearby continuum at the same wavelength to adequate approximation, with $t_{\mathrm{line}}>t_{\mathrm{cont}}$. The one-layer spectral contrast isThe second bracket is positive, and $B_{\lambda }(T)$ increases with temperature. Therefore a cooler upper layer produces absorption, a hotter upper layer produces emission, and an isothermal source-plus-layer produces no line contrast:This is why higher opacity sampling a cooler altitude creates absorption in an outward-cooling atmosphere, whereas an atmospheric thermal inversion can create emission. If both line and continuum are very optically thick in the same layer, their contrast tends to vanish even with a different deeper temperature. Scattering, non-LTE excitation, and nonuniform layers require a more general radiative transfer treatment.

One approach uses exoplanet transmission spectra. Three aerosol signatures are:

A second approach uses the dayside spectrum and brightness obtained with exoplanet secondary eclipses, optionally supported by orbital phase variations. Its three useful aerosol signatures are:

Transmission measures aerosol extinction along the limb; eclipse/reflection measurements test dayside scattering and emission. No single flattened feature proves clouds: small atmospheric scale height, low molecular abundances, stellar heterogeneity and thermal-gradient degeneracies must be considered. Joint wavelength coverage and geometries make the inference stronger.

Three observational signatures constrain exoplanet atmospheric dynamics:

Opacity, exoplanet cloud distributions, and pressure-dependent brightness temperature affect the phase-curve interpretation. These measurements constrain circulation through atmospheric models; a single offset or line shift does not uniquely determine a three-dimensional wind field.

First refine the ephemeris, planetary mass, stellar radius, and stellar variability using exoplanet transit photometry and the radial-velocity method. Then combine observations that probe different regions rather than relying on one spectrum. A present-day programme could use the following complementary measurements; in the 2019 setting of the paper, James Webb Space Telescope observations would have been a future capability.

Together these address aerosols, molecular composition, elemental enrichment, vertical thermal structure, horizontal heat transport, winds, and escape. Repeat key events and monitor stellar activity, since stellar contamination and instrumental trends can imitate atmospheric signals. Use actual brightness, saturation limits, and predicted feature amplitudes to choose observing modes and exposure times.

The wavelength ranges and time-series capabilities are documented in the NIRISS SOSS guide, NIRSpec overview, MIRI spectroscopy guide, and ESO's CRIRES description.

Interpret every logarithm in the empirical profile as base ten with the dimensionless argument $P/(1\mathrm{mbar})$. PutThe upper atmosphere has $T=800\mathrm{K}$ for $x\le 0$, while integration of $dT/d{\log }_{10}P=Ax$ below it givesAssume that the stated $1250\mathrm{K}$ planetary equilibrium temperature is a reasonable brightness temperature at the $1\mathrm{bar}=10^3\mathrm{mbar}$ photosphere. ThenAt $10\mathrm{bar}$, $x=4$, and thereforeThis estimate neglects day-night variation, wavelength-dependent photospheric pressure, and a possible radiative-convective boundary; it treats the retrieved profile as representative of the dayside disk.

A homogeneous interpretation assigns the planet a brightness temperature $T_b$ satisfyingAt a fixed wavelength let $x=hc/(\lambda k_B)$ andInverting the Planck law givesAt $\lambda =20\mu \mathrm{m}$, $x=719.4\mathrm{K}$. With ${\theta }_T=\pi /4$, $T_1=1500\mathrm{K}$, and $T_2=1000\mathrm{K}$, one obtainsThe Rayleigh-Jeans law would give the nearly identical area-weighted estimate $1250\mathrm{K}$.

For blackbody star and planet observed at one common wavelength sufficiently long for the Rayleigh-Jeans law to hold for both,This follows by substituting $B_{\lambda }=2c{k}_{B}T/{\lambda }^4$ in the thermal eclipse depth. A radiative flux ratio alone does not determine the radius without a planetary brightness temperature and a stellar temperature. Spectral peaks interpreted with Wien's displacement law must refer to $F_{\lambda }$, not $F_{\nu }$ or $\lambda F_{\lambda }$.

A satellite test of greenhouse spectral change compares outgoing radiance with radiative transfer predictions. The 2001 study compared IRIS on Nimbus 4 (1970) and IMG on ADEOS (1997). Matching April–June ocean regions, masking land, screening clouds and correcting resolution and wavenumber controlled comparability. Spectral radiance became brightness temperature readouts. Regional comparisons and independent calculations tested interpretation. Carbon dioxide and methane features supported changed absorption; residual clouds, calibration and interannual variability limited background inference.^[ref]

This observational comparison tests radiative transfer; it does not by itself measure equilibrium climate sensitivity or validate a complete climate model.

To understand the readout, write an instrument's spectral channel schematically asHere $I$ is spectral radiance in the viewing direction $\hat{\mathbf{n}}$, $K_j$ is a normalized channel response, $b_j$ is calibration offset and $e_j$ is random error. Subtracting two channel readings isolates an atmospheric difference only after their response functions and sampling are comparable. Matching kernels reduces a false difference caused merely by observing a narrow spectral line at different resolutions; it cannot remove every calibration offset.

Outgoing longwave radiation in an energy budget also integrates radiance over upward directions and over the thermal spectrum, with the appropriate projected-angle factor. A brightness temperature converts one spectral radiance to the equivalent blackbody value through Planck's law; it is not automatically surface temperature. This explains why a spectral difference, a broadband radiative flux difference and surface warming are related observables with different units and different inferences.