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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.

An exoplanet emission spectrum describes the planetary thermal radiation as a function of wavelength. In an opaque exoplanet atmosphere, emerging light at a given wavelength comes predominantly from layers near outward optical depth of order one. Changing molecular opacity changes the layers sampled, so the spectrum jointly encodes composition and the atmospheric pressure-temperature profile.

A strong band is commonly darker when the sampled upper layers are cooler than the continuum-emitting layers. An atmospheric thermal inversion can instead put the band as an emission line. Occultation spectroscopy isolates the dayside spectrum, while exoplanet direct imaging can obtain it by spatially separating planetary and stellar light. A brightness temperature assigned to each band is a useful description of the data, but different bands can correspond to different physical layers.

An exoplanet secondary eclipse occurs when a planet passes behind its host star. Immediately outside the event the detector receives light from both bodies; during complete occultation it receives only stellar light. Their difference isolates planetary light, consisting of reflected stellar radiation, planetary thermal radiation, or both. Its fraction of the combined out-of-eclipse light is the eclipse depth.

Measuring that depth across wavelengths gives occultation spectroscopy of the visible dayside. The thermal eclipse depth connects a simple blackbody model to the radius ratio and brightness temperature. At visible wavelengths, reflected light can instead constrain geometric albedo. Changes in the host star or detector response must be modelled because the planetary signal is a small difference between two much brighter measurements.

An exoplanet thermal phase curve measures the change in planetary infrared brightness through the orbit, after modelling the host star's contribution. Around exoplanet secondary eclipse the visible hemisphere is mostly dayside; around primary exoplanet transit it is mostly nightside. The amplitude therefore constrains the longitudinal thermal contrast, while an offset maximum indicates an asymmetric brightness distribution.

Spitzer Space Telescope observations of HD 189733 b demonstrated that this technique could constrain day-night heat redistribution and a displaced bright region even without resolving the planetary disc.^[ref] A phase curve averages light over a visible hemisphere. Its map is therefore model-dependent, and brightness temperatures in one band need not describe all atmospheric pressures.

HD 189733 b is a transiting hot Jupiter used to study day-night heat redistribution. Observations with the Spitzer Space Telescope at $8\mu \mathrm{m}$ followed the changing combined light over half an orbital period. Subtracting the star's contribution yielded an exoplanet thermal phase curve, from which a broad longitudinal brightness distribution could be inferred.^[ref]

The bright hemisphere reached its maximum visibility before exoplanet secondary eclipse, implying a hot region displaced from the point directly facing the star. A warm nightside supports transport of absorbed energy by the exoplanet atmosphere. These are brightness temperature measurements in one band; they neither resolve small weather features directly nor uniquely specify the full atmospheric pressure-temperature profile.

thermal bremsstrahlung is continuum radiation from electrons accelerated in ion electric fields. For Maxwellian electrons,At high frequency the exponential cutoff constrains $T_e$; X-ray fitting must also model foreground absorption, line emission, detector response and possible multitemperature or nonthermal components. At radio frequencies, $h\nu \ll k_B{T}_e$, an optically thin free-free spectrum is almost flat, with the weak frequency dependence set mainly by the Gaunt factor.

For pure ionized hydrogen, radio free-free opacity givesThe turnover constrains emission measure; a resolved, optically thick source has brightness temperature approaching $T_e$. An unresolved source has an unknown beam filling factor, so its observed brightness need not equal its electron temperature. Angular size and an independent distance give a geometric depth $L$; a uniform model then yields $n_e=\sqrt{\mathrm{EM}/L}$. Clumping instead measures a density-squared average, not the mean density.^[ref]

A free-free spectrum alone does not uniquely determine density and size. The density-depth degeneracy of free-free radiation is explicit: for a resolved uniform column, $n_e\mapsto q{n}_e$, $L\mapsto L/q^2$ leaves both emissivity-integrated brightness and optical depth unchanged at fixed temperature. With independent SZ information and known $T_e$, the electron column is $N_e=y{m}_e{c}^2/({\sigma }_T{k}_B{T}_e)$; thenThese equalities require consistent composition and geometry, and uniform density and temperature. Relating $L$ to angular extent can give distance; asphericity and the gas clumping factor are important limitations of this combined cluster method.

An electron accelerated in an ion's Coulomb field emits bremsstrahlung. Both electron states are unbound, giving free-free radiation rather than a discrete bound-state line. Averaging over a Maxwellian electron distribution gives an optically thin spectral emissivity with dependenceHere $C_{\mathrm{ff}}$ depends on the emissivity convention, and the Gaunt factor supplies the quantum correction. No numerical emissivity normalization is needed for these diagnostics. For a mixture, sum $Z_i^2{n}_i$ over ion species. The bolometric emissivity is proportional to $n_e\sum Z_i^2{n}_i{T}^{1/2}$ with a frequency-averaged Gaunt factor. Thermal electron speeds control the spectrum while the encounter rate supplies its density-squared dependence.

temperature can be measured from the high-energy exponential cutoff at photon energy of order $k_{B}T$, provided the spectrum spans that turnover and foreground absorption and nonthermal contamination are modelled. At radio frequencies $h\nu \ll k_{B}T$, the exponential is nearly one, so the optically thin continuum is a poor thermometer by slope alone. Kirchhoff's law, ${\alpha }_{\nu }=j_{\nu }/B_{\nu }$, and the Rayleigh-Jeans law give ${\alpha }_{\nu }\propto n_e{n}_i{T}^{-3/2}{\nu }^{-2}{\bar{g}}_{\mathrm{ff}}$ for an isothermal thermal plasma. The transfer solution isin the radio limit. A resolved optically thick region therefore measures its electron temperature through brightness temperature, if it fills the beam. The turnover between thick and thin emission constrains the line-of-sight emission measure $\mathrm{EM}=\int n_e^{2}dl$ for a hydrogen-dominated fully ionized plasma. In the thin regime $T_b\simeq T{\tau }_{\nu }$, rather than $T_b=T$. The slowly varying Gaunt factor gives a nearly flat optically thin radio flux-density spectrum, in contrast to the ${\nu }^2$ thick limit at fixed projected area.^[ref]

To obtain density, one also needs geometry. An angular extent and independent distance give a transverse size; an assumed shape provides a depth or volume. For approximately uniform emitting gas with astrophysical volume filling factor $f_V$, $L_{\mathrm{ff}}\propto n_e^2{f}_{V}V{T}^{1/2}$, so $n_e\propto [L_{\mathrm{ff}}/(f_{V}V{T}^{1/2}){]}^{1/2}$. Alternatively a measured column emission measure and assumed path length give an rms density. Clumping raises the mean squared density relative to the square of the mean: the inferred value is not automatically the mean gas density. Free-free brightness alone does not uniquely determine temperature, density and physical size. Resolved imaging, a distance, optical-depth information or an independent column-density diagnostic is needed to remove those degeneracies. Cluster SZ data are complementary because their leading signal depends on $\int n_e{T}_{e}dl$, whereas free-free emission depends on density squared. Metal lines, unresolved temperature structure, foreground absorption, synchrotron emission and background subtraction must be controlled before attributing the continuum to a single thermal component.

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 $1250\mathrm{K}$.

Detector noise and graphical reading. Noise-equivalent power specifies the incident optical power producing signal-to-noise ratio one, with the electrical bandwidth and averaging convention stated. A spectral NEP has units $\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$; a band-integrated NEP has units W. Specific detectivity normalizes the latter by detector area and noise bandwidth:Its usual unit, the Jones, is $\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}{\mathrm{W}}^{-1}$. Removing the expected square-root area and bandwidth dependence helps compare different detector sizes and readout bandwidths, provided noise conditions and spectral response are comparable. This does not normalize away every background, temperature or implementation difference.

Reading the native embedded voltage plot gives approximate means $V_C\simeq 0.331\mathrm{V}$, $V_S\simeq 0.406\mathrm{V}$ and $V_H\simeq 0.493\mathrm{V}$. The one-second root mean square scatter is about $0.006\mathrm{V}$; the plotted peak-to-peak excursions are larger and must not be substituted for an root mean square noise. Independent graphical digitization of the blue trace gives means near $0.331,0.406,0.493$ V and scatters about $0.006$–$0.007$ V, consistent with this estimate. These are readings from a drawing, not exact original data.

The supplied Planck approximation gives $B_C\simeq 0.08590$ and $B_H\simeq 0.12836$ in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}(\mathrm{c}{\mathrm{m}}^{-1}{)}^{-1}$. Thus the noise-equivalent radiance of one one-second sample isHere the calibrating means are used to determine the response slope; their own uncertainty is included explicitly in the final scene estimate below.

Optical throughput and detectivity. A paraxial $f/F$ cone has half-angle approximately $1/(2F)$ and projected solid angle approximately $\pi /(4F^2)$. With $F=2$ and $A_d=10^{-6}{\mathrm{m}}^2$,Using the exact cone angle $\tan \theta =1/(2F)$ gives projected etendue $A_d\pi {\sin }^2\theta =1.85\times 10^{-7}{\mathrm{m}}^2\mathrm{sr}$, within the intended approximation. The radiance noise corresponds to detector power noiseThe wavenumber bandwidth is $1\mathrm{c}{\mathrm{m}}^{-1}$ and must match the radiance's per-wavenumber unit; it is not an electrical noise bandwidth. For the common order-of-magnitude assignment $\Delta f\sim 1\mathrm{Hz}$ to a one-second sample,Area in the last expression is $1\mathrm{m}{\mathrm{m}}^2=0.01\mathrm{c}{\mathrm{m}}^2$. For a specified one-second boxcar with white noise, the one-sided equivalent noise bandwidth is instead $1/(2t)=0.5\mathrm{Hz}$. That convention gives spectral NEP $\simeq 2.5\times 10^{-10}\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$ and $D_{\ast }\simeq 3.9\times 10^8$ Jones. The source does not specify the electrical filter, so the two conventions should not be mixed or assigned spurious precision.

Two-point retrieval and its uncertainty. Assume linear, stable response for the three views. The radiometer two-point calibration isWith target radiances treated as known, differentiating givesNotice $b<0$ for an intermediate scene: raising the hot voltage while keeping the scene voltage fixed reduces the inferred scene radiance. Numerically $a\simeq 0.2621$, $b\simeq -0.1214$ in radiance units per volt.

The scene interpolation fraction is about $0.463$, giving $B_S\simeq 0.1056$ andFor independent white noise in each view, each hundred-sample mean has standard error ${\sigma }_{\bar{V}}=0.006/10=0.0006\mathrm{V}$. The radiance error isUsing the given temperature sensitivity,A scatter estimate closer to $0.007$ V changes this to roughly $0.14$ K. These are random standard errors under the averaging assumptions. Target-temperature/emissivity uncertainty, response drift and atmospheric or surface-emissivity corrections are separate: the retrieved quantity is a brightness temperature, not automatically the true sea-surface temperature.

Pressure broadening is proportional to collision frequency and therefore becomes small in the upper atmosphere. Doppler broadening remains finite at finite temperature, with a characteristic width of order ${\nu }_0\sqrt{k_{B}T/(m{c}^2)}$ up to the stated width convention. Where that exceeds the collisional width, the line core is Doppler dominated. A Voigt profile combines the two mechanisms; natural broadening can contribute at still lower pressures. Thus a Lorentz pressure-broadened profile cannot hold all the way to $p=0$.

At line centre, replacing the fictitious $1/p$ opacity by a finite Doppler-core coefficient removes the divergent column optical depth. The emitting pressure is then finite rather than arbitrarily close to zero. If that level still lies in the isothermal stratosphere and the core is opaque, the central brightness temperature stays approximately $T_{\mathrm{strat}}$, so the flux floor changes little. If it lies below the stratosphere, or the Doppler-core column becomes thin, warmer layers or the surface contribute and the core brightens. The core-to-wing transition changes shape and width, but Lorentz wings formed mainly at higher pressures remain approximately as in part (c). The size of the correction needs the missing spectroscopic widths, but no unique numerical correction is requested.

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.