Astronomy & Celestial Mechanics
the sky, computed
Concept

Angular resolution — where it appears

The smallest angular separation at which two sources can be told apart by a given instrument through a given atmosphere. From the ground it is set by the atmosphere rather than by the aperture, and the two only coincide for apertures under about ten centimetres.

Named by 4 essays across 4 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 The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them.

Resolution without a mirror

Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those.

starlight · Interferometry How much of a background a survey can resolve, against sources per beam. The fraction of a background resolved into individual sources before confusion stops the count, against X, the number of sources brighter than the break in the counts that fall inside one beam, on a logarithmic axis. The curve is computed for counts falling as S^−2.5 above the break and S^−1.5 below it, with a source counted as recoverable while there is less than one source per 30 beams brighter than it. It is a single curve: telescope, wavelength and integration time enter only through X. Half the background is resolved at X ≈ 0.033, and the fall from nearly all to nearly none takes about four decades. The surveys are placed on it at their own beams and counts: Hubble, optical at X = 8.7·10⁻⁵, 100 per cent; Chandra, soft X-ray at X = 6.4·10⁻⁶, 100 per cent; Spitzer, 24 µm at X = 0.012, 54 per cent; Herschel, 250 µm at X = 0.043, 24 per cent; Herschel, 500 µm at X = 0.18, 11 per cent. The points do not sit exactly on the line because each band has its own slopes.

A background that stops breaking into sources

A deep enough image turns a glowing sky into a list of galaxies. Past a certain depth it stops doing so, and the depth is set by how many sources share one beam — which a longer exposure cannot change.

cosmology · Olbers's paradox 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

Named alongside it

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

Adaptive opticsDiffraction limitAperture synthesisAtmospheric turbulenceBaselineConfusion limitCosmic infrared backgroundDeconvolutionDynamical massExtragalactic background lightFried parameterFringe

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