Sixteen small mirrors that twinkle less than one large one

A measurement of a star’s brightness has an error that comes from counting: the photons arrive at random, and the fractional noise on $N$ of them is $1/\sqrt{N}$. For a faint star that is the whole story, and the remedy is collecting area — a bigger mirror gathers more photons and the noise falls as the inverse of its diameter. For a bright star seen from the ground it is not the story at all. A bright star twinkles, and the twinkle — scintillation, produced by turbulence kilometres overhead focusing and defocusing the starlight — changes the flux reaching the telescope by a fraction that does not depend on how bright the star is. It is a fixed percentage, and photons do nothing to it.

Scintillation does depend on the size of the telescope. A larger aperture averages over more of the pattern of bright and dark patches that the turbulence casts on the ground, and the fractional noise falls as the diameter to the minus two-thirds. That is weak: a mirror a hundred times larger in area reduces the twinkle by a factor of under five. Comparison stars do not help unless they are close enough on the sky to share the target’s turbulence, which for a bright target they usually are not.

This essay is about a way round the weak exponent that sounds as if it should not work. Take the area of one large mirror and divide it among many small telescopes, far enough apart that each twinkles independently, and average their measurements. The photon noise is unchanged, because the total area is unchanged. The scintillation is reduced, because independent fluctuations average down as the square root of their number, which is faster than the two-thirds power of a diameter. Sixteen small mirrors twinkle less than one large one with the same light. The catch is in the phrase “far enough apart”, and the wind decides what it means.

The same photons, a different twinkle #

The scintillation noise in a measurement of duration $t$ is approximately

$$\sigma_{\rm sc}^2 \approx 10^{-5}\, C_Y^2\, D^{-4/3}\, t^{-1}\, X^{3}\, e^{-2h/H},$$

with $D$ the aperture in metres, $X$ the airmass, $h$ the observatory’s altitude, $H$ about eight kilometres and $C_Y$ a factor near one and a half that varies with the night’s turbulence. The formula is Young’s from 1967, with its coefficient and airmass exponent revised by later measurements on large telescopes; Young’s original airmass power was 1.75 in $\sigma$, the revision gives 1.5. Either way the shape is the same: weakly dependent on aperture, strongly on airmass, independent of the star.

The photon noise, dashed, is the same for both arrangements: it depends only on the total collecting area. The scintillation is not. The four-metre mirror reduces its twinkle by $4^{-2/3}$, a factor of 2.5 below a single one-metre telescope. Sixteen independent one-metre telescopes averaged together reduce theirs by $\sqrt{16} = 4$. The difference, $16^{1/6} = 1.59$, is the advantage of the array, and it applies to every star bright enough for scintillation to dominate — brighter than about twelfth magnitude in a minute.

The reason the array wins is not mysterious once it is stated. A large mirror averages the scintillation pattern over its own area, and the pattern’s fluctuations on scales larger than the mirror survive the average; there is always a largest scale, and it carries most of the remaining variance. Independent telescopes average over patterns that have nothing in common, so even the large scales average away. A single mirror is a spatial filter with a cut-off at its own size. An array of separated mirrors is a set of independent samples.

Cutting one mirror into many #

The advantage grows with the number of pieces.

The law $N^{-1/6}$ holds while each small telescope is still large compared with the size of the smallest twinkling patches, which is the Fresnel scale $\sqrt{\lambda h}$ — about seven centimetres for visible light and turbulence ten kilometres up. Below that, a small aperture no longer averages anything: its scintillation stops falling with diameter and saturates at the full twinkle of a point. An array of such apertures gains the whole $\sqrt{N}$, and the curve steepens. A thousand telescopes of thirteen centimetres each would twinkle about a quarter as much as a single four-metre mirror.

The exponent is a sixth, so the gain is slow; to halve the scintillation of one large mirror takes about sixty-four pieces. But it is a gain in the one term that more aperture cannot buy cheaply, and it is bought without any extra light.

Across the wind and along it #

Independence is the whole of the argument, and two telescopes twinkle independently only if they see different turbulence. The scintillation pattern at the ground is the shadow of a turbulent layer kilometres up, and its correlation between two points depends on how far apart they are.

Across the wind the answer is reassuring. The correlation between two one-metre apertures, computed from the spectrum of a turbulent layer filtered by each aperture, falls from 0.6 at a quarter of a metre to essentially nothing once the apertures are separated by their own diameter. Telescopes that are not touching already twinkle independently, because the scintillation an aperture of diameter $D$ still sees is carried by patches of size about $D$, and two apertures a diameter apart see different ones.

Along the wind the answer is different, and this is the condition that a naive layout fails. The turbulent layer drifts with the wind at ten or twenty metres a second, carrying its shadow along the ground nearly unchanged over a minute — the approximation of frozen flow. A telescope integrating for a minute sees the brightness averaged along a strip six hundred metres long. A second telescope two metres downwind sees the same strip, shifted by two metres, and its average is almost identical. For the exposures used in photometry, two telescopes on the same line along the wind are not independent until they are hundreds of metres apart.

So the layout matters. Sixteen telescopes in a row across the wind give the full factor of four. Sixteen in a compact four-by-four grid, with the wind along one side, give only the four independent columns — and four columns of one-metre telescopes, each averaging a one-metre strip, do worse than a single four-metre mirror averaging a four-metre strip. Since the wind changes direction, a compact array gains its advantage on some nights and loses it on others. The geometry that is robust is either a line long enough to be across most winds or a spacing larger than the wind’s travel during an exposure.

The same structure explains why the large mirror’s own scintillation follows $D^{-4/3}$ in variance rather than the steeper law a single snapshot would give. Each instant, a mirror of diameter $D$ sees a fluctuation of variance proportional to $D^{-7/3}$; over an exposure the wind drags $vt/D$ independent patches across it, and dividing by their number gives $D^{-4/3}/t$. The time-averaging is a spatial average along the wind, and an array whose members are strung along the wind gets no more out of it than the single mirror does.

Integrating longer never changes the winner #

A natural objection is that longer exposures reduce scintillation anyway, so the arrangement does not matter for a patient observer.

Both scintillation and photon noise fall as the inverse square root of the integration time, so on logarithmic axes the three arrangements give parallel lines. Integration buys the same factor for each and never changes their ratio. To reach a hundred parts per million on an eighth-magnitude star — the precision needed to see a sub-Neptune cross a Sun-like star — takes more than half an hour with a single one-metre telescope, six minutes with a four-metre and a little over two minutes with sixteen one-metre telescopes spread across the wind. Time is the currency of a transit, which lasts only a few hours, so the factor of 2.5 between the second and third arrangements is a factor of 2.5 in the number of transits a night can resolve.

There is a limit the square-root law hides. Real photometric noise from the ground is not white on long timescales: changes in transparency, in focus, in the position of the star on the detector and in the comparison stars’ colours add slow drifts — red noise — that do not average down as $t^{-1/2}$. Those drifts set a floor that neither arrangement removes, and they are the reason ground-based photometry rarely reaches the precision a white-noise calculation promises over hours.

The shallowest transit one night can see #

The figure that summarises the use of all this is the smallest transit depth detectable in a single transit.

For faint stars every curve rises along the photon-noise line, and the space telescope — with less than a twentieth of the four-metre mirror’s area — is worse than any ground arrangement fainter than about tenth magnitude, because the ground telescopes simply collect more light. For bright stars the ground curves flatten into floors set by scintillation, and the floors differ by the factor the array buys. A four-metre telescope’s floor in white noise sits just above the depth of an Earth-sized planet crossing a Sun-like star; an array of the same area sits just below it; the space telescope, with no twinkle at all, falls below both for stars brighter than eighth magnitude and keeps falling.

That is the regime in which space photometry is decisive, and the reason it is: not light, which a ground array can have in abundance, but the absence of the one noise that light cannot beat. For a star brighter than about eighth magnitude, the floor in space is the star itself — its granulation and oscillations — while the floor on the ground is the air.

The penalty for looking low #

Airmass enters scintillation more strongly than anything else.

The longer path through the turbulent layers and the greater distance to them both increase the twinkle, and a star at sixty degrees from the zenith scintillates nearly three times as much as one overhead. A transit that begins with the star low in the east is measured badly in its first hour whatever the telescope. Altitude helps by a modest, fixed factor, because a high site has less air above it in which turbulence can form; a site at 2,400 metres gains about a third over one at sea level. The array’s advantage multiplies both and is independent of each, which is the point: the factor it buys is geometric and survives every other condition.

What the array cannot do #

Small telescopes have weaknesses of their own, and they are why large photometric programmes have not simply abandoned large mirrors.

The first is the faint end. For faint stars the background in the aperture dominates the noise, and an array is at a disadvantage there for reasons that have nothing to do with turbulence. With equal total area and equal image size the sky collected is the same, but small telescopes are usually built for wide fields with coarse pixels, so their images are larger on the sky and each aperture admits more background; and every telescope adds its own detector’s read noise, so sixteen detectors add sixteen times the read-noise variance of one. For faint targets the large mirror wins.

The second is the systematics. Each small telescope has its own detector, its own flat field, its own guiding errors and its own response to temperature, and averaging sixteen of them averages their random noise but not their correlated systematics — a common error in timing or a common colour term in the comparison stars survives the average. The red noise that limits long ground-based light curves is in part this, and it does not fall as $\sqrt{N}$.

The third is resolution. A large mirror can resolve a crowded field, separate a target from a neighbour, feed a spectrograph. The array is a photometer for isolated bright stars and nothing else. Surveys built on it, such as the Next-Generation Transit Survey on Paranal, with a dozen twenty-centimetre telescopes, have combined several of their telescopes on single bright stars and reached precisions of a few hundred parts per million in half an hour, approaching what space missions deliver for the same stars — a demonstration of the averaging, not a replacement for apertures that do other things.

The same atmosphere, read two ways #

The array’s argument has a mirror image in the other thing the atmosphere does to starlight. Turbulence near the ground and in the free atmosphere corrugates the wavefront, and that corrugation is what blurs an image — the seeing that makes a ten-metre mirror resolve like a ten-centimetre one. Scintillation is the same turbulence seen in intensity rather than in phase, and it comes overwhelmingly from the high layers, because a phase corrugation has to propagate some distance before it focuses into bright and dark patches. The low air blurs; the high air twinkles. A site’s seeing and its scintillation are therefore set by different parts of its atmosphere, and a site chosen for one is not automatically good for the other.

The two effects also scale oppositely with aperture in the way that matters. Seeing is a size, fixed in arcseconds, and a larger mirror does not reduce it; only correction does. Scintillation is a fraction of the flux, and a larger mirror averages it down, weakly. That is why a photometrist and an imager want different things from the same telescope: the imager wants the turbulence fixed, the photometrist wants it averaged, and a photometric aperture can afford to be generous — the best aperture for a faint star throws away a tenth of the light to keep out the sky, while for a bright star the aperture is opened wide enough that no image motion or blur can move light across its edge.

There is one more parallel worth drawing. An interferometer replaces one large mirror by several small ones for a different purpose — to resolve what no single mirror could — and in doing so it has to combine the light coherently, keeping track of the phase at each aperture. A photometric array combines its light incoherently, adding counts rather than waves, and needs no phase at all. It is the simplest possible use of many apertures, and it works precisely because it throws away the information that interferometry needs: whatever the turbulence does to the phase is irrelevant, and what it does to the intensity at each aperture is independent from one aperture to the next.

Another way to remove the twinkle #

Averaging independent samples is one way to beat scintillation. Measuring it is another. The twinkle is produced mostly by a few thin layers at known heights, and a star’s twinkling carries the high layer’s signature directly. An instrument that images the pupil of a telescope conjugated to the height of the dominant layer sees the scintillation pattern itself, and a reference star nearby, seen through the same layer at a slightly shifted position, records the same pattern displaced by the angle between them times the layer’s height. Dividing one by the other removes the twinkle the two share. Such conjugate-plane photometry has been demonstrated to reduce scintillation severalfold when one layer dominates, and loses its advantage when the turbulence is spread through many heights.

The two approaches are complementary. The array needs no knowledge of the turbulence but needs space and many instruments; the conjugate-plane method needs one telescope but a bright reference and a turbulence profile that cooperates. Neither touches the other limits of ground photometry — transparency, red noise, the sky — which is why the best ground light curves of bright stars still fall short of space by more than the scintillation alone accounts for.

What the calculation assumes #

The scintillation formula is an average over nights and sites; on a given night $C_Y$ can be half or double its median, and the airmass dependence varies with the height of the turbulence. The correlation across the wind is computed for a single thin layer in weak scintillation with a Kolmogorov spectrum; real turbulence is spread through several layers moving at different speeds and directions, and the correlation along a line is a weighted mixture of their individual correlations. The along-wind correlation assumes frozen flow over a whole exposure, which is approximately true for a minute and less true for longer.

Photon noise assumes a V-band filter with a fixed throughput and no sky background, appropriate for bright stars. And the transit-depth figure uses white noise only, at five times the noise in a single transit of three hours. Real detection thresholds are set by the red noise and by the number of transits observed, and the ground curves would sit higher on both counts.

Still open: the cost of a factor of N to the one-sixth #

The physics of the array is settled; what is not settled is whether it is the best use of money and sites. A large mirror’s cost grows faster than its area, so a given area is cheaper in small pieces, and the pieces can be bought off the shelf. Against that, every piece needs a mount, a camera, a dome or enclosure and a share of the calibration effort, and the systematics of many detectors are harder to control than those of one.

The trade turns on a number that has not been measured well: how much of the noise in long ground-based light curves of bright stars is scintillation, which the array reduces, and how much is correlated systematics, which it does not. Arrays with many small telescopes now running at good sites have the data to answer it — light curves of the same bright stars from one telescope, from several and from all of them, at many airmasses and wind directions. Whether the averaged noise follows $\sqrt{N}$ down to the scintillation floor, and whether that floor shifts with the wind’s direction across the array as frozen flow predicts, will say how far small mirrors on the ground can follow the precision that space photometry has made routine, and where the twinkle finally stops mattering.