Astronomy & Celestial Mechanics
the sky, computed
Concept

Source counts — where it appears

The number of sources per unit area of sky brighter than a given flux, as a function of that flux. Their slope reveals how the sources are distributed in space and time, and they set how deep a survey can go before confusion.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The dipole a 370 km/s motion has to put into counts of distant sources. The amplitude of the dipole in the number of sources per unit solid angle expected from the Sun's motion at 369.82 km/s relative to the microwave background, β = 1.234e-3, split into its two parts: aberration, 2β, which crowds sources towards the direction of motion, and the Doppler boost, x(1 + α)β, which brightens sources there and lifts fainter ones above the flux limit. For radio sources, with counts steepening as S^(−1) and spectra falling as ν^(−0.75), the expected dipole is 0.0046; for mid-infrared quasars, with counts steepening as S^(−1.7) and spectra falling as ν^(−1.26), the expected dipole is 0.0072, and the measured one is 0.0155 — 2.16 times larger, which would need a speed of 797 km/s. A dipole of a few parts in a thousand needs a catalogue of more than a million sources to see at all. The disagreement is not with the direction, which lies close to the microwave background's, but with the size, and it is not yet explained: either the samples carry a systematic nobody has found, or the matter and the radiation do not share one rest frame on these scales — in which case the assumption that the universe looks the same from everywhere is wrong in a way it has never been caught being wrong before.

The Sun's speed counted in quasars comes out twice too large

The microwave background is warmer in one direction by a part in a thousand, and that dipole is read as the Sun's motion through it at 370 kilometres per second. The same motion must crowd the counts of distant galaxies and quasars towards the same direction by an amount that can be calculated exactly. When a million quasars were counted, the direction agreed and the size came out more than twice too large — as though the Sun were moving at 800 kilometres per second relative to the matter.

cosmology · Expansion 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 histogram of a confused map, for three faint-end slopes. The distribution of pixel values in a simulated map in which every beam holds 0.2 sources brighter than the break in the counts, for counts identical above the break (falling as S^−3.8) and falling as S^−b below it with b = 1.3, 1.5, 1.7. Pixel values are in units of the break flux per beam, measured from the map's own mean as a real map's are and no instrumental noise has been added. No single source below the break can be picked out of any of these maps, and yet the histograms differ: b = 1.3 gives a peak 0.15 below the mean, a width of 0.566 and a skewness of 2.5; b = 1.5 gives a peak 0.15 below the mean, a width of 0.629 and a skewness of 1.6; b = 1.7 gives a peak 0.13 below the mean, a width of 0.617 and a skewness of 1.5. Shallower faint counts leave a map dominated by the rarer bright sources, with a sharp peak and a long positive tail; steeper ones fill each beam with many faint sources and the histogram moves towards a Gaussian. The shape of the histogram is therefore a measurement of the counts below the level at which anything can be counted.

The histogram of a sky too crowded to count

When every beam holds more than one faint source, a map stops being a list. It remains a measurement — the histogram of its pixels carries the number of sources nobody can pick out, down to the depth where there are so many that their sum turns Gaussian.

cosmology · Olbers's paradox

Named alongside it

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

Confusion limitCosmic infrared backgroundExtragalactic background lightAberrationAngular resolutionCentral limit theoremCMB dipoleCosmological principleDiffraction limitDoppler boostingIsotropyKinematic dipole

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