A day side as hot as bare rock

The most direct way to find out whether a planet has an atmosphere is to read its spectrum, and for a planet the size of the Earth around a star like the Sun that is far beyond any existing telescope. Around the smallest stars, the red dwarfs, the problem is easier. The star is small and dim, so a rocky planet blocks a larger fraction of its light in transit and emits a larger fraction of the system’s infrared light. Its year lasts days, so it transits and is eclipsed dozens of times a year. Even so, a spectrum of a rocky planet’s atmosphere, molecule by molecule, is at the edge of what the James Webb Space Telescope can do, and for most targets it is out of reach.

There is a cruder question that can be answered first: is there an atmosphere at all? It has a clean answer, because an atmosphere changes something simpler than a spectrum. It changes how hot the day side is. A planet with no air re-radiates the starlight it absorbs from the place where it absorbed it, and its day side is as hot as a surface can be. An atmosphere, even a thin one, carries some of that heat away to the night side. The day side’s temperature, measured from the dip in infrared light as the planet passes behind its star, is therefore a test of whether there is air, and it can be made with a single measurement in a single band.

The hottest a day side can be #

A rocky planet this close to a red dwarf is almost certainly tidally locked, turning once per orbit so that one hemisphere faces the star permanently. On that day side, a surface with no atmosphere and negligible heat capacity on the timescale of the orbit is in instantaneous balance with the starlight falling on it: each patch emits exactly what it absorbs, as thermal radiation, at a temperature set by the local angle of the star above the horizon. The patch under the star, where the light falls straight down, is hottest; patches towards the terminator, lit at a grazing angle, are cooler.

Seen from the observer just before secondary eclipse, when the full day side faces the telescope, the brightness of that hemisphere is a weighted average of those temperatures, and the average works out to a definite number. For a planet with no reflection, at distance $a$ from a star of radius $R_\star$ and temperature $T_\star$,

$$T_{\rm day} = T_\star\sqrt{\frac{R_\star}{a}}\,(1-A_B)^{1/4}\,f^{1/4}, \qquad f = \tfrac23,$$

where $A_B$ is the Bond albedo, the fraction of starlight reflected. The factor $f = 2/3$ for bare rock is the largest it can be. Any mechanism that moves heat around — an atmosphere with winds, an ocean, a thick layer that stores heat — lowers the day side’s temperature, down to $f = 1/4$ if the heat is spread evenly over the whole planet. The reflection lowers it too.

An atmosphere spreads the heat #

How much an atmosphere lowers the day side depends mostly on how much of it there is. A thin atmosphere radiates its heat to space before winds can carry it far; a thick one carries it efficiently. Circulation models of tidally locked rocky planets have been distilled into a scaling for the redistribution factor in terms of the surface pressure and the planet’s equilibrium temperature.

For TRAPPIST-1b, the innermost of the seven planets around the red dwarf TRAPPIST-1, bare rock would give a day side of 508 kelvin. An atmosphere of a tenth of a bar, roughly that of Mars at the low end and a small fraction of the Earth’s, barely changes it. One bar brings it down by several tens of kelvin; ten bars, by more than a hundred. The planet’s day side was measured at 503 kelvin, from five eclipses observed at a wavelength of fifteen microns. That leaves little room for any atmosphere thicker than about half a bar, unless the surface is unusually reflective, which a hot, dark basaltic surface would not be.

The measurement itself is a careful extraction. The planet is seen when it disappears: the combined infrared light of star and planet dips by about 0.09 per cent when the planet passes behind the star, and the depth of that dip is the planet’s emission as a fraction of the star’s. Converting the depth into a temperature requires the star’s own infrared flux, and a cool red dwarf at fifteen microns is far from a simple blackbody — its atmosphere is laced with molecules that make its spectrum lumpy. The published temperatures use model spectra of the star, checked against the star’s measured flux. Treating the star as a blackbody at its effective temperature instead would put TRAPPIST-1b far above the bare-rock limit, a physical impossibility that says only that the approximation is poor.

Why an eclipse rather than a transit #

The usual way to probe an exoplanet’s atmosphere is the transmission spectrum: the planet’s apparent size measured colour by colour as it crosses the star, larger at wavelengths where its atmosphere absorbs. For hot giants it has been spectacularly successful. For rocky planets around red dwarfs it has a serious difficulty, and the difficulty is the star. Red dwarfs are covered in starspots and bright faculae, and a transiting planet crosses some of them and not others. The light it blocks is therefore not a fair sample of the star’s disc, and the mismatch imprints colour-dependent features on the transit depth that mimic an atmosphere — or hide one. For a planet whose apparent radius depends on colour because of its air, the star’s spots can produce the same dependence. For a small planet the spurious signal can be as large as the real one.

The eclipse is largely immune. When the planet passes behind the star, the change in light is the planet’s own emission, and the star’s spots and faculae are present before and after in the same proportions. The star’s variability adds noise, but not a systematic feature at the wavelengths that matter. The eclipse also measures a temperature, a physical quantity with a hard upper limit, rather than a spectral feature whose absence could mean no atmosphere or a cloudy one. A flat transmission spectrum of a small planet is ambiguous between those two. A day side at the bare-rock limit is not, because clouds would reflect light and lower the day-side temperature further. It cannot be cloudy and as hot as bare rock at the same time.

Four planets against their limits #

The same test has now been applied to several small planets around red dwarfs, at different wavelengths with different instruments, and the results can be put on one scale by dividing each measured day-side temperature by its own bare-rock limit.

Three of the four are at the bare-rock limit within their uncertainties. LHS 3844b, a planet about 30 per cent larger than the Earth on an 11-hour orbit, was the first: its day side measured at 1,040 kelvin, and its night side was undetectable. GJ 367b, smaller than the Earth and orbiting its star in eight hours, sits at the limit too. These planets are as hot as airless rock can be, and the simplest reading is that they are airless rock.

TRAPPIST-1c is the exception, and the one that shows the limits of the method. Its day side, 380 kelvin against a bare-rock limit of 434, sits about one in eight below the limit, beyond its uncertainty. That excludes a thick, Venus-like atmosphere of carbon dioxide, which would have cooled the day side much more and darkened it at fifteen microns as well. It does not exclude a thin atmosphere, and it does not require one.

One temperature, two unknowns #

The reason TRAPPIST-1c’s measurement is ambiguous is that the day-side temperature depends on two unknowns through their product.

The temperature fixes $(1 - A_B)\,f$ and nothing else. A dark surface with some air and a bright surface with none can give exactly the same day side. For TRAPPIST-1c the bright-bare-rock explanation needs a surface reflecting about forty per cent of the starlight, much brighter than fresh basalt. Weathered or frost-covered surfaces could reach that, and so could some mineral compositions, but they are not what a hot, airless rock exposed to a red dwarf’s radiation would be expected to look like. The dark-rock-with-air explanation needs an atmosphere of perhaps a fraction of a bar that is not rich in carbon dioxide.

Breaking the degeneracy needs a second measurement that depends on the two factors differently. The reflected light, measured at shorter wavelengths, constrains the albedo directly, though for a planet around a red dwarf the reflected light is extremely faint. A second thermal band gives the shape of the emitted spectrum, which differs between a bare surface and an atmosphere. And the night side, as the next section shows, does not depend on the albedo at all.

Why fifteen microns #

The wavelength of the TRAPPIST-1 measurements was chosen for a second reason besides the planets’ emission. Carbon dioxide, expected to be the dominant gas in the atmosphere of a rocky planet that has lost its water and kept its volcanic outgassing, has a strong absorption band centred at fifteen microns.

In the band, an atmosphere is opaque, and the emission that escapes comes from high, cold layers rather than from the hot surface. A planet with a carbon-dioxide atmosphere therefore looks dark at fifteen microns even if its surface is hot. The measurement is doubly sensitive. An atmosphere would lower the whole day side’s temperature by carrying heat away, and if it contained carbon dioxide it would darken the band further by absorbing. A bright fifteen-micron eclipse rules out both effects at once, and that is what was seen for TRAPPIST-1b. This is the same principle as reading an atmosphere’s temperature profile from its emission, used here not to read the profile but to detect whether there is a profile at all.

The design of the follow-up programme reflects this. A large allocation of telescope time has been set aside for measuring fifteen-micron eclipses of rocky planets around red dwarfs, a survey whose single question is whether each planet’s day side is as hot as bare rock, because that question can be answered for many planets with modest time each.

The night side has no albedo #

The cleanest test is not the day side at all. A thermal phase curve follows the planet’s infrared emission around its whole orbit, from just after transit, when the night side faces the observer, through eclipse, when the day side does.

Bare rock with no heat transport has a night side that cools to the faint glow of whatever internal heat the planet has, effectively zero against its day side. An atmosphere that carries heat across the terminator warms the night side, and the phase curve’s minimum rises. That rise does not depend on the albedo, since the night side receives no starlight to reflect, so a detected night-side glow means heat transport, and on a rocky planet heat transport means air. LHS 3844b’s phase curve, measured with the Spitzer Space Telescope, showed a night side consistent with zero and a day side at the bare-rock limit, with the brightest point at the substellar point rather than shifted eastward by winds: three independent signs of no atmosphere from one curve.

Phase curves are expensive, since they need the planet watched continuously around a full orbit, and the night side’s signal is small. For planets on orbits of hours they are practical. For the TRAPPIST-1 planets, with orbits of days around a faint star, they are at the limit of what can be done.

The airless worlds that can be visited #

The bare-rock limit is not only a theoretical bound; it is measured on the airless bodies of the solar system. The Moon, which keeps one face turned towards the Earth but rotates relative to the Sun once a month, has a day side whose temperature at the point under the Sun reaches about 390 kelvin, close to what instantaneous re-radiation predicts for its distance and albedo. Its night side falls below 100 kelvin within days, because the lunar regolith is a poor conductor and stores little heat. Mercury, locked to the Sun in a three-to-two resonance rather than one-to-one, reaches 700 kelvin at its hottest and about 100 at night. Both behave as bare rock should, and spacecraft and telescopes measuring their thermal emission see day sides at the predicted limit to within the effects of surface roughness and thermal inertia.

Those effects are the corrections the exoplanet measurements will eventually need. A rough surface, with craters and boulders, puts some of its area facing the Sun more directly than the average and some in shadow, and re-radiates preferentially back towards the Sun. Seen at full phase this makes it appear slightly hotter than a smooth sphere would. A surface with significant thermal inertia carries some heat past the terminator into the night, shifting the hottest point a little towards the afternoon. On the Moon both effects are measured; on a rocky exoplanet seen only as a point of light, they are folded into the uncertainty of what bare rock should look like. That uncertainty is a few per cent of the temperature, comparable to the measurements’ own errors, and it is why a day side within a few per cent of the limit is called consistent with bare rock rather than proof of it.

Why so many airless worlds #

That three of the first four rocky planets tested around red dwarfs appear airless is not surprising, and it may not be representative of all rocky planets. These planets are the easiest to test precisely because they are close to their stars, hot, and exposed. Red dwarfs are active for billions of years, flaring frequently and emitting strong ultraviolet and X-ray radiation, and they were brighter still when young. Close-in planets receive that radiation at intensities hundreds of times what the Earth receives from the Sun, and an atmosphere heated from above can escape at rates that strip it in a fraction of the system’s age. The stellar wind, dense and close, erodes what the radiation loosens.

A useful way to organise the expectation is a line in the plane of a planet’s escape velocity against the radiation it receives: planets on one side keep their atmospheres, those on the other lose them. In the solar system the line separates the bodies with atmospheres — Venus, the Earth, Titan — from those without, such as Mercury and the Moon. The planets tested so far lie on the airless side of an extrapolation of that line, and the result is consistent with it. The planets that matter most, rocky ones in the temperate zone of their stars where water could be liquid, lie closer to it and are harder to test. TRAPPIST-1c’s ambiguous result, near the line and below the bare-rock limit, is the kind of measurement on which the question will turn.

What the figures assume #

The bare-rock limit assumes a surface that re-radiates instantaneously, with no heat stored and released later, and a uniform albedo; real regolith has a finite thermal inertia that shifts the hottest point slightly towards the afternoon and a reflectivity that varies with angle. The redistribution scaling is a fit to a set of circulation models for atmospheres of particular compositions, and a real atmosphere’s heat transport depends on its composition, clouds and rotation. The spectrum figure is a sketch: the band’s depth and width depend on the amount of carbon dioxide and on the temperature profile above the surface, and are drawn here only to show where in wavelength the effect falls. The phase curve treats the atmospheric case as two uniform hemispheres and so misses the eastward shift of the hot spot that winds produce. The measured temperatures are taken as published, each derived with its own model of its star.

Still open: whether any rocky planet around a red dwarf has kept its air #

The first results all point one way, and the question is whether that way is universal. If every rocky planet close to a red dwarf is airless, the most common kind of rocky planet in the Galaxy — red dwarfs are three-quarters of all stars, and most have rocky planets — is a world without air, and the search for habitable environments has to concentrate on the rarer planets around Sun-like stars. If some keep their atmospheres, the boundary between those that do and those that do not will say how atmospheres survive a red dwarf’s youth.

The planets that would settle it are the cooler, more distant ones, which receive less radiation but whose day sides are cooler and whose eclipses are shallower, harder to measure at fifteen microns. TRAPPIST-1c sits where the answer begins to be possible and gives an ambiguous result. Measurements of planets further out in the same system, and of temperate planets around slightly larger red dwarfs, are under way. They will need the night side and a second thermal band to separate a thin atmosphere from a bright surface. The test is crude, and it was chosen because it can be made at all. Its first answer, that the hottest rocky planets around red dwarfs are bare, was the expected one. The answer for the cooler planets is the one that matters, and it has not been found.