Astronomy & Celestial Mechanics
the sky, computed
Series

Seeing — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Resolution stops improving at 10 cm of aperture. Angular resolution against aperture at 500 nm, both logarithmic. The falling line is diffraction alone, 1.03 λ/D, which is what a telescope in vacuum delivers and has no floor. The curve is the same telescope under an atmosphere of Fried parameter r₀ = 10 cm, combining diffraction and seeing in quadrature: it follows the diffraction line while D < r₀ and then bends onto a plateau at 0.98 λ/r₀ = 1.01″. An amateur's 100 mm at 0.1 m would resolve 1.062″ above the air and delivers 1.47″ through it; a metre at 1 m would resolve 0.106″ above the air and delivers 1.02″ through it; the VLT at 8.2 m would resolve 0.013″ above the air and delivers 1.01″ through it; the ELT at 39 m would resolve 0.003″ above the air and delivers 1.01″ through it. At 39 m the atmosphere is costing a factor of 371: the aperture is 390 coherence lengths across and every one of the 152,100 patches it collects arrives with a phase of its own. What the extra aperture still buys is photons and speckles, and those two are what adaptive optics and speckle interferometry respectively spend to get the falling line back.

    A ten-metre mirror that resolves like a ten-centimetre one

    The atmosphere delivers a wavefront in patches about ten centimetres across, and an aperture larger than a patch collects patches rather than detail. Resolution stops improving at that size — and what the extra aperture keeps buying is photons and speckles, which is why there are two entirely different ways out.

    part 1 · sky
  2. 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.

    part 2 · sky
  3. 196 nanometres of wavefront, and a Strehl that depends entirely on the colour. Above, the error budget of an adaptive-optics system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is 196 nanometres and is dominated by the two largest; removing the smallest term entirely would improve it by under six per cent, which is why an optimisation programme that does not know which term is largest achieves nothing. Below, what that residual delivers: the Strehl ratio, the fraction of the light in the diffraction core, against wavelength. It is the exponential of minus the square of the residual measured in radians, and a radian is a wavelength, so the same physical error is four times smaller in phase at two microns than at half a micron. The system drawn here delivers 1 per cent of its light into the core at 550 nanometres and 73 per cent at 2200, and nothing about it changed between the two.

    An error budget added in quadrature

    Adaptive optics does not deliver a resolution. It delivers a fraction of the light in the diffraction core, and that fraction is the exponential of minus a sum of squares. Five independent failures of the correction add in quadrature, the largest one decides everything, and the same hardware is useless in the visible and excellent in the infrared.

    part 3 · sky
  4. 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.

    part 4 · sky
  5. 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.

    part 5 · sky
  6. 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.

    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.

    part 6 · sky

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