Astronomy & Celestial Mechanics
the sky, computed
Concept

Extrapolation — where it appears

The use of a fitted relation outside the range the data cover. It is unavoidable where a survey's completeness falls to zero, and the value it returns depends on the functional form assumed as much as on anything that was measured.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

spaceflight · Escape Giant planets per logarithmic interval of orbital distance, and who can see where. Giant planets (a third of Jupiter's mass to thirteen Jupiters) per star per unit of the natural logarithm of orbital distance, against distance, round Sun-like stars. The curve is a broken power law representative of the long radial-velocity surveys: rising as a^0.8 to a peak at 3 AU and falling beyond it as a^−1, which integrates to 2.7 per cent of stars inside 1 AU and 12 per cent in all. The shaded bands are where each method has the sensitivity to measure it: radial velocities out to about 7 AU, where two decades of monitoring cover one orbit, and direct imaging beyond about 10 AU, where a young giant can be separated from its star. The peak sits near the outer edge of the radial-velocity window, where a planet has completed one or two orbits in the data, and the fall beyond it lies entirely outside that window — so the peak's position is where the two methods have to be joined rather than a place either can measure alone.

A peak that neither survey can see alone

Giant planets become commoner with distance from their star out to a few astronomical units, and then rarer. The rise is measured by radial velocities and the fall by direct imaging, and the peak between them sits where neither method is comfortable.

exoplanets · Occurrence rates

Named alongside it

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

Occurrence rateCompletenessEta earthFalse-positiveHabitable zoneSurvey completenessDark matter haloDefinitional uncertaintyDetection thresholdDirect imagingError propagationEscape velocity

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