Dust that does not dim with distance

A telescope built to photograph an Earth-like planet around another star has to solve a contrast problem before a resolution problem. In reflected visible light an Earth twin is ten billion times fainter than a Sun-like star, and the star’s light has to be suppressed to that level at a separation of a tenth of an arcsecond. The engineering of that suppression — coronagraphs, starshades, subtraction of the star’s own halo, nulling interferometers — has been the centre of the subject for two decades.

Suppose it succeeds perfectly. The planet is still not alone in its resolution element. The Solar System is full of fine dust, ground from asteroids and shed by comets, spread in a thick disc through the inner planets’ orbits; sunlight scattered from it is the zodiacal light, visible from a dark site as a cone rising from the horizon after sunset. Other planetary systems have dust of the same kind, called exozodiacal dust, and an Earth in the habitable zone of another star sits inside it. The dust scatters its star’s light across every resolution element around the planet, the planet’s included.

And the dust has a property the planet does not.

A surface does not fade #

A point source dims as the inverse square of its distance, because its light spreads over a sphere. An extended surface of uniform brightness does not dim with distance at all, when brightness is measured per unit solid angle, because the area of surface packed into each unit of solid angle grows as the square of distance in exactly compensating proportion. That is why the surface brightness of a galaxy is the brightness distance cannot touch, except through the expansion of the universe; and it is why a wall looks equally bright from across a room and from its far end.

A telescope’s resolution element is a fixed solid angle, about $(\lambda/D)^2$ — a few ten-thousandths of a square arcsecond for a 6 metre mirror at visible wavelengths. The exozodiacal dust falling within it contributes a fixed amount of light, set by the dust’s surface brightness, wherever the system is. The planet within it contributes light that falls as the square of distance. The ratio of dust to planet therefore grows as the square of distance, and there is a distance beyond which the dust is brighter than the planet whatever the contrast of the instrument.

The numbers make it concrete. A standard unit, one zodi, is the surface density of the Solar System’s own dust at the Earth’s distance from the Sun. Seen from outside, such dust at the habitable zone of a Sun-like star has a surface brightness of about 22 magnitudes per square arcsecond in visible light. An Earth twin at ten parsecs is about magnitude 29.6. In a resolution element of $(0.019'')^2$, one zodi contributes about magnitude 30.6 — four-tenths of the planet’s light. At three zodis, it matches the planet near nine parsecs.

The Solar System seen from ten parsecs #

It is worth running the comparison for the one system whose dust is measured in detail. Seen from ten parsecs, the Solar System would show an Earth at magnitude 29.6, a Venus somewhat brighter but closer to the Sun, and a zodiacal cloud at one zodi by definition — about forty per cent of the Earth’s light in each resolution element of a 6 metre telescope.

That symmetry is a curiosity with a practical edge. The observer’s own zodiacal light cannot be removed by any choice of target, and its brightness depends on where in the Solar System the telescope sits and which way it looks: brightest toward the Sun and in the ecliptic plane, faintest toward the ecliptic poles. A telescope at the second Lagrange point, a million and a half kilometres beyond the Earth, sees about the same foreground as the Earth does, because it is at the same distance from the Sun. Only a telescope far out in the Solar System, beyond the asteroid belt where the zodiacal cloud thins, would escape it, and the New Horizons spacecraft, looking back from beyond Pluto, has measured how dark the sky is there. For the telescopes that will actually search for Earths, the local foreground is a fixed tax, and the target’s dust is the variable one.

Time grows faster than the dimming #

A detection is a signal-to-noise problem, and the dust adds noise without adding signal. The exposure time needed to reach a given signal-to-noise ratio goes as the total light in the resolution element — planet, dust, leaked starlight, local zodiacal light — divided by the square of the planet’s light. When the planet dominates, doubling the distance quarters the planet’s light and multiplies the time by four. When the background dominates, the numerator no longer falls with distance, and doubling it multiplies the time by sixteen.

The Solar System’s own zodiacal light is present for every target, as a foreground: the telescope looks out through the Earth’s neighbourhood, and the dust there scatters sunlight into every image. It is a little fainter than one zodi of exozodiacal light at the habitable zone, and it cannot be escaped by choosing a target, only reduced by looking away from the ecliptic or by placing the telescope far from the Sun. It is why even a dust-free target system’s exposure time grows faster than the square of distance.

The fourth-power regime is the important one, because it sets how steeply the cost of each more distant target rises. A survey for Earth-like planets does not look at one star; it looks at as many as its total observing time allows, and the stars are spread through space with a density that is roughly uniform over a few tens of parsecs. How far out the survey can reach decides how many stars it can reach, and the fourth power decides how far.

A horizon set by dust #

Fix a time budget per star — here, three times the exposure a three-zodi system at ten parsecs needs — and ask how far away a target can be before the budget is exceeded.

The fourth root is gentle, and the gentleness is misleading. A horizon that shrinks by forty per cent sounds tolerable; the number of stars within it shrinks as the cube.

That is the arithmetic behind a sentence in every study of a future telescope to find Earths: the expected number of detected Earth-like planets depends on the exozodi distribution as strongly as on the aperture. The number of temperate rocky planets per Sun-like star is itself uncertain by a factor of a few, and multiplying it by a sample size that the dust could reduce by a further factor of several leaves the expected yield of a multi-billion-dollar mission uncertain by an order of magnitude. Measuring the dust first is cheaper than finding out afterwards.

What aperture buys #

A larger mirror helps twice. It collects more light, which shortens every exposure in proportion to its area, and it shrinks the resolution element, which reduces the dust in it in proportion to its area again. For a planet the second effect does nothing, since a point source is entirely within the element whatever its size; for the dust it is the whole effect.

In the dust-dominated limit the exposure time falls as the fourth power of the diameter, and that is the quantitative case for the largest mirror that can be launched. The concept studies for a habitable-worlds telescope compared designs of four, six and eight metres, and the expected yield of characterised Earth-like planets rose steeply with size for exactly this reason. It rose most steeply in the scenarios with the most dust, and the observatory now being planned was sized with a six-metre-class mirror partly on the basis of the dust levels that had just been measured.

Why stars have dust, and how much #

Zodiacal dust does not last. Grains in orbit around a star are pushed by its light and dragged by the reaction of their own re-emitted radiation, and that drag spirals the smallest grains inward into the star in a few hundred thousand years at the Earth’s distance, or they are ground finer by collisions and blown out. A star with zodiacal dust today has a source replenishing it now: asteroids colliding, comets shedding. A cascade of collisions turns a small mass of large bodies into a large area of small grains, and the area is what scatters light. The dust level is a measure of how active a system’s small bodies are.

The Solar System’s own level is low. Its zodiacal cloud is replenished mostly by Jupiter-family comets breaking up, and its brightness is thought to be modest compared with what a young or dynamically active system might have. Whether the Sun is typical is exactly the question, and it was asked directly by a survey with the Large Binocular Telescope Interferometer between 2014 and 2018. Nulling interferometry at ten microns, which suppresses the star and passes extended emission, measured the warm dust around several dozen nearby Sun-like stars. The best fit to the survey was a median level of about three zodis — a little dustier than the Solar System — with a minority of stars having tens to hundreds.

That result shaped the planning of the next generation. At three zodis typical, a six-metre telescope can survey enough Sun-like stars to expect a handful to a few dozen Earth-like planets, depending on how common they are. At ten, the sample would be halved; at a hundred, gone. The measured median was near the optimistic end of what had been feared, and the minority of very dusty stars can be identified in advance and skipped.

In the thermal infrared, the order changes #

At visible wavelengths the planet and the dust both scatter starlight. At ten microns both glow with their own heat, and the comparison is different. An Earth at 255 kelvin radiates most of its energy near ten microns and is only about ten million times fainter than a Sun-like star there, rather than ten billion — the contrast problem is a thousand times easier. The dust, re-emitting the starlight it absorbs, glows at the same wavelengths, and because warm dust radiates efficiently in the mid-infrared, one zodi at the habitable zone outshines an Earth there by a factor of several hundred when integrated over the whole disc.

That is why the thermal-infrared route to Earths is an interferometer: only a long baseline gives a resolution element small enough, at ten microns, to hold the dust in it to a manageable fraction, and the nulling interferometer’s rotating fringe pattern is designed to separate a point source from a smooth disc. The same property that made nulling the right instrument to measure exozodiacal dust makes dust the dominant noise when nulling is used to look for planets. Either route to an Earth, scattered light or thermal emission, is limited by the dust at a similar level; they simply meet it at different wavelengths.

The survey found one correlation worth carrying into target selection. Stars already known to have cold dust in an outer belt, the analogue of the Kuiper belt, detected by its far-infrared emission, were much more likely to have warm dust in their habitable zones as well — as though dust or comets from the outer belt were being delivered inward, or as though a system’s small bodies were simply more numerous or more stirred throughout. Stars with no cold dust had, on average, very little warm dust. Since cold dust is easy to detect and warm dust hard, the far-infrared census of nearby stars already made by space telescopes becomes a first, cheap filter: a star with a bright outer belt is a poor target for an Earth search, and its dust can be predicted before a coronagraph ever looks at it.

A larger mirror also helps in a way the exposure-time figure does not show. The habitable zone of a star at twenty parsecs lies about fifty milliarcseconds from it, and a coronagraph cannot see closer to its star than a few times $\lambda/D$. A 6 metre mirror reaches in to about sixty milliarcseconds at visible wavelengths; an 8 metre mirror to about forty-five. For the more distant stars in a sample, the aperture decides not only how long the exposure takes but whether the habitable zone is outside the coronagraph’s blind spot at all — a second horizon, set by angle rather than by dust, that also recedes with a bigger mirror.

The dust is not smooth #

Everything above treats the dust as a uniform glow, and it is not. The Solar System’s zodiacal cloud has a ring of dust trapped in resonance just outside the Earth’s orbit, co-orbiting with the Earth, with a trailing clump behind it; similar structures, sculpted by planets, are expected around other stars. A clump of dust near a planet’s position in the image can mimic a planet, and a ring can produce a pattern that a subtraction algorithm mistakes for a companion. Structure matters more than brightness in one respect: a smooth background adds noise that averages down with time, while a structured one adds a false signal that does not.

Structure is also information. A resonant ring or clump in exozodiacal dust points to the planet that traps it, and in systems where the dust is too faint to image directly, its asymmetries can reveal planets that are themselves too faint. The same dust that sets the horizon for finding Earths is a way of finding planets that cannot be seen.

What the figures leave out #

The counts in the figures are simplified in ways that move the numbers without changing the scalings. The dust is taken to be face-on and uniform over a resolution element; real discs are inclined, which brightens them along the line of sight by the inverse cosine of the inclination and adds forward scattering that brightens one side. The planet’s brightness depends on its phase and its albedo, and an Earth at quadrature, where it is easiest to separate from its star, reflects less than half the light of a fully illuminated one. The leaked starlight is represented by a single contrast floor; real coronagraphs have a contrast that varies with separation and with the star’s angular size, and it is often the leak rather than the dust that dominates at the smallest separations. And the exposure times are relative: the absolute time for a given telescope depends on its throughput, its detector and the wavelength, none of which enter the ratios drawn here.

The count of reachable stars assumes every star has the same dust. In reality the dust level is a distribution, and a survey chooses its targets — the nearest and least dusty first — so the realistic yield is higher than the uniform-dust curves suggest for a given median, and more sensitive to the fraction of very clean systems than to the median itself.

Still open: the dust closest to the planets #

The nulling survey measured warm dust at ten microns, which is emitted by grains at a few hundred kelvin — dust at and somewhat inside the habitable zone. Visible-light imaging of an Earth sees the same dust in scattered light, and the conversion from one to the other depends on the grains’ sizes and composition, which are not known for any system but the Sun’s. Whether a star measured at three zodis in the thermal infrared is also at three zodis in scattered light, at the separation of its habitable zone, is an assumption rather than a measurement. It will be tested by the first coronagraph flown with enough contrast to see exozodiacal light directly, a demonstration instrument on a space telescope due to launch within a few years, and the answer will set how much of the dust the survey telescope after it has to see through. If scattered light turns out brighter than the thermal measurements imply — if the grains are more reflective, or the dust extends further inward than the warm emission shows — the horizons drawn above shrink further, and the case for a larger mirror strengthens with them. If it is fainter, the survey can reach deeper than its planners dared assume.