Astronomy & Celestial Mechanics
the sky, computed
Concept

Eta earth — where it appears

The frequency of Earth-size planets in the habitable zones of Sun-like stars, per star. Published values span a factor of thirty, and the spread is mostly a matter of where the boundaries of the definition were drawn rather than of measurement error.

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

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

A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there.

The planets that were not seen

An occurrence rate is a count divided by a probability, and the probability can be a five-hundredth. Everything difficult about saying how common planets are lives in that denominator.

exoplanets · Occurrence rates Four defensible boxes, and a factor of 2.9 between the answers. Above: the occurrence surface in starlight received against planet radius, with four published definitions of "an Earth-size planet in the habitable zone" drawn on it as rectangles. The surface is a stated parameterisation — two lognormal populations at 1.3 and 2.4 Earth radii, with the radius valley at 1.9 between them, normalised so that the whole of it comes to 0.5 planets per star between one and four Earth radii inside a hundred days. Integrating it over the four boxes gives 17%, 8%, 6%, 10% — a factor of 2.9 between the conservative zone and a broad definition, before any error bar is attached to any of them. The dashed curve is what a transit survey can actually see: at one year around a Sun-like star the smallest detectable planet is 1.6 Earth radii, so the lower-left corner of every box contains no detections at all and the rate there is an extrapolation of a fitted surface rather than a count of anything. Below: the same integral with only one corner moved. Holding the flux range fixed and sliding the radius bound from 1.5 to 1.75 Earth radii — a quarter of an Earth radius, well inside the uncertainty of a measured planetary radius — changes the answer by 43 per cent, which is larger than every error bar quoted with any of these numbers. The published values of η⊕ span two per cent to sixty; roughly a factor of 2.9 of that is definition, and the rest is how far each author was willing to extrapolate past the dashed line.

The part of a rate that is a definition

Published values for the frequency of Earth-size planets in habitable zones span two per cent to sixty. The spread is not measurement error — it is where the box was drawn, on a surface that is steepest exactly at the corner every author has to choose, and outside the last detection.

exoplanets · Occurrence rates

Named alongside it

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

CompletenessExtrapolationFalse-positiveHabitable zoneOccurrence rateDefinitional uncertaintyDetection thresholdError propagationReliabilitySmall number statisticsSurvey completenessSurvey yield

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