A radius that depends on the colour it is measured in

A planet with no atmosphere has one radius. A planet with an atmosphere has as many radii as there are wavelengths, because at each colour the light is blocked out to whatever height the gas becomes opaque there.

So a transit measured in the blue and a transit measured at 1.4 microns give different depths for the same planet, and the difference between them is a measurement of what the atmosphere is made of.

Why the signal is a scale height #

An atmosphere in hydrostatic equilibrium thins exponentially with height, with an e-folding distance

$$H = \frac{kT}{\mu g},$$

set by the temperature, the mean molecular weight and the surface gravity. For a hot Jupiter — 1,400 K, hydrogen-dominated so $\mu = 2.3$, and a surface gravity of about 9 m/s² because the planet is enormous and not very heavy — that is 538 km, against a planetary radius of 98,000 km.

At a wavelength where the gas absorbs strongly, the atmosphere becomes opaque higher up; where it is transparent, the light gets deeper before being blocked. The difference between “strongly absorbing” and “transparent” is typically five to ten scale heights, because opacity changes by orders of magnitude across a molecular band and the density falls by a factor of $e$ per scale height.

So the change in transit depth across a band is

$$\Delta\delta \approx \frac{2 R_p\, n H}{R_\star^2},$$

which for the numbers above is 153 ppm per scale height, or of order 1,000 ppm for the strongest features. Against a transit of 14,000 ppm, that is a seven per cent modulation of a signal that is itself one part in seventy.

This is the smallest routine measurement in the field, and the reason it is possible at all is that it is differential: the same star, the same instrument, minutes apart, with only the wavelength changed. Nothing about the star’s absolute brightness needs to be known, which matters — the star’s radius is the dominant uncertainty in the planet’s radius and cancels almost entirely out of a ratio of depths.

Which planets are worth attempting #

The expression above says everything about target selection, and it is worth reading off.

$H$ is proportional to $T/\mu g$. So the signal is large for hot planets, for planets with low gravity, and for atmospheres of light molecules. A hot Jupiter satisfies all three: it is at 1,000–2,000 K because it is close in, its gravity is low because it is puffy, and it is hydrogen-dominated because it is massive enough to have kept its hydrogen.

Reverse each one and the signal collapses. An Earth-like atmosphere is $\mu = 29$ rather than 2.3, twelve times heavier, and at 288 K rather than 1,400. Its scale height is 8.5 km, and against a Sun-sized star the whole spectral modulation is under one part per million.

The escape is the same one the transit method always uses: shrink the star. Around an M dwarf a temperate rocky planet’s signal is boosted by the square of the radius ratio — a factor of seventy for TRAPPIST-1 — which turns an impossible measurement into a merely very difficult one. Every current attempt at a rocky planet’s atmosphere is around a small star for exactly this reason.

What the flat spectra mean #

The single most common outcome of a transmission observation is a featureless spectrum: the depth is the same at every wavelength, to the precision available.

That is the interpretation, and it took some years to be accepted. Clouds and hazes are grey — opaque across a wide range of wavelengths — so a deck high in the atmosphere truncates every molecular feature at the same level and leaves a flat line. Aerosols are expected in these atmospheres on general grounds: at 1,000–2,000 K, silicates, iron and various sulphur compounds condense, and photochemistry produces hydrocarbon hazes.

The consequence is severe. A flat spectrum says almost nothing about composition, and roughly half the well-observed hot Jupiters produce one. The information is not merely noisy; it has been physically blocked.

Two partial escapes exist. The first is to observe at longer wavelengths, where aerosol opacity falls and the deck becomes transparent — which is a large part of what JWST was built to exploit. The second is to observe at very high spectral resolution from the ground, resolving individual lines whose narrow cores form far above any cloud deck.

The lines themselves #

The features in the figures are the same absorption lines that stellar spectroscopy has read for a century, seen in a different geometry.

What is detectable depends on wavelength. In the optical, the alkali metals dominate: the sodium doublet at 589 nm and potassium at 770 nm, both with enormously broad pressure-broadened wings in a hot atmosphere. In the near infrared, water has bands at 0.95, 1.15, 1.4 and 1.9 µm, and the 1.4 µm band is the workhorse because it sits where Hubble’s near-infrared spectrograph is most sensitive. Further out, carbon monoxide at 2.3 and 4.6 µm and carbon dioxide at 4.3 µm are the strongest tracers of the carbon-to-oxygen ratio.

That ratio is the prize. Different ices condense at different distances in a protoplanetary disc, so the gas a planet accretes has a carbon-to-oxygen ratio that depends on where it grew — which means a spectrum, unlike a mass or a radius, is a statement about formation location, and therefore a direct test of whether the planet moved. Pinning a planet’s C/O to better than a factor of two is currently the field’s most active argument.

What is actually being sampled #

A transmission spectrum probes the terminator — the ring of atmosphere around the planet’s limb, seen edge-on — and not the day side or the night side.

That has two consequences. The path length through the atmosphere at the limb is long, of order $\sqrt{2\pi R_p H} \approx 20{,}000$ km for the numbers above, which is what makes a thin shell detectable at all: the light crosses forty times more gas than a vertical path would.

And the terminator is not a single place. A tidally locked hot Jupiter has a permanent day side and night side, so its morning and evening terminators can differ by hundreds of kelvin and carry different chemistry and different clouds. A transmission spectrum averages the two, and asymmetries between ingress and egress — measurable now, barely — separate them.

The number of photons required #

It is worth putting the precision requirement beside the other measurements in this collection, because it is easy to say “a few hundred parts per million” without registering what that costs.

A relative photometric precision of $10^{-4}$ needs at least $10^8$ photons collected in the band, and in practice ten to a hundred times more, because the limiting noise is rarely Poissonian. Detector persistence, pointing jitter moving the spectrum across pixels of different sensitivity, thermal drifts, and the star’s own variability all enter at the level being measured.

The standard defence is the same in every case: measure the difference, not the value. In-transit spectra are divided by out-of-transit spectra taken minutes earlier through the same optics. Almost everything that is not the planet cancels. What does not cancel is anything that changes on the timescale of the transit and depends on wavelength — which is a short list, and it is exactly the list of systematics that has occupied the field for twenty years.

What was actually measured #

Sodium in HD 209458 b, 2002. The first detection of an exoplanet atmosphere. Hubble measured the transit depth in a narrow band around the sodium doublet and found it deeper by $2.3\times10^{-4}$ than in the surrounding continuum — a 0.023 per cent difference on a 1.6 per cent transit. Sodium had been predicted as the strongest optical feature four years earlier.

The escaping hydrogen, 2003. The same planet observed in Lyman-α showed a transit depth of about 15 per cent — ten times the optical depth — implying that hydrogen fills a region larger than the planet’s Roche lobe and is escaping. The measurement is not of a bound atmosphere at all but of a tail.

Water, and its absence, 2013–2016. A Hubble survey of ten hot Jupiters found water in some and flat spectra in others, with a continuum between. The correlation is with aerosol opacity rather than with actual water abundance — a conclusion reached by combining the optical slope, which measures scattering, with the infrared band amplitude.

Carbon dioxide in WASP-39 b, 2022. JWST’s first transmission spectrum showed a $4.3\ \mu$m carbon dioxide feature at 26 standard deviations — a signal of about 500 ppm, measured on a planet 200 parsecs away. The same data showed sulphur dioxide, which is a photochemical product and therefore evidence of chemistry driven by the star rather than of primordial composition.

And the rocky planets. TRAPPIST-1’s planets have been observed repeatedly. The current results are consistent with no detectable atmosphere on the innermost ones, which is itself a measurement: a hydrogen-dominated envelope would have produced a signal far larger than the limits allow, so those planets do not have one.

The molecular weight, which is what a small planet’s spectrum is really about #

For a hot Jupiter the atmosphere is hydrogen and the question is what trace species it contains. For a small planet the first question is different and much cruder: what the atmosphere is made of in bulk, which is to say what its mean molecular weight is.

That matters because the scale height carries $\mu$ in the denominator, so the signal size is itself the measurement. A sub-Neptune with a primordial hydrogen envelope has $\mu \approx 2.3$ and a spectral modulation of hundreds of parts per million; the same planet with a secondary atmosphere of water, carbon dioxide or nitrogen has $\mu$ between 18 and 44, a scale height smaller by a factor of ten to twenty, and a modulation at the level of tens of parts per million or less.

So the two hypotheses differ by an order of magnitude in amplitude before any molecular feature is identified, and the first thing a spectrum of a small planet settles is which regime it is in. A detection of any feature at the hundreds-of-parts-per-million level rules out a heavy atmosphere; a non-detection at that level rules out a light one, provided the precision is good enough.

The awkwardness is the third possibility. A hydrogen atmosphere with a high cloud deck produces the same flat line as a heavy atmosphere with no clouds at all, and distinguishing them means either reaching the tens-of-parts-per-million precision at which a heavy atmosphere’s own features appear, or observing at wavelengths where the clouds are transparent.

That is the current frontier and it is a precision problem rather than a conceptual one. The distinction between a stripped rock, a steam atmosphere and a cloudy mini-Neptune is a distinction between three amplitudes, and the observations that will settle it are the ones now being attempted on the small planets around the nearest small stars.

There is a further complication that the amplitude argument alone does not capture, and it is specific to the small stars these planets orbit. A heavy atmosphere is expected to be thin in extent as well as in signal, so its features form deep, where the pressure broadening is strong and the bands are wide and shallow rather than narrow and deep. A light atmosphere’s features form high, and are correspondingly sharper. So the two hypotheses differ in the shape of the spectrum as well as in its amplitude — and shape survives an error in the reference radius, which amplitude does not.

That is the same structural preference this collection keeps arriving at: where a measurement can be made to depend on a shape rather than on a level, it should be, because the level carries every calibration error and the shape carries almost none. It is the reason the strongest claims about small-planet atmospheres are made from the relative depths of two bands rather than from the absolute depth of one.

A quantity that appears in the denominator of the signal is measured by how large the signal is, which is an unusually direct arrangement and an unusually unforgiving one: a null result is informative only if the noise is understood.

Where the picture stops #

Stellar contamination is the dominant systematic for small stars. A planet crossing a spotted star samples a different part of the stellar spectrum than the disc average, and the resulting “transit light source effect” produces wavelength-dependent depth changes that mimic planetary features. On an active M dwarf it can exceed the planetary signal entirely, and correcting it requires a model of the star’s surface that nobody has.

The reference radius is unmeasurable. The spectrum gives relative depths, so the absolute pressure level to which the radii correspond is unknown, and a spectrum can be shifted vertically by trading the reference radius against the abundances. The degeneracy is real and is one reason abundances are quoted with generous errors.

Retrieval is model-fitting. Turning a spectrum into abundances means fitting a forward model with assumed temperature structure, chemistry and cloud parameterisation. Different retrieval codes on the same data have disagreed by an order of magnitude in inferred water abundance.

And a transmission spectrum cannot see a surface. It probes the pressures where the atmosphere becomes opaque along a very long slant path, which for a hot Jupiter is around a millibar — far above anything one might call a surface, on a planet that does not have one.

Reading a spectrum backwards #

The step from a spectrum to a composition is called retrieval, and it is worth describing because it is where the honest uncertainty lives.

A forward model takes an assumed temperature–pressure profile, a set of molecular abundances, a cloud prescription and a reference radius, and computes the transit depth at every wavelength by integrating the opacity along every slant path through the atmosphere. Retrieval runs that model tens of millions of times inside a sampler, exploring the parameter space until the posterior distribution of the parameters is mapped.

Three features of the result recur, and they are properties of the problem rather than of any code.

Abundances trade against the reference radius. Raising the assumed radius at the reference pressure and lowering every abundance produces nearly the same spectrum, so what a single band constrains is a product rather than an abundance.

Abundances trade against clouds. A muted water feature is a low water abundance or a high cloud deck, and separating them needs either the optical scattering slope or a second band with different cloud sensitivity.

And the temperature structure is mostly assumed. Transmission is only weakly sensitive to the vertical temperature profile, so a profile has to be imposed — and the scale height, which sets the whole signal, is proportional to that temperature.

The consequence is that the well-determined quantity is usually a ratio of abundances rather than any absolute one, which is fortunate: the carbon-to-oxygen ratio, the quantity formation models actually predict, is exactly a ratio.

The generalisation #

Reading a composition from light that has passed through something, rather than from light emitted by it, is a technique the subject uses at every scale.

The solar chromosphere was discovered this way, in the flash spectrum seen for two seconds at the start and end of a total eclipse. The interstellar medium’s composition is read from narrow absorption lines superimposed on the spectra of distant hot stars. The Lyman-α forest — thousands of absorption lines in a quasar’s spectrum — maps intergalactic hydrogen along a line of sight billions of light years long, and is one of the primary probes of cosmological structure.

The exoplanet case is the hardest of these by a wide margin, because the absorbing column is a thin annulus and the background source is a hundred thousand times brighter. It also happens to be the only one in which the absorbing object’s geometry is known exactly, which is what makes the extra depth interpretable as a height.

The signal is one scale height, and the two things that set a scale height are worth moving one at a time.

Where this goes next #

Transmission is not the only way to reach the atmosphere. The planet also has its own light — it is not merely a silhouette — and subtracting two brightness measurements a few hours apart isolates it.

Later rungs on this anchor: retrieval and its degeneracies. The transit light source effect. High-resolution cross-correlation spectroscopy. The carbon-to-oxygen ratio and formation location. Aerosols and condensation chemistry. Terminator asymmetry. Escaping atmospheres in Lyman-α and helium 10830. Secondary eclipse spectroscopy. Phase curves and heat redistribution. And the atmosphere of a rocky planet, which is where the whole technique is trying to get to.