A zodi measured in heat and needed in light

An Earth-like planet around a nearby Sun-like star, imaged in reflected visible light, is seen against a glow: the starlight scattered by dust in the star’s own inner system, the analogue of the zodiacal light that brightens the Solar System’s ecliptic before dawn. That glow is a surface brightness and does not dim with distance, so around more distant stars it fills more of the resolution element the planet occupies, and the exposure time needed to detect an Earth grows steeply with the amount of dust. A survey designed to characterise a few dozen Earth-like planets — inside the band of stellar flux where liquid water could exist — has to know, star by star, how much exozodiacal dust each target carries.

That knowledge exists, in one form. A survey of nearby stars with the Large Binocular Telescope Interferometer measured the warm dust of several dozen of them, and found that most Sun-like stars carry no more than a few times the Solar System’s level — a median of a few “zodis”, with a minority far dustier. The result was a relief for mission planning, and it is the number every yield calculation for a future Earth-imaging telescope now uses.

But the survey did not measure the visible light a coronagraph will see. It measured the dust’s thermal glow at a wavelength of eleven microns, through a nulling interferometer whose interference pattern cancels the star and suppresses everything close to it. The “zodi” it reports is a conversion: the ratio of the measured glow to the glow the Solar System’s own dust would produce around the same star, computed with the Solar System’s radial profile. Turning that into the scattered light at the habitable zone is a second conversion. Both rest on assumptions, and the hero figure shows the first of them.

Where a ten-micron zodi comes from #

At ten microns a dust grain glows according to its temperature, and the temperature of a grain in equilibrium with starlight falls with distance from the star — as the inverse square root of distance, for a grain that absorbs and emits like a blackbody, from about 280 kelvin at the Earth’s distance from a Sun-like star to 500 kelvin at a third of that. The emission at a fixed wavelength rises steeply with temperature in this range, because ten microns lies on the short-wavelength side of the emission peak for grains cooler than about three hundred kelvin — the side where, as Wien’s law and the steep Planck tail both say, a modest rise in temperature multiplies the output. So the glow at ten microns comes overwhelmingly from the hottest dust that is present in quantity.

For dust distributed like the Solar System’s, half of the unsuppressed glow comes from inside a third of an astronomical unit — closer to the star than Mercury. The interferometer then removes part of that. Its fringe pattern has a dark stripe through the star, and the transmission rises with angle from the star, reaching its first maximum at an angle of about half the wavelength divided by the baseline. For a star at ten parsecs and the instrument’s 14.4-metre baseline, that is a few tenths of an astronomical unit, so the innermost and hottest dust is suppressed along with the star.

What survives is a signal that peaks around three-quarters of an astronomical unit and draws half of itself from inside 0.69 AU. That is a measurement of warm dust near and inside the habitable zone, weighted toward its inner edge. The scattered visible light that matters for a planet search comes from the dust at the planet’s own projected separation — at the habitable zone itself — and the measurement reaches it only through the assumption that the dust’s density falls with radius the way the Solar System’s does, gently, as the $-0.34$ power of distance.

One measured zodi, six brightnesses #

The assumption can be tested by asking what other dust distributions would have produced the same measurement, and how bright each would be at the habitable zone.

Smooth profiles that differ only in slope convert within a few tens of per cent of the reference. A steeper disc, with more of its dust close in, produces more of the measured glow from the hot inner region and therefore needs less dust overall, so it is fainter at the habitable zone than the reference predicts; a flatter disc is slightly brighter. If exozodiacal dust always followed a power law between its sublimation radius and the outer system, the conversion would be uncertain by perhaps a factor of 1.5, and the zodi census would carry over to visible light reasonably well.

Structured discs do not. A narrow ring of dust at three-tenths of an astronomical unit — the kind of ring that could be fed by sublimating comets, or held by a close-in planet — can produce the whole measured signal while leaving the habitable zone almost empty; the conversion is under one per cent. A ring at the habitable zone itself, cooler than the dust the reference profile would put inside it, needs nearly five times as much dust to produce the same glow, and the visible brightness at the planet’s position is nearly five times the prediction. A disc with an inner hole — cleared by a planet, or by the dust’s own drag having moved it inward and destroyed it — is two and a half times brighter at the habitable zone than its measured zodi level says.

The span of the figure is a factor of several hundred between the extreme cases and a factor of seven among the plausible ones. That is larger than the spread of zodi levels found among the survey’s Sun-like stars.

A conversion that changes with distance #

The interferometer’s suppression is a fixed angle on the sky, not a fixed distance from the star, so the part of a disc it hides depends on how far away the star is.

The weighting itself moves with distance, and drawing it for a star twice as far away makes the point directly.

For a star at five parsecs the null’s first bright fringe falls at about a fifth of an astronomical unit, and the instrument sees almost all of the warm dust; for a star at twenty-five parsecs it falls at about one AU, and much of the inner disc is hidden. A disc that differs from the reference profile is therefore converted differently at different distances, and the direction depends on its shape. For a disc with an inner hole the reference profile wrongly assumes hot dust that the instrument would have seen nearby but cannot see far away, so the conversion error is largest for the nearest stars — which are precisely the stars an Earth survey would observe first and longest.

This does not make the zodi census wrong. It makes it a measurement of a specific, distance-dependent weighting of each disc, which the reference profile then translates into a single number. For a population of discs with the reference shape, the number is right everywhere. For any particular disc, how far it is from the reference shape decides how far the number is from the visible brightness at the habitable zone, and the census does not say how far that is.

What the grains are made of #

The second conversion concerns the grains themselves. A grain’s visible scattered light is proportional to its albedo — the fraction of the starlight it reflects. Its thermal glow is proportional to the fraction it absorbs, which is one minus the albedo, and to how hot the absorbed energy makes it.

Albedo matters in the obvious direction: brighter grains scatter more light for the same glow, and grains of albedo 0.3 instead of 0.1 are four times brighter at the habitable zone for the same ten-micron zodi. The temperature matters in the opposite direction and more sharply. Grains smaller than a micron or so absorb visible starlight efficiently but radiate poorly at wavelengths much longer than their size, so they must reach a higher temperature to shed the energy they absorb. Solar System zodiacal grains are mostly tens of microns across and run close to blackbody temperatures; dust around other stars could be dominated by smaller grains if it were produced by a recent collision rather than by a steady trickle from comets and asteroids — the kind of event that briefly pushes a disc above the ceiling its own grinding normally imposes. Grains twenty per cent hotter than blackbodies glow at ten microns more than twice as brightly per unit area, so the same measured zodi corresponds to less than half as much dust, and half as much visible light.

Neither the albedo nor the temperature of any exozodiacal dust has been measured. The Solar System’s own zodiacal dust has an albedo of roughly 0.1 to 0.2 and temperatures close to blackbody — values measured from inside it, by spacecraft and from the Earth — and those are the values the conversion borrows.

Two conversions that multiply #

The profile and the grains enter as separate factors, and they multiply. A star whose dust has an inner hole and grains of moderately high albedo could be ten times brighter at the habitable zone than its measured zodi level predicts; one whose dust is mostly in a hot inner ring of small grains could be a hundred times fainter. Both would appear in the census as typical.

For an Earth survey the asymmetry of those errors is what matters. A star whose visible dust is overestimated is observed for longer than necessary, which wastes time but still yields its planet. A star whose visible dust is underestimated is observed for the planned time, the planet is not detected at the planned confidence, and the observation has to be repeated or the star dropped — and the yield calculation, which assumed the census level, has overestimated the number of planets the mission will characterise. Because the exposure time grows roughly in proportion to the dust brightness once the dust dominates the background, a factor of several in the conversion is a factor of several in the time a star costs.

The census still constrains the distribution in a useful way. Its upper limits on the dustiest stars hold under any reasonable conversion, because the conversion would have to be implausibly extreme to hide a hundred zodis of visible dust behind a measured few. What it cannot do is assign a reliable visible level to each star at the factor-of-two precision a schedule needs.

Why it was measured in heat at all #

It is fair to ask why the measurement was not simply made in visible light, and the answer is the reason the whole problem exists. Imaging exozodiacal dust in scattered light around a nearby Sun-like star requires a contrast of better than a hundred millionth at a few tenths of an arcsecond — nine orders of magnitude half an arcsecond apart, nearly the same contrast as imaging the planets themselves. No instrument has reached it, even with the star subtracted using its own images. At ten microns the contrast between a star and its warm dust is far less extreme, because the star’s light falls steeply into the infrared while the dust’s peaks there, and an interferometer on a large ground-based telescope could reach the levels needed.

So the warm dust was measured in the only band where it could be, and the conversion to the band where it matters is a model. That is not unusual — many astronomical quantities are measured in one band and needed in another — but it is unusually consequential here, because the mission it informs is designed around a single number per star and the number is set before the mission flies. The dust that reaches the habitable zone is moved light, absorbed in one band and re-emitted in another, and the ratio between the bands depends on what does the moving.

The reference disc, measured from inside it #

The whole conversion is anchored to one disc, and it is worth recalling how that disc was measured, because it was measured in a way no other disc ever will be. The Solar System’s zodiacal cloud was mapped by infrared satellites orbiting inside it, which saw its thermal glow in every direction and fitted a three-dimensional model to the brightness of the whole sky; by spacecraft travelling outward through it, whose dust counters and sky photometers measured how its density falls with distance from the Sun; and from the ground, in the visible light of the zodiacal band and the gegenschein. The radial slope, the albedo, the temperatures and the asymmetric structures — the resonant ring the Earth traps just outside its orbit, the trailing clump behind it, the bands from asteroid families — all come from that vantage point.

So the “zodi” is a unit defined by the best-measured debris disc there is, and the conversion inherits that precision only for discs that resemble it. The Solar System’s dust is supplied mainly by Jupiter-family comets, with a smaller share from asteroid collisions, and is dragged inward by sunlight on a timescale of tens of thousands of years. Other systems with different populations of small bodies, different planets and different ages have no reason to strike the same balance. A disc fed by a single disintegrating comet close to its star would be a narrow inner ring; a disc in a system with a giant planet just outside the habitable zone could have a gap; a young system still clearing its planetesimals could be dominated by small, hot collisional grains. Each of these is a conversion factor that differs from one.

What planets do to the dust #

There is one structure that is not merely possible but expected, and it complicates the search in a more particular way than a change of brightness. Dust drifting inward past a planet is caught temporarily in the planet’s outer resonances, and accumulates into a ring just outside the planet’s orbit with a gap or a clump at the planet’s position. The Earth does this to the Solar System’s dust: its resonant ring raises the zodiacal brightness near the Earth’s orbit by about ten per cent, and the clump trails the Earth by a few tens of degrees.

Around another star, an Earth-like planet would sculpt its own exozodiacal dust in the same way, raising the visible brightness exactly at the habitable-zone separation the conversion is applied to, and imprinting a clump that rotates with the planet. A ten-micron measurement, dominated by the hotter dust inside, would see none of this. The visible brightness at the planet’s position would then exceed even a correctly converted zodi level, and the clump — a few per cent of the dust’s brightness, at the planet’s distance — could be mistaken for a planet or could hide one. The structure that makes the conversion wrong is a structure the planet itself creates.

What the calculation leaves out #

The disc is seen face-on. An inclined disc puts more dust along each line of sight near the star’s projected position and changes both the ten-micron weighting and the visible brightness at a given projected separation. Inclination is unknown for most of the surveyed discs.

The grains scatter isotropically. Real zodiacal dust scatters strongly forward, so the visible brightness depends on the scattering angle at the planet’s position and therefore on the disc’s inclination; for a face-on disc the scattering angle is near ninety degrees, where forward-scattering grains are faint.

The null is idealised. The transmission is the azimuthal average of a two-aperture fringe pattern, ignoring the finite apertures’ own response and the instrument’s calibration of stellar leakage. The real instrument’s sensitivity map differs in detail but has the same structure.

The star is the Sun. For hotter stars the habitable zone moves outward and the dust is hotter at a given separation in AU; the weighting shifts accordingly. The shape of the argument is unchanged.

The profiles are illustrative. The rings, gaps and slopes chosen span the structures seen in the Solar System and in resolved cold debris discs. Whether exozodiacal dust has such structure is unknown, which is the point.

Still open: the first visible zodi #

The conversion will be tested directly for the first time when a coronagraph in space reaches the contrast needed to see exozodiacal light around a nearby star in scattered light. A demonstration instrument on a space telescope now being built is designed to reach close to that level for the brightest, dustiest nearby stars — the ones whose ten-micron zodi levels are highest and best measured.

Each star observed both ways gives one measurement of the conversion: a visible brightness at a known separation, beside a ten-micron zodi level. A handful of such pairs would say whether the Solar System’s profile and grains are typical — whether the conversion is near one, with a modest scatter, or whether it varies by the factors the figures allow. If it is near one, the census can be used as it stands, and the yield estimates for Earth-imaging missions stand with it. If it scatters by a factor of several, the first year of any such mission will have to be spent measuring the dust it was meant to see past.