Generator

The magnitude-scale generator

The magnitude scale, plotted

The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

3 essays call magnitude-scale. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Starlight

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.

Photometric against true redshift for a simulated sample. The redshift fitted from broad-band magnitudes against the true redshift, for 300 simulated galaxies — half old, half young — measured to a signal-to-noise of 10 in their brightest band in six optical bands, and fitted with the same two templates they were made from. 217 fall within 15 per cent in 1 + z of the diagonal, with a scatter of 0.053 in (the fitted z − z)/(1 + z) among them. 83 are catastrophic failures, more than 15 per cent off, and they are not scattered at random: they sit in clumps far from the diagonal, where a galaxy's break was mistaken for the other kind of break. The scatter can be improved by measuring better; the outliers are a different kind of error, and averaging does not remove them.

Starlight

A redshift that is wrong by three

Most galaxies in the largest surveys will never have a spectrum. Their distances are read from five or six magnitudes, and the reading usually lands close. When it fails it does not fail by a little — a nearby galaxy is placed ten billion years away, because two different breaks in two different spectra fell into the same filter.

$How small an outlier fraction the mean redshift can bear. The shift in a tomographic bin's mean redshift, f·Δz, against the outlier fraction f, for outliers displaced by Δz = 0.5, 1.5, 2.5, both logarithmic. The dashed line is the requirement for a bin at z = 0.8, 0.0036. Outliers half a unit of redshift away must be fewer than 0.72% of the bin; outliers two and a half units away, fewer than 0.14% — or known to that precision, since a known fraction can be corrected. The best photometric redshifts reach outlier fractions of a few per cent, ten to a hundred times above these lines.$ Starlight

A mean that a few per cent of strays can drag

A lensing survey sorts its galaxies into slices of redshift by their colours, and what its cosmology needs from each slice is a single number, the mean redshift, to two parts in a thousand. A few per cent of the galaxies in a slice are catastrophic outliers — young galaxies at redshift three mistaken for old ones at one — and each moves the mean by its full error. Three per cent of strays two redshift units away move it twenty times more than the survey can tolerate, and finding them needs spectra of exactly the galaxies whose spectra fail.

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