A spectrum flattened by cloud, or by nothing

A transiting planet with an atmosphere is bigger at some wavelengths than at others: where a molecule absorbs, the atmosphere is opaque higher up, and the planet’s silhouette is larger. Measuring the transit depth as a function of wavelength therefore measures the atmosphere’s composition.

The signal is small. The relevant length is the atmospheric scale height — the height over which the pressure falls by a factor of e — and a strong absorption band raises the apparent radius by a few of them. For a hot Jupiter that is a few hundred kilometres out of seventy thousand, so the transit depth changes by a part in a thousand of itself.

A planet’s radius depends on the colour it is measured in, and that dependence is the measurement. This essay is about the fact that the dependence is all there is: the curve’s shape is exquisite and its height is unknown.

Why there is no absolute level #

The apparent radius at a wavelength is the height at which the slant optical depth through the atmosphere reaches about unity. That height depends on the abundance of the absorber, on the temperature, on the gravity, and on the radius at some reference pressure — and nothing in the observation fixes the last of these.

The reason is that the atmosphere has no bottom in the data. Below the level where the atmosphere becomes opaque at all wavelengths, the planet is simply a disc, and the transit depth there is the same whatever the composition. So the observation constrains the shape of the radius-versus-wavelength curve and its overall level only through the transit depth’s absolute value, which is itself uncertain at the level of the stellar radius and the limb darkening.

In practice the reference radius is treated as a free parameter in a retrieval, and it is degenerate with the abundance because both act to move the whole curve up or down: more absorber raises the features, and a larger reference radius raises everything.

It is worth writing the relation down once, because it makes the degeneracy explicit. The apparent radius at wavelength $\lambda$ is approximately

$$R(\lambda) = R_0 + H\ln\!\left(\frac{\chi\,\sigma(\lambda)\,P_0}{\tau_{\rm eq}}\right),$$

with $H$ the scale height, $\chi$ the absorber’s mixing ratio and $R_0$ the radius at a reference pressure $P_0$. The abundance appears inside a logarithm and the reference radius appears outside it as an additive constant, so multiplying the abundance by ten and lowering the reference radius by $2.3H$ give the identical curve. That is not an approximate degeneracy that better data will resolve; it is an exact one, broken only by information the single-band shape does not contain.

It helps to carry one worked set of numbers through, because the sizes involved decide which of the escapes below are worth building an instrument for. Take a hot Jupiter of radius one hundred thousand kilometres around a star of eight hundred and thirty thousand: the transit depth is the square of the ratio, about one and a half per cent. At thirteen hundred kelvin, a gravity of nine metres a second squared and a hydrogen-dominated composition, the scale height is a little over five hundred kilometres. A band that raises the apparent radius by five scale heights therefore adds twice the radius times that height, divided by the star’s area — about seven hundred parts per million on top of fifteen thousand.

So the feature is five per cent of the transit depth, and the transit depth is itself measured against a stellar radius known to a few per cent. The quantity that carries the composition is smaller than the uncertainty in the quantity it sits on, which is the arithmetic behind everything in this essay. It is also why the differential measurement is made at all: the same spectrum measured in one exposure against itself cancels the stellar radius exactly, at the cost of cancelling the level along with it.

The cloud deck #

A cloud or haze adds a grey opacity — one that does not depend on wavelength — at some pressure. Above the cloud the atmosphere is transparent and the features are visible; below it nothing is seen.

The effect on the spectrum is to truncate the features from below, flattening them. A spectrum with small features is therefore consistent with two quite different atmospheres: a clear one with a low abundance, and a cloudy one with a high abundance whose features are hidden.

The cloud’s effect is worth distinguishing from the abundance’s, because they are not the same kind of flattening. A low abundance shrinks the features while keeping their shape — the bands are simply weaker. A cloud clips them, so the band cores are flat-topped while their wings are unaffected. In principle that difference is measurable and in practice it requires signal-to-noise that most spectra do not have, which is why the two are usually reported as degenerate rather than as distinguishable in principle.

And there is a fourth parameter that is often overlooked and that acts the same way: the temperature. The scale height is proportional to it, so a hotter atmosphere has larger features at the same abundance, and a retrieval that fits the temperature as well is fitting four parameters to one curve. The temperature at the terminator is not the equilibrium temperature and is not measurable from the transmission spectrum independently, so it is either fixed at an assumed value — which imports an assumption — or fitted, which widens everything. Two methods and one density is the same arithmetic in a different currency: more unknowns than independent constraints, resolved by importing one.

What breaks it #

Four things break the degeneracy, and their relative power is worth knowing.

A wide wavelength range. A cloud is grey and a molecule is not, so observing across a range wide enough to include both a strongly absorbing region and a nearly transparent one separates them: the cloud raises the baseline everywhere and the molecule raises it only in bands. That is the main argument for observing from the optical to the mid-infrared rather than in one band.

A Rayleigh slope. Scattering by small particles rises steeply towards the blue, and the slope’s steepness measures the temperature divided by the mean molecular weight. A measured Rayleigh slope therefore constrains a combination that the bands alone do not, and it anchors the level.

A resolved line core. At high spectral resolution the core of a strong line probes far higher in the atmosphere than the broad-band feature, where the cloud is below and cannot hide it. High-resolution cross-correlation spectroscopy detects a molecule’s presence by its pattern of hundreds of lines and is nearly immune to a grey continuum — which is why it has succeeded on planets whose low-resolution spectra were flat.

And a mass. The scale height depends on the gravity, so an independent mass tightens the relation between the observed feature amplitude and the abundance. That is one of the reasons a planet with a radial-velocity mass is a far better atmospheric target than one with only a radius.

There is a fifth route that has become important and does not fit the pattern of the other four: measure the planet’s mass loss. A planet losing its atmosphere shows an enormously deeper transit in a resonance line of an escaping species, and the depth constrains the density of the escaping gas rather than the abundance at the terminator. It is a different quantity, but it is a quantity from the same atmosphere measured with a completely different sensitivity to clouds — clouds sit at millibar pressures and the escaping gas is at nanobars, far above anything condensable. So the two measurements bracket the atmosphere from opposite ends, and a planet with both is far better characterised than one with either.

What was actually measured #

Three of them, and the third changed the interpretation of the first two.

Flat spectra, everywhere. The first surveys of hot Jupiter transmission spectra found that a large fraction were featureless or nearly so, at a time when clear solar-composition atmospheres had been predicted. The interpretation was clouds, and it was correct in outline and unconstrained in detail.

A sodium line, resolved. The first detection of an exoplanet atmosphere at all was a sodium absorption feature, and it was later resolved at high spectral resolution, showing the line’s core and its shape. That measurement is immune to the cloud degeneracy because it probes above any plausible cloud deck, and it demonstrated the technique that has since found several molecules in atmospheres whose broad-band spectra are flat.

Water abundances, spanning two orders of magnitude. Retrievals of water abundance in hot Jupiters gave values from far sub-solar to solar, and the spread was widely interpreted as a real diversity in formation. Reanalyses that treated the cloud and reference-radius parameters more carefully — and that used wider wavelength coverage — brought most of them towards solar, and attributed the earlier spread to the degeneracy rather than to the planets.

What a retrieval actually does #

The word retrieval suggests an inversion — a procedure that reads a composition off a spectrum. It is not one. A retrieval is a forward model of a spectrum, given a composition, run inside a sampler that explores the parameter space and reports the region consistent with the data.

That distinction matters for reading the result. The output is a posterior distribution over ten or fifteen parameters, and the number quoted for an abundance is a one-dimensional projection of it. When two parameters trade against each other the joint posterior is a long curved ridge, and projecting a ridge onto one axis produces a wide, skewed interval whose width is the degeneracy rather than the noise. Improving the data narrows the ridge across its short direction and does almost nothing to its length.

There is a further consequence that is easy to miss and that determines what the quoted lower limits mean. Along the degenerate direction the likelihood is nearly flat, so the posterior’s shape there is set by the prior — and the prior on an abundance is conventionally uniform in the logarithm over ten or twelve decades. A flat likelihood over a log-uniform prior returns the prior, so a retrieval reporting an abundance “consistent with sub-solar values down to a part in a million” is frequently reporting its own prior with the data’s ridge cut out of it. The tell is that the reported interval barely changes when the error bars are halved.

None of that is a criticism of the method, which is the correct one; it is a statement about what the method can return when the data constrain a combination rather than a parameter. The honest presentations quote the two-dimensional posterior — abundance against cloud pressure, or abundance against reference radius — and let the ridge be seen, because the ridge is the result.

Where the picture stops #

Three of them, and the second is the one that most affects small planets.

The retrieval’s priors matter. With three degenerate parameters and a curve, the posterior’s shape depends on the priors placed on the abundance, the cloud pressure and the reference radius — and those priors are choices. Two groups analysing the same spectrum with different priors report different abundances with overlapping error bars, and the difference is not a disagreement about the data.

A high mean molecular weight mimics a cloud. The scale height is inversely proportional to the mean molecular weight, so an atmosphere of water or carbon dioxide rather than hydrogen has features several times smaller. For a small planet, where the atmosphere might be either, a flat spectrum is consistent with a cloudy hydrogen atmosphere and with a clear heavy one — and distinguishing them is the central question about that entire population.

And the terminator is not one place. A transmission spectrum probes the ring of atmosphere at the day–night boundary, which has a temperature gradient across it and may have clouds on one side and not the other. The spectrum is an average over that ring, and fitting it with a one-dimensional model attributes structure to composition that belongs to geography.

A fourth limit is worth adding because it is the one that limits the best current data. Stellar activity contaminates a transmission spectrum: a starspot on the disc but not behind the planet makes the star fainter and the transit deeper, and since spots are cooler they are darker in the blue, so unocculted spots imprint a slope that mimics Rayleigh scattering. Correcting for it requires knowing the spot covering fraction, which is not measurable from the transit itself. For active stars — which include most small, cool planet hosts — that contamination is comparable to the atmospheric signal, and several claimed detections of scattering hazes have been reinterpreted as stellar.

Why the missing level is the interesting part #

The general lesson is worth separating out because it recurs whenever a measurement is differential.

A differential measurement is precise and has no zero. A transit depth is a ratio and inherits the star’s radius; a transmission spectrum is a difference of ratios and inherits nothing at all, which is why it can be measured to parts per million and interpreted to an order of magnitude. The precision belongs to the differential quantity and the accuracy belongs to whatever supplies the level.

That structure is the same one that makes an equivalent width depend on a continuum nobody observed and a spectroscopic gravity depend on a diagnostic that constrains a combination. In each case the observable is a difference, the difference is measured beautifully, and turning it into a physical quantity requires a reference that is not in the data.

The escape is always the same in form: find a second observable whose dependence on the missing quantity is different. A Rayleigh slope, a resolved line core, a wide wavelength baseline and an independent mass are four such observables here, and every one of them is a different measurement rather than a better version of the first.

A closing observation about how the field has responded, because it is a good example of a degeneracy driving instrument design. The response was not better photometry — the transit depths were already measured to a few parts per million — but wider coverage and higher resolution, both of which add a differently-behaved observable rather than improving the existing one. Instruments built after about 2015 for this purpose are specified by their wavelength range and their resolving power rather than by their photometric stability, and the reason is the arithmetic in this essay: with three parameters and one curve, the return on precision is zero and the return on a second axis is everything. The same conclusion was reached about planet radii, where the effort moved from the planets to the stars for the same structural reason.

Say what the degeneracy costs in scientific terms rather than in error bars, because that is what decides whether it is worth the instrument. The quantity the field wants from an atmosphere is not the water abundance but the ratio of carbon to oxygen, because that ratio records where in the disc the planet accreted its gas relative to the ice lines of water and carbon monoxide. A ratio is a difference of two logarithms, so the reference-radius offset cancels out of it exactly and the cloud pressure very nearly does. The degenerate parameter drops out of the quantity of interest, which is why the field has moved to quoting ratios and why a spectrum too flat to give an abundance can still give a formation history.

Where the ladder goes next #

The natural next rung is the emission spectrum: what is learned from the planet’s own light when it disappears behind the star, which measures a temperature profile rather than a slant opacity and has a completely different set of degeneracies. The rung after that is the high-resolution route, where a molecule is identified by the pattern of hundreds of lines moving with the planet’s own orbital velocity, and where a grey continuum is irrelevant by construction.