Astronomy & Celestial Mechanics
the sky, computed
Concept

Poisson statistics — where it appears

The statistics of counting rare independent events, in which the spread of a count about its mean is the square root of the mean. It sets the precision of any measurement that comes down to counting — photons, craters, detections or stars.

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

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 The age a handful of craters allows. The probability of each age for a 100 square-kilometre surface on Mars, given a count of 0, 3, 10, 30, 100 craters 50 metres across or larger, per unit of the logarithm of age, with a uniform prior on age. No craters: younger than 45 Myr at 95 per cent, and nothing more; 3: 55 Myr, from 31 Myr to 88 Myr; 10: 158 Myr, from 116 Myr to 212 Myr; 30: 457 Myr, from 382 Myr to 547 Myr; 100: 1,502 Myr, from 1,359 Myr to 1,660 Myr. Each tenfold increase in the count narrows the range by about the square root of ten, from a factor of 2.8 with three craters to 1.22 with a hundred. The counting error is only part of the story: the chronology converting a crater density into an age is uncertain by a factor of about 1.6 on Mars before any crater is counted.

A young surface dated by a handful of holes

The surfaces that decide whether a planet is still geologically alive — the freshest lava on Mars, the smoothest ice on Europa — are the ones with fewest craters to count. Their ages are Poisson estimates from a few dozen holes, they cannot be younger than the area and crater size allow, and a fraction of a per cent of older ground in the counting area can double them. What survives best is not when a surface formed but which of two surfaces formed first.

orbits · Surface chronology

Named alongside it

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

Central limit theoremChronology functionConfusion limitCosmic infrared backgroundCrater countingExtragalactic background lightImpact fluxMarsP(D)Production functionRadio backgroundResurfacing

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