Astronomy & Celestial Mechanics
the sky, computed
Concept

Seeing — where it appears

The blurring of a star's image by atmospheric turbulence, quoted as the full width at half maximum of the resulting profile. It spreads the light over many pixels, so a photometric aperture has to trade the flux it collects against the sky and read noise it adds.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

sky · Seeing A bright star wants a wide aperture and a faint one wants 0.68 of the seeing. Signal-to-noise of simple aperture photometry against the aperture radius, in units of the seeing's full width at half maximum (1″), each divided by what optimal pixel weighting achieves for the same star, for stars of V = 12, 17, 20, 23 observed for 60 s through a 1 m telescope under a sky of 21 mag/arcsec². A small aperture loses starlight; a large one admits sky, and the balance depends on which dominates. For a bright star its own photons are most of the noise, so a wider aperture keeps gaining light almost for free and the best radius is large — 1.63 FWHM at V = 12, reaching 100.0 per cent of the optimum. For a star fainter than its sky the best radius shrinks to 0.680 FWHM and the best aperture reaches only 90.5 per cent of what weighting each pixel by its share of starlight divided by its variance achieves. That residual is exact in the background-limited limit: the best aperture captures 71.5 per cent of the light and 0.902 of the optimal signal-to-noise, so optimal weighting is worth 11 per cent in signal-to-noise, or 23 per cent in exposure time, and no more. The image is taken to be Gaussian; a real point-spread function has broader wings, which makes a fixed aperture a little worse and the optimal weights harder to know.

The best aperture throws away a tenth

Aperture photometry counts every pixel inside a circle equally and every pixel outside it not at all. For a faint star against its sky the best circle is two-thirds of the seeing wide, catches 71.5 per cent of the light, and reaches 90.2 per cent of the signal-to-noise that weighting each pixel by what it is worth achieves — a loss of 23 per cent in exposure time that no algorithm can beat by more.

starlight · Photon noise 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.

sky · Seeing 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.

sky · Seeing The edge of the Sun, computed. The brightness of the Sun at 500 nanometres across its limb, as a fraction of the brightness at the centre of the disc, against height above the level where the photosphere's radial optical depth is one, from the radiative transfer along each line of sight through a grey model atmosphere. Inside the disc the brightness falls gently with limb darkening; at the edge it falls to nothing over about two hundred kilometres. The steepest point of the fall — the inflection point, which is what a solar radius conventionally means — is 335 km above the radial τ = 1 level, because a ray grazing the limb passes through 270 times more atmosphere than a vertical one and becomes opaque that much higher up, at about 336 km. The brightness falls to one per cent 555 km above the τ = 1 level and to a thousandth at 694.

The Sun is three hundred kilometres larger seen edge-on

The radius of the Sun measured from images and eclipses and the radius measured from its oscillations disagree by about three hundred kilometres, and neither is wrong. A line of sight grazing the limb passes through hundreds of times more atmosphere than a vertical one and becomes opaque higher up — so the edge a telescope sees sits above the surface the waves feel, and every other definition of the edge, from the detector's threshold to the colour of the light, moves it again.

sky · Phases and eclipses 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.

sky · Seeing

Named alongside it

The objects these essays reach for when they reach for this one.

Fried parameterAdaptive opticsAtmospheric turbulenceIsoplanatic angleLucky imagingStrehl ratioAnisoplanatismDiffraction limitThe point-spread functionSpeckleAngular resolutionAperture photometry

All concepts