Astronomy & Celestial Mechanics
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Exoplanets

Molecules seen only through a change of temperature

The heat a planet gives off, measured colour by colour as it disappears behind its star, carries the fingerprints of the molecules in its atmosphere — but only because the atmosphere's temperature changes with height. Water absorbs where the air cools upward and emits where it warms, and in an atmosphere at one temperature every molecule vanishes from the spectrum, however much of it there is. An emission spectrum is a sounding of a temperature gradient, and the molecules are what it is taken with.

Assumes Exoplanet atmospheres, Limb darkening and Line formation.

14 min read 6 figures Light is the only messengerMeasuring the unreachable

When a transiting planet passes behind its star, the system’s light drops by the planet’s own contribution, and subtracting the brightness during the eclipse from the brightness just before it isolates a body nobody has resolved. Measured colour by colour across the infrared, that subtraction gives the planet’s emission spectrum. For hot Jupiters it is a few parts per thousand of the star’s light, measured to a few tens of parts per million, and it is the most direct route to the atmosphere of a planet around another star that there is.

The spectrum shows molecules: the broad bands of water near 1.4, 1.9, 2.7 and 6.3 microns, carbon monoxide at 4.6, carbon dioxide at 4.3, methane in the cooler planets. It is natural to read those bands the way a transmission spectrum’s are read, as the amount of each molecule present. That reading is wrong in a specific way. An emission spectrum does not measure how much of a molecule there is. It measures how the atmosphere’s temperature changes with height, and the molecules are visible only through that change.

Brightness temperature across the infrared, for three profiles. The brightness temperature of the planet's emission against wavelength from 1 to 12 microns, for the three temperature profiles, with water and carbon monoxide bands in the opacity. In a band the opacity is high, so the light comes from higher, lower-pressure layers; in the windows between bands it comes from deeper. Over a profile falling outward the bands are cooler than the windows: at 6.3 microns 1457 K against 1570 K at 3.6. Over the inverted profile they are hotter: 1851 against 1538. Over the isothermal profile every wavelength has the same brightness temperature and the spectrum is a blackbody, whatever molecules are present.
Fig. 1 The brightness temperature of a hot Jupiter’s emission against wavelength, for one composition and three temperature profiles. Over a profile that cools outward, the water and carbon monoxide bands are cooler than the windows between them. Over an inverted profile, which warms outward, the same bands are hotter. Over an isothermal profile every wavelength has the same brightness temperature and the molecules have disappeared.

Every colour has its own surface

The key is where each wavelength’s light comes from. A surface is a depth: the light leaving an atmosphere at a given wavelength comes mostly from the level where the optical depth at that wavelength, measured inward from space, is about two-thirds. Above it the gas is too thin to emit much; below it the light cannot get out. At a wavelength where a molecule absorbs strongly, that level is high up, in thin gas; in a window between bands it is deep down, where the gas is dense.

So an emission spectrum is a set of readings of temperature at different heights. Each wavelength reports the temperature of its own photosphere, and the photospheres of strongly absorbed wavelengths sit higher. If the temperature falls with height, as it does in the Earth’s lower atmosphere and in most planetary atmospheres heated from below, the bands report cooler temperatures than the windows and appear as dips. If the temperature rises with height, as it does in the Earth’s stratosphere where ozone absorbs sunlight, the bands report hotter temperatures and appear as peaks. And if the temperature does not change with height, every photosphere has the same temperature, and the spectrum is a blackbody, whatever molecules are present.

This is the same physics that makes the Sun’s limb darker than its centre, and the same one sentence — light leaves from where its optical depth is about two-thirds — covers both. A sight line entering the Sun at an angle stops higher up, in cooler gas; the darkening measures the temperature gradient, not the edge. An emission spectrum makes the same measurement with wavelength instead of angle: the opacity varies instead of the path, and the photosphere moves up and down through the temperature gradient accordingly.

Three profiles, one planet

A hot Jupiter’s temperature profile depends on where the starlight is absorbed. If the atmosphere is fairly transparent to starlight, the light penetrates deep, heats the lower layers, and the atmosphere cools outward. If something high up absorbs starlight strongly — titanium and vanadium oxides in the hottest planets, or atomic iron, or photochemical haze — the upper layers are heated directly and the profile inverts.

Three temperature profiles for one irradiated planet. Temperature against pressure, with pressure decreasing upward on a logarithmic scale from a hundred bar to a millionth of a bar, for a hot Jupiter heated by its star, from the two-stream grey solution. The three profiles differ only in γ, the ratio of the atmosphere's opacity to starlight to its opacity to its own heat: γ = 0.3, falling outward, from 1689 K at 0.1 bar to 1449 K at the top; γ = 1, nearly isothermal, from 1598 K at 0.1 bar to 1598 K at the top; γ = 3, inverted, from 1461 K at 0.1 bar to 1884 K at the top. When starlight penetrates deep (γ small) the atmosphere is heated from below and cools outward like the Earth's troposphere; when something high up absorbs starlight strongly (γ large) the upper layers are hotter than the lower, an inversion like the Earth's stratosphere.
Fig. 2 Temperature against pressure, from a hundred bar at the bottom to a millionth of a bar at the top, for a hot Jupiter heated by its star, from an analytical two-stream solution. The only difference between the three profiles is the ratio of the atmosphere’s opacity to starlight to its opacity to its own heat. Below one, the atmosphere cools outward; above one, it inverts; equal to one, it is isothermal. The profiles cross near a hundredth of a bar.

The analytical solution drawn is due to Tristan Guillot and is the simplest one that captures both regimes. It treats the atmosphere’s opacity as grey in two broad bands, one for starlight and one for the planet’s own heat, and the single ratio between them decides the shape. Real atmospheres are not grey in either band, and full radiative-transfer models find profiles that bend more, but the direction of the gradient is set by the same comparison: whether starlight is absorbed above or below the level from which the planet radiates.

The spectrum each profile makes

Converted to what a telescope measures — the fraction of the system’s light that disappears at eclipse — the three profiles give three very different spectra from the same composition.

Eclipse depth across the infrared, for three profiles. The fraction of the system's light that disappears when the planet passes behind its star, in parts per million, against wavelength from 1 to 12 microns, for a hot Jupiter a tenth of its star's radius with the three temperature profiles and the same composition. The same water and carbon monoxide bands appear as dips over the profile falling outward and as peaks over the inverted one — at 6.3 microns 1222 against 1904 ppm — and nearly vanish over the isothermal one. An emission spectrum measures how temperature changes with height; the molecules are visible only through that change.
Fig. 3 The eclipse depth in parts per million against wavelength for a planet a tenth of its star’s radius, for the three profiles and one composition. The water and carbon monoxide bands are dips over the profile that cools outward and peaks over the inverted one; at 6.3 microns the depth is 1222 parts per million for the first and 1904 for the second. Over the isothermal profile the spectrum is smooth.

The figure makes the danger concrete. A flat, featureless emission spectrum does not mean a planet has no water. It could equally mean an atmosphere with plenty of water and very little temperature gradient — or a cloud deck high enough to hide the gradient, which does the same thing from the other side. And the size of a feature does not mean much water: it means a large enough change in temperature between the band’s photosphere and the window’s, which can come from a steep gradient with little water or a shallow one with a lot.

That was learned the hard way. In 2008 emission measurements from the Spitzer Space Telescope of the hot Jupiter HD 209458b were read as showing water in emission, and so a thermal inversion, and the idea that the hottest planets were inverted by gaseous titanium oxide became standard. Later observations with better calibration and more wavelengths found no inversion in that planet at all; the original features had been partly instrumental. Genuine inversions were found in 2017 in hotter planets, such as WASP-121b, where the water band at 1.4 microns appears in emission with a clear detection — the first unambiguous stratosphere on a world around another star.

Clouds enter the same way, from the other side. A cloud deck is an opacity that is nearly the same at every infrared wavelength, so above a thick deck the photospheres of the bands and the windows are pushed up toward the same height, and the temperature difference between them shrinks. A high cloud on a planet that cools outward flattens the emission spectrum just as an isothermal atmosphere would, and the two are distinguished only by the overall brightness: a cloudy day side radiates from higher, cooler layers at every wavelength and looks colder. So a featureless emission spectrum has at least three readings — no gradient, a high cloud, or the ion continuum of the hottest planets — and the planet’s temperature, its phase curve and its transmission spectrum are what choose among them.

The lowest layers of the profiles drawn are nearly isothermal for a different reason. Deep in a hot Jupiter the starlight has been absorbed and the only heat flowing outward is the planet’s own internal heat, which for a planet a few billion years old is tiny compared with what the star delivers. The atmosphere there settles to nearly one temperature over a wide range of pressure, and an emission spectrum cannot see into it: no wavelength is transparent enough. The deep atmosphere, where the planet’s inflated radius and interior heat would be recorded, is below the reach of the method.

A product that one feature cannot split

The trade between abundance and gradient can be drawn directly. More of an absorbing molecule makes its band more opaque, lifting the band’s photosphere higher; in an atmosphere that cools outward, a higher photosphere is a cooler one, so the band deepens. A steeper profile does the same thing without any change of abundance.

One feature, many atmospheres: abundance against gradient. The contrast between the brightness temperature in the 6.3-micron water band and in the 3.6-micron window, against the water abundance relative to a reference, on a logarithmic axis, for three temperature profiles falling outward with different steepness (γ = 0.15, 0.3, 0.6, steeper for smaller γ). More water lifts the band's photosphere higher, into cooler gas, and deepens the feature; a steeper profile does the same thing for a fixed abundance. A band 60 K cooler than the window is produced by 0.5 times the reference abundance with γ = 0.15, 0.8 times the reference abundance with γ = 0.3, 2.5 times the reference abundance with γ = 0.6. A single feature measures a product of abundance and gradient, and only features at several wavelengths, probing several heights, can separate them.
Fig. 4 The difference between the brightness temperature in the 6.3-micron water band and in the 3.6-micron window, against the water abundance, for three profiles cooling outward with different steepness. A band sixty kelvin cooler than the window is produced by half the reference abundance with the steepest profile, 0.8 times it with the middle one, and 2.5 times it with the shallowest. One feature measures a product of abundance and gradient.

The degeneracy is the emission-spectrum counterpart of the one that afflicts transmission spectra, where the planet’s reference radius trades against abundance. It is broken the same way: by features at many wavelengths. Different bands of the same molecule have different strengths, so their photospheres sit at different heights; a spectrum spanning several bands samples the profile at several heights, fixes its shape, and then the abundance follows from where the band photospheres lie. The more of the spectrum is measured, the fewer ways there are to reproduce it.

Thermometers at different heights

The picture of one photosphere per wavelength is a simplification. Each wavelength’s light comes from a range of heights, weighted by how much each layer emits and how much of that emission escapes.

The height each wavelength's light comes from. The contribution function — how much each layer adds to the emergent light — against pressure, for 3.6 µm, 1.6 µm, 6.3 µm, over the profile falling outward, each normalised to its peak. The light at 3.6 microns comes from about 0.069 bar, at 1.6 microns comes from about 0.038 bar, at 6.3 microns comes from about 0.0033 bar: the clearest window sees deepest, the water band highest. Each wavelength samples a layer about a decade of pressure thick, centred where its optical depth reaches about one, and the layers overlap. An emission spectrum is a set of thermometers hung at different heights — a coarse sounding of the atmosphere, with a resolution in height set by how much the opacity varies across the spectrum.
Fig. 5 The contribution function — how much each layer contributes to the emergent light — against pressure, for a window at 3.6 microns, a weak absorption at 1.6 and the strong water band at 6.3, over the profile that cools outward. The window’s light comes from near seven hundredths of a bar, the weak absorption’s from four hundredths, the band’s from three thousandths. Each is about a decade of pressure thick, and they overlap.

That thickness sets the vertical resolution of the sounding. An emission spectrum can distinguish temperatures at levels a decade apart in pressure — a scale height or two — and no finer, however precise the measurement, because each wavelength averages over a layer that thick. The atmosphere’s temperature structure is therefore recovered as a smooth curve with a handful of independent points on it, spread between about a bar and a thousandth of a bar, and the finer structure that general circulation models predict is below that resolution.

The same principle is how the Earth’s atmosphere has been sounded from satellites for fifty years. Instruments measuring the carbon dioxide band near fifteen microns at several wavelengths across its wing record temperatures at several heights, because the band’s opacity varies across it and each wavelength’s contribution function peaks at a different level. The profiles used in weather forecasting are inverted from exactly that kind of spectrum. For an exoplanet, the method is the same and the sounding is coarser, done on a disc-averaged spectrum of a planet that cannot be resolved.

The heights are a property of the opacity, not of the temperature, and that separation is what makes the sounding work.

The height each wavelength's light comes from. The contribution function — how much each layer adds to the emergent light — against pressure, for 3.6 µm, 1.6 µm, 6.3 µm, over the profile inverted, each normalised to its peak. The light at 3.6 microns comes from about 0.044 bar, at 1.6 microns comes from about 0.015 bar, at 6.3 microns comes from about 0.003 bar: the clearest window sees deepest, the water band highest. Each wavelength samples a layer about a decade of pressure thick, centred where its optical depth reaches about one, and the layers overlap. An emission spectrum is a set of thermometers hung at different heights — a coarse sounding of the atmosphere, with a resolution in height set by how much the opacity varies across the spectrum.
Fig. 6 The same three contribution functions over the inverted profile. They peak at the same pressures as before — the heights are set by how opaque each wavelength is, not by how hot the gas is — but now the highest layer, sampled by the water band, is the hottest, and the band appears in emission. Changing the profile moves the temperatures read at fixed heights; changing the abundance moves the heights.

That is the logic every retrieval exploits, and it is worth being clear about what a retrieval is: a search over thousands of combinations of abundances, temperature profiles and cloud properties for the ones whose computed spectra match the measured one within its errors, reporting the spread of the matches as the uncertainty. A change of abundance slides the contribution functions up or down together, all by the same factor in pressure. A change of profile leaves them where they are and changes what they read. With enough wavelengths — enough independent heights — the two can be separated, because the first shifts every reading along the same curve and the second changes the curve. With two or three wavelengths they cannot, and the early emission measurements, from broadband photometry at a handful of wavelengths, were in exactly that regime.

Why the hottest planets invert

Whether a hot Jupiter has a stratosphere depends on whether some species high in its atmosphere absorbs starlight strongly, and the leading candidates are the oxides of titanium and vanadium, which absorb broadly in the visible and dominate the spectra of cool stars. In a planet’s atmosphere they exist as gases only if the gas is hot enough everywhere they travel. On a tidally locked planet, gas circulating to the cold night side can pass below the condensation temperature of those oxides, and there they form solid grains and rain out of the upper atmosphere — a cold trap. Only planets hot enough that even the night side stays above the condensation point should keep them aloft.

That predicts a transition with temperature, and one is seen: planets with day sides below about 2000 kelvin show water in absorption, and many above it show water in emission or a featureless spectrum, as the ratio in the figures above rises through one. At the very highest temperatures, above about 2500 kelvin, water itself is split into hydrogen and oxygen on the day side and a new opacity source, the negative hydrogen ion — the same ion that makes the Sun’s continuum — flattens the spectrum again. So the sequence from absorption to emission to blankness runs across the hot Jupiters as their temperature rises, and a flat spectrum at the top end means something quite different from a flat spectrum at the bottom.

The day side, not the planet

Everything above refers to the hemisphere facing the star at the moment of eclipse, when the planet is behind the star and its day side faces the observer. A hot Jupiter is tidally locked, with a permanent day side and night side, and the two can differ by a thousand kelvin. The eclipse spectrum is a day-side spectrum, averaged over a hemisphere that is not uniform: hotter at the substellar point, cooler toward the terminator, and shifted by winds that carry heat eastward.

Measuring the spectrum at other phases of the orbit — a spectroscopic phase curve — gives the temperature profile as a function of longitude, and the latest measurements show profiles that are inverted on the day side and not on the night side, as expected if the inversion is caused by molecules that absorb starlight and are destroyed or condensed where there is none. Those spectra also show that the day side’s emission is not a single profile but a blend, which adds a further way for a spectrum to be misread: two regions with different gradients, averaged, can mimic a single region with a gentler one.

What was measured

An emission spectrum is a difference of two brightnesses, and its precision is set by photon counting. A hot Jupiter’s eclipse is two or three hours long; the star’s light is measured before, during and after, and the depth is the difference, at each wavelength, of a few parts per thousand. The Hubble Space Telescope’s infrared spectrograph measured the 1.4-micron water band this way to tens of parts per million for a few dozen planets; Spitzer’s broadband cameras at 3.6 and 4.5 microns gave two more points for many more; and the James Webb Space Telescope now measures continuous spectra from one to twelve microns, resolving the bands and the windows between them for the first time. The photon budget of such a measurement is the same one that limits transmission spectra, applied to a signal that is larger for hot planets and far smaller for temperate ones.

What those measurements find is consistent with the physics drawn here, and in one case they show it with unusual clarity: a hot Jupiter observed over a whole orbit, whose water band appears in emission on the day side and in absorption on the night side, as the inversion switches on and off with the illumination. The molecules did not change between one hemisphere and the other. Their temperature gradient did, and the spectrum followed it.

What the model behind the figures leaves out

The figures use an analytical grey profile and a parametric opacity with bands of fixed shape; the brightness temperature is the temperature where each wavelength’s optical depth reaches two-thirds. That is exact in the limit where the source function varies linearly with optical depth, and good to within tens of kelvin for real profiles, but it omits scattering, clouds, the dependence of molecular abundances on temperature and pressure through chemistry, and the fact that the bands themselves are forests of thousands of lines rather than smooth humps. Retrieval codes that fit real spectra include all of those, and the degeneracies they meet are larger than the one drawn here, not smaller.

The star is a blackbody in the figures and is not in reality; its own spectrum, with its own absorption bands, divides every eclipse depth, and a mismatch between the stellar model and the star can masquerade as a planetary feature. And the eclipse depth is measured, not the brightness temperature: the conversion requires the planet’s radius, from the transit, and the star’s spectrum, from a model, and each adds its uncertainty to every temperature quoted.

Still open: a stratosphere on a rocky planet

For temperate rocky planets the whole question changes scale. A rocky planet’s emission is a few parts per million of its star’s light, and the first measurements, made with the James Webb Space Telescope at fifteen microns for planets around small cool stars, have asked only whether the planet has an atmosphere at all — whether its day side is as hot as bare rock would be, or cooler because an atmosphere carries heat to the night side. A carbon dioxide band at fifteen microns, measured in emission, would say more: it would appear as absorption if the atmosphere cools upward and as emission if it has a stratosphere, and the Earth has one only because of ozone, made from the oxygen that life produces. The Sun has one too, of a different kind — its chromosphere, where the temperature climbs outward, is why the Sun’s limb brightens in the far ultraviolet rather than darkening. A thermal inversion on a rocky planet around another star would be the first measurement of an atmosphere’s vertical structure there, and depending on what caused it, possibly more than that.

Where this goes next

One whole orbit. The system's total brightness through one orbit: the transit at phase 0, the slow rise and fall of the planet's illuminated hemisphere between, and the secondary eclipse at phase 0.5 where the planet's own light is removed. The transit is 1.05%; the secondary eclipse is 1800 ppm, about 6 times shallower. The planet is seen when it disappears Half an orbit after the transit the planet passes behind its star, and the light that vanishes is the planet's own. Subtracting two brightnesses taken hours apart isolates a body nothing has ever resolved. Three planets that are the same spectrum. A model transmission spectrum, in scale heights of apparent radius, drawn three times: once as it is, once with the reference radius raised by 0.45 scale heights and the abundance reduced to compensate, and once with a cloud deck truncating the features. The three differ by 0.21 scale heights root-mean-square against features of 2.1, which is well inside the error bars of any real observation. The reason is structural rather than observational: a transmission spectrum measures a difference in apparent radius with wavelength and never an absolute radius, so the level is a free parameter, and shifting the level trades against the abundance almost exactly. Adding a cloud deck adds a third parameter that flattens features and trades against both. Three unknowns and one curve is why the quoted abundance uncertainties from transmission spectroscopy are so much larger than the photometric precision suggests. A spectrum flattened by cloud, or by nothing A transmission spectrum measures how a planet's apparent radius changes with wavelength, and never the radius itself. That missing level is a free parameter, it trades almost exactly against the abundance of whatever is absorbing, and a cloud deck adds a third unknown to a curve that constrains two. The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in. The light that is missing from the edge The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

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Abundance degeneracyBrightness temperatureContribution functionEmission spectrumSecondary eclipseTemperature profileThermal inversionWater band