Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A twinkle that knows the high air and not the low

Lucky imaging keeps the frames in which a reference star is sharp, and that choice happens to favour moments when the high turbulence was quiet, which is why it reaches further across the field than it should. A star's twinkling reports the high layer directly. Choosing frames by the twinkle alone gives a field that is equally sharp everywhere and sharp nowhere, because the twinkle cannot see the air at the ground. Choosing by both buys a little more field for a little less centre, and only where the high air comes and goes.

Assumes Seeing.

15 min read 6 figures Light is the only messengerScales that defeat intuition

A ground-based telescope looks through turbulent air, and the sharpness of a star’s image changes from one moment to the next as the turbulence along the line of sight rearranges itself. Lucky imaging exploits that: it takes thousands of exposures short enough to freeze the air, measures the sharpness of a reference star in each, and keeps only the few per cent in which the star happened to be sharp. In those frames the telescope works close to its diffraction limit, and the method has resolved thousands of close binary stars and checked the hosts of transiting planets for companions that would dilute their transits.

What the selection keeps is a statement about the air along one line of sight, and a target a few arcseconds away looks through different air at altitude. The reach of the luck across the field turned out to be longer than the geometry of a single fixed layer allows, for a reason that was a property of the selection: when the strengths of the turbulent layers vary from frame to frame, the frames that are sharp on the reference are disproportionately frames in which the high layer was quiet, and a quiet high layer is good for the whole field. The selection was choosing the high layer without being asked to.

That raised an obvious question. The high layer can be measured directly, from something the reference star does that its sharpness only hints at: it twinkles. Choosing the frames by the twinkle might choose them for the field instead of for the star.

Lucky frames chosen by the twinkle instead of the sharpness. The mean Strehl ratio of a target against its separation from the reference star, drawn as the high layer's shift in units of the aperture (32 arcseconds at the right for a 2.5-metre telescope and a layer at 8 km), for the best 2 per cent of 1600 frames chosen three ways: sharpest on the reference (orange), least scintillation of the reference's light (green), and best on both ranks together (blue); half the turbulence at the ground, both layers varying in strength frame to frame. Chosen by sharpness, the frames give 0.52 on axis and 0.34 at the edge. Chosen by scintillation, 0.22 and 0.22: the same everywhere, because the high layer — the only one whose blur differs between directions — has been selected directly, and lower everywhere, because the ground layer, which blurs every direction alike, has not been selected at all. Both ranks together give 0.44 and 0.34, a little more at the edge than sharpness alone at some cost at the centre. All frames average 0.10.
Fig. 1 The mean Strehl ratio of a target against its separation from the reference star, for the best 2 per cent of 1,600 frames chosen by the reference’s sharpness (orange), by its scintillation (green) and by both ranks together (blue), with half the turbulence at the ground and both layers varying frame to frame; 32 arcseconds at the right for a 2.5-metre telescope and a layer at 8 km. Sharpness gives 0.52 on axis and 0.34 at the edge; scintillation 0.22 everywhere; both, 0.44 and 0.34. All frames average 0.10.

Why a star twinkles, and which air makes it

Twinkling — scintillation — is a fluctuation in a star’s total brightness, not in its sharpness, and it needs distance to develop. A turbulent layer imposes a pattern of phase delays on the starlight passing through it — small fluctuations in the same refractive index whose smooth gradient bends light round the Earth near the horizon. Directly below the layer the light’s intensity is unchanged; the phases are scrambled but the energy has not moved. Over the kilometres between the layer and the ground, the phase pattern focuses and defocuses the light, like the bright and dark ripples on the bottom of a swimming pool, and the telescope sits in a pattern of intensity that shifts as the layer blows past. The further the light has travelled since the layer, the stronger the pattern.

So scintillation is made almost entirely by high turbulence. A layer at the ground, or in the telescope dome, scrambles the phases just above the mirror and leaves the light no distance in which to turn phase into intensity; it blurs the image and does not make the star twinkle. A layer at eight or ten kilometres, where the jet stream’s wind shear generates turbulence, makes the star twinkle strongly. For a large telescope, which averages the intensity pattern over its aperture, the twinkling of a bright star depends on each layer’s strength weighted by roughly the square of its height.

That weighting is the useful coincidence. The size of the patch of sky over which the turbulence is the same in every direction — the isoplanatic angle, which sets how far from the reference a target can be and still be corrected — is set by each layer’s strength weighted by the five-thirds power of its height. Scintillation and isoplanatism weight the layers in nearly the same way, which is why instruments that profile the turbulence at observatories use scintillation to estimate the isoplanatic angle. The twinkle of a bright star is close to a direct measurement of how wide the corrected field would be.

Even, and not sharp

The opening figure is the answer, and it is not the one hoped for. Frames chosen by the reference’s scintillation give a target the same sharpness at every separation out to thirty arcseconds: a Strehl ratio of 0.22 on axis and 0.22 at the edge. The selection has done exactly what it should for the high layer — the frames are ones in which it was quiet, so the target’s blur barely depends on where it is — and nothing at all for the ground layer, which blurs every direction alike and which the twinkle cannot see.

With half the turbulence at the ground, that half is selected only by chance. The frames chosen by scintillation have an ordinary ground layer, and the Strehl ratio that an ordinary ground layer allows is low. Frames chosen by the reference’s sharpness, by contrast, are sharp because both layers happened to be quiet; they start higher on axis, at 0.52, and fall to 0.34 at the edge because their high layer was quiet only partly by design.

So the twinkle on its own makes the field even rather than sharp. That is not useless — an image whose blur is the same everywhere is easier to measure than one whose blur changes with position, for photometry of crowded fields especially — but it is not what was wanted. What was wanted was the sharpness of the lucky frames carried further across the field, and for that the ground layer still has to be selected somehow.

Instruments that read the air from the twinkle

Using a star’s twinkle to measure the turbulence above a telescope is not new; it is how observatory sites are chosen. A double star whose two components are a few arcseconds apart casts two intensity patterns on the telescope’s pupil, from two directions, and turbulence at a given height shifts one pattern relative to the other by an amount proportional to that height. Correlating the two patterns reveals a peak for each layer at its own shift, and the heights and strengths of the layers can be read off the correlation — a technique that sounds the atmosphere with starlight, as the shape of a setting Sun sounds the air near the horizon. Smaller instruments do it with a single bright star, by comparing the twinkle in apertures of different sizes, each sensitive to turbulence at a different range of heights.

Those instruments established the profiles on which the two-layer model here is built: a strong ground layer of a few hundred metres, a weaker but significant layer near the tropopause, and between them a variable assortment. They also established the variability, on time scales from seconds to hours. What they average away, by necessity, is the frame-to-frame behaviour on millisecond scales, because their correlations need many frames to build up. The lucky-imaging question sits exactly in that gap.

Choosing on both

The natural compromise ranks every frame twice, once by the reference’s sharpness and once by its twinkle, and keeps the frames best on the sum. The sharpness rank selects the ground layer, along with whatever the high layer was doing; the twinkle rank selects the high layer specifically.

Sharpness at the centre traded for sharpness across the field. The target's mean Strehl ratio at the largest separation (32 arcseconds) against the reference's own, in the best 2 per cent of frames chosen by a score that weights the reference's sharpness by a fraction α and its scintillation by 1 − α, from α = 0 (scintillation alone) to α = 1 (sharpness alone), each standardised. Scintillation alone (green) gives 0.22 at the centre and 0.22 at the edge, worse at both; sharpness alone (orange) gives 0.52 and 0.34. The edge is sharpest at α = 0.5, 0.368, for 0.43 at the centre. Between that point and sharpness alone the curve is a frontier — more of the field for less of the centre — and the choice of a point on it is a choice about what the image is for, a star and its close companion or a field.
Fig. 2 The target’s Strehl ratio far off axis against the reference’s own, for frames chosen by a score weighting the reference’s sharpness by α and its scintillation by 1 − α. Scintillation alone gives 0.22 and 0.22; sharpness alone, 0.52 and 0.34. The edge is sharpest at α = 0.5, at 0.368, with 0.43 on axis.

The figure traces the family of such rules. Scintillation alone sits at the bottom left, worse at both centre and edge. Sharpness alone sits at the top of the centre axis. Between them, an equal weighting gives the sharpest edge of any rule in the family — 0.37 against 0.34 — at the cost of nearly a fifth of the central sharpness. From that point to sharpness alone the rules form a frontier, each trading some of the centre for some of the field, and which point is chosen depends on what the image is for. A survey for close companions of bright stars wants the centre; an image of a globular cluster’s core wants the field.

The gain at the edge is modest, about eight per cent in Strehl ratio, and it is bought with a larger loss at the centre. The twinkle adds information, but less than the hope suggested, because most of what the selection by sharpness already achieved at the edge was the same choice of a quiet high layer, made indirectly.

A measurement that has to be good

A frame’s scintillation is not known exactly. It is estimated from the fluctuation of the reference star’s total brightness within the few milliseconds of the exposure, and that estimate is noisy — from the photons, and from the averaging of the intensity pattern over the aperture.

How well the twinkle has to be measured. The target's mean Strehl ratio at the largest separation drawn (32 arcseconds), in the best 2 per cent of frames chosen by the reference's scintillation (green) or by both ranks (blue), against the error in measuring the scintillation in one frame, as the width of a log-normal factor. Chosen by sharpness alone the frames give 0.339 there (dashed orange), and all frames 0.101 (dotted). Scintillation alone, perfectly measured, gives 0.263, below selection by sharpness at every error, because it never sees the ground layer. Both ranks together give 0.381 with a perfect measurement, 0.327 with a factor-of-1.6 error and 0.271 with a factor-of-e error; they beat sharpness alone at the edge only for errors up to about 0.3. The scintillation of a few milliseconds of starlight is itself noisy, from the photons and from the averaging over the aperture, so whatever the twinkle adds is available only for a reference bright enough to measure it well.
Fig. 3 The target’s Strehl ratio at the largest separation for frames chosen by scintillation alone (green) and by both ranks (blue), against the error in one frame’s scintillation, as the width of a log-normal factor. Sharpness alone gives 0.339 there (dashed). Both ranks give 0.381 with a perfect measurement, 0.327 with a factor-of-1.6 error and 0.271 with a factor-of-e error; they beat sharpness alone only for errors up to about 0.3.

The advantage is fragile. With a perfectly measured twinkle, choosing on both beats choosing on sharpness at the edge by about an eighth. With an error of a third in the natural logarithm — a scintillation known to within about a third of itself in each frame — the advantage is gone, and with larger errors choosing on both is worse than choosing on sharpness alone, because the twinkle rank is now adding noise to a selection that was working.

Measuring a star’s scintillation to a third in a few milliseconds needs a bright star, brighter than the references lucky imaging usually uses; the method’s strength has been that a star of fifteenth or sixteenth magnitude is enough to judge sharpness. The twinkle is a measurement of second order — a variance of a brightness rather than a brightness — and it costs more light to measure well.

When the high air comes and goes

The whole idea depends on the high layer’s strength changing from frame to frame, so that there are quiet moments to choose. If it does not change, there is nothing for the twinkle to pick.

The gain from the twinkle, against how much the layers vary. The target's Strehl ratio far off axis in frames chosen by both ranks (blue) and by scintillation alone (green), each divided by its value in frames chosen by the reference's sharpness, against the frame-to-frame variation of each layer's strength, as the width of a log-normal factor, with half the turbulence at the ground. 0.15: 0.74 and 0.59; 0.3: 0.79 and 0.64; 0.45: 0.94 and 0.60; 0.6: 1.00 and 0.66; 0.8: 1.06 and 0.61; 1: 1.05 and 0.67. With this much turbulence at the ground scintillation alone never catches up, while the combination climbs from 0.74 to 1.05 as the variation grows: the twinkle adds information about the field only when the high layer's strength changes enough from frame to frame for there to be quiet moments worth choosing, and even then the gain is a few per cent. The counts behind each point are a few dozen frames, so the curves carry a few per cent of noise of their own.
Fig. 4 The target’s Strehl ratio far off axis in frames chosen by both ranks (blue) and by scintillation alone (green), each divided by its value with frames chosen by sharpness, against the frame-to-frame variation of each layer’s strength, with half the turbulence at the ground. The combination rises from 0.74 at a variation of 0.15 to 1.05 at 1.0; scintillation alone stays near 0.6.

The combination’s gain grows with the variation. When the layers’ strengths vary by fifteen per cent from frame to frame, adding the twinkle rank costs a quarter of the edge’s sharpness, because it dilutes the sharpness rank without adding information worth having. When the layers vary by a factor of e, it gains five per cent. Real layers have been measured varying by factors of two on time scales of seconds, which puts the method near the break-even point — a gain that exists, is small, and depends on the night.

Scintillation alone never catches up at this share of ground turbulence, whatever the variation. It is always choosing half the turbulence and ignoring the other half.

A site whose turbulence is high

The decisive number is how much of the turbulence sits near the telescope, because that is the part the twinkle cannot see.

The gain from the twinkle, against how much turbulence is at the ground. The target's Strehl ratio far off axis in frames chosen by both ranks (blue) and by scintillation alone (green), each divided by its value in frames chosen by the reference's sharpness, against the fraction of the turbulence at the ground, with both layers varying by a log-normal factor of width 0.6. 20 per cent at the ground: 1.13 and 0.96; 35 per cent at the ground: 0.90 and 0.74; 50 per cent at the ground: 1.00 and 0.66; 65 per cent at the ground: 0.99 and 0.53; 80 per cent at the ground: 0.82 and 0.38. Choosing on both beats sharpness alone at the edge when 20 per cent of the turbulence is at the ground. Scintillation says nothing about the ground layer, so the more of the turbulence is near the telescope, the less the twinkle has to choose with; at a site whose turbulence is mostly high, it nearly matches the sharpest frames at the edge on its own.
Fig. 5 The same ratios against the fraction of the turbulence at the ground, with both layers varying by a factor of width 0.6. With 20 per cent at the ground, choosing on both gives 1.13 of the sharpness-selected value at the edge and scintillation alone 0.96; with 80 per cent at the ground, 0.82 and 0.38.

At a site where most of the turbulence is high — a mountain summit where the ground layer is thin and the dome is well ventilated — the twinkle has most of the turbulence to choose with, scintillation alone nearly matches the sharpest frames at the edge, and choosing on both gains more than a tenth. At a site dominated by ground turbulence, the twinkle has little to work with and the selection is worse for consulting it. The good astronomical sites are characterised partly by how thin their ground layer is, and it is at those sites that the high layer is most of the problem and the twinkle most of the answer.

The same consideration decides the value of a much more ambitious remedy. Adaptive optics that corrects only the ground layer — a single deformable mirror conjugated to the telescope’s pupil, guided by an average over several stars — gives a corrected field as wide as the high layers allow, which is many arcseconds. A ground-layer correction followed by lucky selection on the twinkle would divide the work cleanly: the mirror removes what the twinkle cannot see, and the selection chooses the moments when what the mirror cannot correct is quiet.

Frame by frame

The scatter behind those averages shows what the twinkle does and does not know.

Frame by frame, the twinkle against the sharpness far away. Each of 1600 frames: the reference star's measured scintillation, on a logarithmic axis, against the target's Strehl ratio at the largest separation (32 arcseconds), with half the turbulence at the ground and a measurement error of 0.3 in the natural log of the scintillation. The correlation between the logarithm of the scintillation and the far target's sharpness is −0.40; between it and the reference's own sharpness, −0.38. The twinkle tracks the star and the edge of the field about equally, because it reports the high layer's share of each frame's blur and both directions suffer that share; what separates them is the ground layer, common to both and invisible to the twinkle, which is why scintillation alone can make a field even but not sharp.
Fig. 6 Each of 1,600 frames: the reference star’s measured scintillation against the target’s Strehl ratio far off axis. The correlation of the scintillation’s logarithm with the far target’s sharpness is −0.40; with the reference’s own sharpness, −0.38.

Frame by frame the twinkle is a weak guide: frames of low scintillation are sharper on average, but the scatter is large, and the correlation with the far target’s sharpness is no stronger than with the reference’s own. That is the ground layer again. Both the reference and the far target suffer the high layer’s share of each frame’s blur, and the twinkle reports that share; what separates a sharp frame from a dull one, for either of them, is often the ground layer, which the twinkle cannot see. The twinkle predicts the field’s evenness better than it predicts its sharpness.

Small telescopes, large telescopes

The method sits in a narrow range of telescope sizes, and the twinkle pulls in the opposite direction from the luck. Lucky frames exist only when the telescope is not much larger than the patches of coherent air: the chance that the whole aperture is momentarily flat falls exponentially with the number of patches across it, and beyond a ratio of telescope diameter to Fried parameter of about seven to ten, lucky frames are too rare to collect. On a ten-metre mirror that resolves like a ten-centimetre one without correction, the method fails outright.

Scintillation is the reverse. A small telescope sees the intensity pattern’s individual ripples and its twinkle is large and dominated by the smallest, fastest structures; a large one averages over many ripples and its twinkle is small but stable and closely tied to the high layers’ strength. Measuring a frame’s scintillation well therefore favours a large aperture, and measuring it quickly favours a bright star. The two-and-a-half-metre telescope drawn here is near the upper end of where lucky imaging works and near the lower end of where scintillation is a clean measurement of the high layer, which is one reason the gain is marginal: the method and its proposed improvement want different telescopes.

The twinkle is also, elsewhere, a noise. For bright stars observed from the ground it sets the floor under precise photometry, which is why fast photometers searching for the brief shadows of small bodies passing in front of stars work hardest against it, and why the error budgets of adaptive optics count the high layers twice — once in the residual blur at the edge of the field and once in the guide star’s own fluctuating brightness. Here it is turned from a nuisance into a measurement, and it turns out to be a better measurement of the field’s evenness than of its sharpness.

What the model leaves out

The atmosphere here is two layers, one at the ground and one at a single height, each a Kolmogorov phase screen whose strength varies by an independent log-normal factor from frame to frame. Real profiles have several layers at different heights, with strengths that vary together when weather systems pass and independently otherwise. A middle layer at three or four kilometres would twinkle weakly and limit the field moderately, and it would weaken the twinkle’s value as a proxy for the isoplanatic angle. The scintillation is modelled as the high layer’s strength times a noise factor, rather than computed from the intensity pattern through the aperture; the frames are independent, when real exposures a few milliseconds apart share turbulence. And the Strehl ratio is computed from the phase variance after removing tilt, which is right for a short exposure’s core and leaves out the halo.

None of these changes the central point. A twinkle is made by distance, so it measures the air that is far from the telescope and not the air that is near, and selection on it can only ever choose the part of the turbulence it can see.

Why the twinkle is still worth having

The twinkle is not wasted even where it loses as a selection rule. The record of each frame’s scintillation, kept alongside the images, is a record of the high layer’s behaviour over the whole run, and it can be used after the fact: to estimate each frame’s isoplanatic angle, to model how the point-spread function changes across the field in the frames that were kept, and to correct the photometry of targets far from the reference for the blur they actually suffered. A lucky image whose field-dependent blur is known is more valuable than one whose blur is merely smaller, for exactly the crowded-field work the method is reaching towards.

That is a use that needs no compromise with the selection, and it suggests that the right place for the twinkle is not in the choice of frames but in the description of the frames chosen.

Still open: what the turbulence profile looks like on millisecond scales

Every number here rests on an assumption about how the layers vary: independently, by a factor of a given size, from one frame to the next. Profiles of the turbulence measured with scintillation-based instruments average over seconds to minutes, which is long enough to show layers strengthening and weakening but too long to say whether, within a second, the high layer has the quiet moments a frame-selection scheme needs. The data that would settle it — simultaneous millisecond records of a bright star’s scintillation and of the wavefront it delivers, at a good site, through a full night — are what a selection on the twinkle should be designed against, and they are within reach of instruments that already exist.

Where this goes next

A lucky frame stays lucky further from the star when the high air was quiet. The mean Strehl ratio of short exposures of a target at a separation from an on-axis reference star, against the shift of the high turbulent layer between the two lines of sight in units of the aperture, s = θh/D — 0.50 of an aperture is 32 arcseconds for a 2.5-metre telescope and a layer at 8 km. The atmosphere has 50 per cent of its turbulence at the ground and the rest high, with D/r₀ = 7, and 1600 simulated frames. Frames are selected by the reference star's sharpness. When the two layers' strengths vary from frame to frame (log-normal, width 0.6), the best 2 per cent start at a mean Strehl of 0.523 and keep 0.339 at the largest separation; when the strengths are steady, the best frames start at 0.321 and fall to 0.105. In the first case the Strehl never falls to six-tenths of its on-axis value within the range drawn; in the second it does so at s = 0.208, 13 arcseconds. Selecting frames by one star selects, preferentially, the moments when the high layer was weak — and a weak high layer is weak for every star in the field. Luck that reaches as far as the quiet air Lucky imaging keeps the rare short exposures in which a reference star happens to be sharp, and a sharp star says the air was flat along one line of sight. A target a few arcseconds away looks through different air. It turns out to be sharper in those frames than the geometry of a single turbulent layer allows, because choosing a lucky instant chooses, preferentially, an instant when the high turbulence was weak — and weakness, unlike the pattern of the turbulence, is shared across the field. How sharp a single short exposure is, for three sizes of telescope. The distribution of the Strehl ratio — the peak brightness of a star's image as a fraction of a perfect telescope's — for 600 independent short exposures through simulated Kolmogorov turbulence, with image motion removed, for telescopes 4, 7, 10 times the Fried parameter r₀ across. The screens' mean phase variance matches the analytic value for tilt-removed turbulence to within 2 per cent. At D/r₀ = 4 the exposures spread over the whole range and 31 per cent are better than 0.37 — a phase error under one radian, a nearly diffraction-limited image. At 7 the distribution has slid to low Strehl ratios and none of the 600 are that good; at 10, none. The best frames at the larger ratios are still several times sharper than the average, which is what practical selection keeps. The atmosphere is not uniformly bad: every so often, for a few milliseconds, the turbulence over a small enough aperture happens to be nearly flat, and those instants are a diffraction-limited telescope for free. A sharp image kept by throwing the rest away The atmosphere does not blur every instant equally. Now and then, for a few milliseconds, the air over a small telescope is almost flat, and a short exposure taken then is as sharp as the mirror allows. How often that happens falls as the exponential of the square of the telescope's size — which is why the method works on a two-metre mirror in the red and nowhere else. A patch 1.5″ wide in the visible and 8″ at 2.2 µm. The isoplanatic angle against wavelength, for r₀ = 10 cm at 500 nm, a turbulence layer at 5 km and a wind of 20 m/s. This is the angle over which one measurement of the wavefront is still valid, and it is the hardest of adaptive optics' three limits: at 1.5 arcseconds in the visible, the guide star has to be inside a patch a hundredth the size of the full Moon. Everything scales as λ^6/5 because r₀ does — measured off the curve at λ^1.200 — so the patch grows to 8 arcseconds at 2.2 µm, and its area by the square of that. With 0.1 stars per square arcminute bright enough to guide on, the fraction of sky reachable goes from 0.018 per cent to 0.5 — a factor of 28. That single curve is why the first working systems were infrared, why a laser is fired to make a star where there is none, and why the laser still does not solve it: a beam launched from the telescope wanders with the same atmosphere it is meant to measure, so it cannot sense the overall tilt, and a natural star is still needed for that. The correction has to be faster than the air Making a large telescope resolve like a large telescope means measuring the wavefront and undoing it. Three numbers bound how well that can work and none of them is the mirror — a frequency of hundreds of hertz, a patch a second and a half wide, and the chance of a bright enough star inside it.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

AnisoplanatismFried parameterIsoplanatic angleLucky imagingScintillationSeeingStrehl ratioTurbulence profile