Astronomy & Celestial Mechanics
the sky, computed
Concept

Lutz kelker bias — where it appears

The systematic error in absolute magnitudes calibrated from parallaxes, which come out too faint for stars measured with noisy parallaxes. A sample chosen by measured parallax keeps stars whose errors made them seem nearer, and the error grows with the fractional parallax error.

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

Also named here as parallax zero point — the same set of essays touches all of them, so they are one junction rather than several.

One over a noisy parallax, at three precisions. The distribution of the distance obtained by inverting a parallax, for a star truly at 100 parsecs measured with fractional errors of 5, 10, 20 per cent. At five per cent the distribution is nearly symmetric and inverting is harmless. At twenty per cent it is strongly skewed: the mean sits at 105 parsecs rather than 100, and the tail runs to distances several times the truth, because a parallax scattered a little towards zero is a distance scattered a long way outward. The asymmetry is a Jacobian and nothing else — the parallax measurement is unbiased and symmetric throughout. Above about twenty per cent the mean of the distribution stops existing at all, because the density falls only as the inverse square of the distance and the integral of d times that diverges.

The distance is not one over the parallax

A parallax is measured with symmetric errors and a distance is one over it. Inverting a noisy positive quantity is not a change of units — it is a change of distribution, and the one that comes out is skewed, biased outward, and above about twenty per cent error has no mean at all.

starlight · Parallax A calibration sample, and the stars a precision cut keeps. Measured parallax against apparent magnitude for one star in four of a simulated sample of 8000 standard candles of absolute magnitude 0.5 and intrinsic scatter 0.15 mag, spread through a disc 15 kpc in radius and 300 pc thick and catalogued to magnitude 16. The parallax axis is stretched near zero so the negative measurements show. The solid curve is the parallax a star of exactly the calibrated luminosity would have at each magnitude, and the dashed curve is the cut ϖ/σ = 5: stars above it are the "good parallaxes" a calibration usually keeps. It keeps 3242 of the 8000, 41 per cent, and at faint magnitudes the curve it cuts through runs through the middle of the stars rather than below them, so the ones kept there are the ones whose errors pushed their parallaxes up. 45 stars have negative parallaxes; no inversion can use them, and a fit in parallax space uses them like any other.

A calibration chosen by the measurement it calibrates

A standard candle is calibrated on stars with good parallaxes, and "good" is decided by the parallax itself — so the calibrators are the stars whose errors happened to push them nearer. Correcting each star's distance afterwards does not repair it, because the correction needs a picture of where the sample's stars really lie, and that picture depends on the luminosity being calibrated. What works is to choose nothing, keep every star with the negative parallaxes included, and fit the candle by predicting the parallaxes rather than inverting them.

starlight · Parallax

Named alongside it

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

ParallaxParallax zero pointPriorCatalogueDistance estimationDistance ladderJacobianMalmquist biasPosteriorSelection functionStandard candleSystematic error

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