Astronomy & Celestial Mechanics
the sky, computed
Concept

Adaptive optics — where it appears

Measuring the wavefront distortion the atmosphere imposes and applying its negative with a deformable mirror, faster than the distortion changes. It is bounded by three numbers, none of them the mirror: the loop rate the wind demands, the angle over which one measurement stays valid, and the availability of a star inside it.

Named by 8 essays across 3 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 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.

sky · Seeing Contrast against separation, which is where direct imaging lives. Planet-to-star brightness ratio against apparent separation, both logarithmic, for a system 10 parsecs away. The reflected-light curves are A_g(R_p/a)² and fall as the inverse square of the orbit; the thermal curve is the ratio of two Planck functions at 10 µm and does not, which is why every imaged planet so far is young and hot rather than merely large. The vertical lines are diffraction limits λ/D — nothing inside a telescope's own line is reachable by it at any contrast at all. An Earth at ten parsecs sits at 6.3e-10, which is 4 orders of magnitude below the faintest planet yet imaged. The four imaged planets are plotted at their measured near-infrared contrasts rather than at 10 µm, because the near infrared is the band they were found in — which is itself part of the argument, since a young planet is hot enough to be bright where its star is not.

Nine orders of magnitude, half an arcsecond apart

Photographing a planet is not a resolution problem. It is a contrast problem, and the contrast is set by an inverse square that punishes exactly the planets a telescope can most easily separate.

exoplanets · Direct imaging 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.

sky · Seeing 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 black holes that can be weighed are the heavy ones. 4000 synthetic galaxies spread evenly through a sphere of 120 megaparsecs, with black-hole masses scattered by 0.45 dex about an M–σ relation of slope 4.24 (faint points), and the 511 whose spheres of influence exceed 0.1 arcseconds (filled), which are the ones a dynamical measurement can resolve. At a given dispersion a heavier black hole has a larger sphere and is resolvable further away, in a larger volume, so the resolved sample is dominated by holes above the relation. Fitted to the resolved galaxies alone, the relation comes out with normalisation 8.74 and slope 2.95 (dashed), against 8.11 and 4.20 for all of them (solid) — 0.63 dex too high at 200 km/s, and more at low dispersions.

The black holes that can be weighed are the heavy ones

A black hole's mass is measured from the motions of stars or gas only inside its sphere of influence, the region where its gravity beats the galaxy's, and that region has to be resolved. At a given galaxy, a heavier hole rules a larger sphere and can be weighed further away, in a much larger volume — so the sample of weighed holes is drawn from the top of the distribution, and the relations built from it describe the heavy end of every galaxy rather than the typical one.

galaxies · Galactic nuclei How much too small a planet looks when a second star shares its light. The factor by which a planet's true radius exceeds the radius inferred from its transit, against the mass of an unresolved companion star in units of the Sun's, for a planet transiting a Sun-like primary and for one transiting the companion, with the radius computed in both cases from the diluted depth and the primary's radius. Logarithmic vertical axis. A planet on the primary is diluted by the companion's light alone, and the factor is √(1 + F₂/F₁), never more than √2 = 1.41, reached for an equal twin; for a companion of half a solar mass, whose 3943 K surface gives 4.5 per cent of the primary's light in a Kepler-like band, it is 1.022. A planet on the companion is diluted by the primary's much larger light and measured against the wrong star's radius, and the factor is (R₂/R₁)√(1 + F₁/F₂): 2.6 for the half-solar-mass companion and 4.3 for one of 0.3 solar masses. The two curves meet at √2 for twins. The stars are a main-sequence sketch — power-law radii, a smoothed luminosity law and blackbody spectra — good to tens of per cent in flux ratio.

The Earth-sized bin is where diluted planets land

A star with an unresolved companion shares its aperture with that companion's light, and every planet found around either star is measured too small. The error runs one way only, so the smallest radius bin receives planets from every larger one and returns none. With a companion beside three stars in ten, more than a quarter of the planets called Earth-sized are something larger.

exoplanets · Detection bias

Named alongside it

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

Diffraction limitFried parameterStrehl ratioAtmospheric turbulenceIsoplanatic angleSeeingAngular resolutionAnisoplanatismCone effectLaser guide starLucky imagingSpeckle

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