Water that is on the star and not the planet

A transmission spectrum is the planet’s apparent size as a function of wavelength. At a wavelength where its atmosphere is opaque the planet blocks more light, and so the transit is deeper. Read colour by colour, the depth is a spectrum of the atmosphere, written in very small differences. For a hot Jupiter the differences are hundreds of parts per million. For a rocky planet in a temperate orbit around a small red dwarf, the best case for finding air on an Earth-sized world, they are a few tens to a hundred.

Every such measurement assumes that what the planet covers is a fair sample of the star. That assumption is stated so rarely that the effect of its failure took decades to get a name. It was seen in the early transit studies of active stars, and then in 2018 Benjamin Rackham, Dániel Apai and Mark Giampapa worked through what it means for the stars that matter most: the transit light source effect. Their result was that on the smallest stars the effect may be larger than any atmosphere it is trying to measure, and that it has the same shape.

A depth is a ratio with the wrong denominator #

A transit’s depth is the light lost divided by the total light. The planet blocks a chord across the disc, and the light lost is the brightness of that chord times the planet’s area. The total light is the whole disc, including all its spots and bright patches.

If the disc is uniform those two brightnesses are the same and the depth is the true area ratio, $D=(R_p/R_\star)^2$. If the chord happens to be clean, while a fraction $f_s$ of the disc elsewhere is covered by spots of spectrum $S_s$, then the disc’s total light is lower than the chord’s brightness suggests, and the observed depth is

$$D_{\rm obs} = \varepsilon(\lambda)\,D,\qquad \varepsilon(\lambda)=\frac{1}{1-f_s\left(1-S_s(\lambda)/S_{\rm ph}(\lambda)\right)} .$$

The factor is never measured directly and there is no reference for it within a transit: nothing about the light curve says the depth is off. If the planet crossed an actual spot, the crossing would show as a bump in the light curve and could be dealt with. The case that does the harm is the one with nothing to see.

Because a spot is cooler than the photosphere, its spectrum is redder. The ratio $S_s/S_{\rm ph}$ is farthest below one in the blue. So the multiplier is largest in the blue, and the planet appears larger at short wavelengths. That is the exact behaviour of a planet with a high-altitude haze, for which scattering by small grains grows towards the blue. A fitted slope in a transmission spectrum may be a haze on the planet, or a clean strip on the star.

The bright patches pull the other way #

Stars also have faculae, bright regions associated with magnetic fields. If the chord misses those, it is darker than the disc average rather than brighter, and the factor is below one. Bright patches therefore pull the depth down, again most in the blue.

The obvious rescue is to measure the star’s total spectrum and work out what fraction is spot and what fraction is facula. This turns out to be harder than it sounds. Disc-integrated spectra constrain a sum of contributions, and with temperatures and fractions all free, several combinations give nearly the same result. Worse, the models of a red dwarf’s atmosphere used for the fit are themselves imperfect in exactly the wavelengths where the correction matters. A disagreement of a few per cent between observed and model spectra is common for these stars, and that is a figure large enough to drive the fit by itself.

Where the star writes water #

On a star cooler than about 3500 K, the coolest gas in the photosphere and particularly in spots has water vapour in it. Spots on these stars are the same gas at a few hundred degrees less, and they hold more. The depth of a water band in a spot’s spectrum can be twice that of the photosphere. The spot’s spectrum therefore dips farther below the photosphere’s inside the band, the ratio $S_s/S_{\rm ph}$ falls there, and the multiplier rises.

So a planet with no atmosphere whatever, transiting a clean chord on a spotted red dwarf, appears larger in the water bands. Its depth has a bump at 1.4 micrometres and another near 1.9 that are not on the planet at all.

The fingerprints are not identical. Hot water on a star and cool water on a planet have somewhat different band shapes when resolved well, and stellar and planetary bands have different widths. A high-resolution instrument can separate them in principle. But the planets of most interest are faint and small, a spectrum at a resolving power of a hundred is what is available, and at that resolution the two are a close match.

This has already happened. On the first planets surveyed by the James Webb Space Telescope around small stars, for instance in the TRAPPIST-1 system, some transmission spectra have shown slopes and bands that vary between visits. An atmosphere on a planet does not change between two transits a few days apart. A pattern of spots on a star that has rotated does. The first results from those programmes indicate that stellar contamination, rather than instrument precision, is setting the limit on the search.

How large a false band can be #

The size of the error depends on three things: how many spots there are, how much cooler they are, and how red the star is. In the simple version it grows in proportion to the coverage, since the denominator is close to one for coverage well under a half.

The arithmetic turns against small stars for a reason that is intrinsic to why they are studied. The depth of a transit by an Earth-sized planet grows as one over the star’s area. So the favourable stars for atmospheric work, the smallest ones, are also the ones with the largest depth for the error to multiply: a factor of a few per cent on a depth of seven thousand parts per million is a hundred or two parts per million. That same depth increase raises the atmospheric signal, but only in proportion, so the ratio of signal to contamination is not improved. Then, the smallest stars are also the most magnetically active, and on them spots are expected to cover tens of per cent of the surface. It is the most unlucky possible arrangement: the conditions that make a small planet’s air measurable are the same ones that hide it.

A Sun-like star with a Jupiter transiting it is in a very different position. The depth is one per cent, the spot coverage a per cent or less at most, the spot’s contrast is high but its coverage low, and it holds no water at the Sun’s temperature. Contamination there is a small correction to a large signal, of the order of tens of parts per million at the very bluest wavelengths. That is the reason it was not seen as a central problem for a decade.

Why a steady star proves nothing #

It seems that the coverage could be measured from how much the star’s brightness varies. A spot rotating into view dims the star, and a typical light-curve amplitude is a per cent or so. A star which appears to be steady ought to have no spots, and it might seem a good choice for a target.

This is a serious fallacy, and it is worth being explicit about why. Rotation sees only the part of the surface pattern that differs between one side and the other. A uniform scatter of spots looks the same from every direction and produces no variation at all, whatever the total coverage.

For a random pattern of $N$ equal spots the amplitude falls as $\sqrt{f/N}$ or so, as the sum of a random walk. A quiet light curve therefore rules out only a small number of big spots. It says little about many small ones, and on a fully convective red dwarf, where magnetic fields are organised differently, many small spots are what is thought to be there. Rackham and colleagues compute that typical red dwarfs with a one-per-cent light curve may carry coverage ten or more times higher.

There is a second blindness for pole-on stars, for which a polar spot cap never rotates in or out of view. Indeed, such a pattern is thought to be common among rapid rotators. And the coverage varies on timescales of many rotations and with a magnetic cycle, so a value inferred from one season may not apply to the next. Taken together, the photometry can set a floor on what is on the star, and it sets almost no ceiling.

Three ways round it, each with a price #

Observe at longer wavelengths. The multiplier falls towards the infrared because a spot and the photosphere both approach the same slope in the Rayleigh–Jeans limit. Beyond several micrometres it is smaller by a factor of a few. That helps the slope. It leaves the molecular bands of the star in place, however, and it also moves away from where a planet’s water and methane bands are strongest.

Use the transit itself. Direct crossings of spots show up as bumps, and bright patches as dips, in the transit light curve. These do constrain the spot’s temperature and sizes along the chord and hence the factor. They are rare and cannot speak for the rest of the star, but several analyses extract useful information from them.

Compare transits. If a planetary signal repeats at the same depth over visits and the star’s pattern does not, then averaging over many transits reduces stellar variations but not the planet’s. This is the most powerful method, and it has a catch: it reduces only the part of the error that changes. A steady population of spots makes a permanent offset, and no number of transits can remove that. There is also the option of watching multiple planets around a single star, for which the error is the same multiplicative factor and each atmosphere is different: the same logic used for a star’s density applies here. A bare inner planet could serve as a calibration for the star’s contamination, if it is known to be bare, and hot rocks do appear to be bare. This is the design behind the most recent programmes.

What the picture cannot show #

The figures use Planck functions, and a single imposed band. Real red-dwarf spectra are forests of molecular lines in which the opacity and the emergent flux are not smooth at any wavelength, and that would have to be computed by atmosphere models in three dimensions, for temperatures at which those models still do not reproduce the observed spectra. So the numbers here are only the right order of magnitude.

The geometry has also been reduced. Limb darkening means the chord is not sampled at uniform brightness. Spots near the limb contribute less than spots at disc centre. A spot straddling the chord only partly is a mixture. And the factor $\varepsilon$ depends on which region of the star faces the observer at that moment; in the real measurement it varies from one transit to another, and the essay’s formula gives only the averaged version.

Finally, there is a conceptual limit. The formula is a correction to a ratio that needs knowledge of a chord spectrum that is never observed on its own. It has to be estimated from a model of the star. Thus any measurement of a planet’s atmosphere on these stars is a joint measurement of the atmosphere and of the star, in which there is no clean way to assign the signal to one or the other without additional assumptions. That has the same shape as the star subtracted to find a planet in velocity, and it is resolved in similar ways.

A planet that measures a star #

There is a pleasant reversal in the end. Transit light source contamination is a nuisance in atmospheric work, but if a planet is known to have no spectrum of its own, then the wavelength dependence of its depth is a direct measurement of the star’s unresolved surface. A dark airless planet is a probe of what fraction of a star is spotted, one that needs no assumption about how spots are arranged, and that no photometric monitoring could give. It samples one chord of the surface for a few minutes and compares that chord with the whole, at every wavelength, with a known geometry.

That is an unusual method for a stellar measurement. It gives something on stars a thousand times too small to resolve, which is a direct reading of the contrast between a clean patch and the star’s mean. So a survey of bare rocks would produce, as a by-product, the first census of spot coverage on the coolest stars.

How the star’s own spectrum can be used against itself #

Since the contaminating signal comes from the star, a spectrum of the star should in principle contain enough information to remove it. In practice this is the most actively pursued route, and its difficulty shows what is at stake. Out of transit, a telescope measures the sum over the disc of the photosphere, spots and faculae, each weighted by area. A model grid gives a spectrum at each temperature. A fit then gives the fractions and temperatures. If the model were correct, the factor $\varepsilon(\lambda)$ would follow immediately, with no assumption about the planet.

The grid is the weak point. For stars of about 2,600 kelvin, model spectra disagree with observed ones by several per cent in the near infrared, which is many times the accuracy needed. When such a fit was run on real spectra from the James Webb Space Telescope, the best-fitting combination often had a spot fraction that made little physical sense, indicating that the fit was compensating for a model deficiency. Empirical templates have been suggested instead, in which spectra of many quiet and active stars of similar type are compared directly. They would remove the dependence on the grid, but they would also bring the assumption that spots on one star look like the disc of another.

Multiple visits give extra leverage here too. Since the star’s pattern changes with rotation, any out-of-transit spectrum taken at different rotational phases shows variations produced by a known change in the surface mix. The shape of those variations gives directly a spectrum of the difference between spotted and unspotted regions, independent of a grid. The same reasoning is what lets radial-velocity observers subtract a star’s own variation from a planet’s signal. But it gets only the part of the pattern which varies. The uniform part remains invisible, and that part is the one the light curves cannot reach either.

There is also a limit on what any correction can achieve, and it should be stated without optimism. The correction is multiplicative and it is applied to a depth of thousands of parts per million. To reach ten parts per million on the planet’s signal requires that $\varepsilon$ be known to better than one part in several hundred, at every wavelength. That is a stronger requirement than any stellar atmosphere model has met, by about a factor of ten. No single advance in instrumentation will change that, since the problem is not one of photons. It is a problem of not knowing a star’s surface well enough, which is the problem that observations of these planets were trying to circumvent.

Still open: whether a temperate red-dwarf planet can be read at all #

The central question is now practical and very pointed. If stellar contamination in these systems reaches a hundred parts per million and is stable in part, then what can be said about a temperate rock whose atmospheric signal has about the same size? The optimistic view is that models of the stars will improve, that observing many transits will average down the variable part, and that the comparison with bare inner planets will take care of the rest. The pessimistic view is that a stable offset cannot be separated from an atmosphere of the same spectrum, and that the transmission method on the coolest stars has reached a limit set by the stars themselves.

The most hopeful alternatives avoid the star’s disc altogether, by way of heat measured at secondary eclipse, where a planet’s own emission is compared with a star that is simply absent. Whether that alone can establish the presence of an atmosphere on a planet with temperate surface temperatures is the other half of the same open problem. A programme of about five hundred hours on the James Webb Space Telescope is now being spent on it. Its first result, on this argument, may be a better knowledge of the stars.