Astronomy & Celestial Mechanics
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The planet-radius-distribution generator

A gap where planets should be
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 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.

9 essays call planet-radius-distribution. 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.

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. Exoplanets

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.

 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. Exoplanets

A gap in a histogram that says how planets are built

Between the super-Earths and the sub-Neptunes there is a radius at which planets are markedly rarer. The gap is not a gap in what can be detected, and its slope with orbital period names the process that made it.

 A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous. Exoplanets

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

 How long a planet's hydrogen lasts, against how much of it there is. The time a young Sun-like star's saturated X-ray and ultraviolet output would take to remove a planet's whole hydrogen envelope, at 100 times the Earth's insolation, against the envelope's share of the planet's mass, for cores of 3, 5, 8 Earth masses. It is computed with energy-limited escape and an interior fit for the envelope's thickness, and it is not monotonic. A heavy envelope takes long to remove because there is a lot of it. A very light one takes long because the planet is small and intercepts little light. In between the time peaks, and it peaks where the envelope has swollen the planet most for its mass: 3 Earth masses at an envelope of 1.9 per cent, 129 Myr, where the envelope is 1.31 times as thick as the core's radius; 5 Earth masses at an envelope of 2.8 per cent, 364 Myr, where the envelope is 1.30 times as thick as the core's radius; 8 Earth masses at an envelope of 4.0 per cent, 946 Myr, where the envelope is 1.30 times as thick as the core's radius. The peak is what makes a valley. A planet above it losing gas moves up the curve, its remaining envelope lasting longer and longer, and settles; a planet below it moves down, lasting less and less, and loses everything. The horizontal lines are the saturated phase, 100 Myr, and the 300 Myr of saturated-equivalent exposure the whole history delivers: a core whose peak lies below the second line cannot keep any envelope at all. Exoplanets

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

 How planet occurrence rises with the host star's iron, by kind of planet. Planet occurrence relative to its value at solar metallicity, on a logarithmic axis, against the host star's iron abundance [Fe/H] (zero is the Sun; one unit is a factor of ten). Each class is a power law, occurrence ∝ 10^(β[Fe/H]). Giant planets within a few astronomical units follow β = 2, the relation found in the first large radial-velocity survey to measure it: 12 per cent of stars at [Fe/H] = +0.3 have one, against 3 per cent at solar and under 0.3 per cent at −0.5 — a factor of 158 across the range drawn. The small planets found by transits rise far more gently; the slopes drawn, β = 0.5 for sub-Neptunes and 0.2 for super-Earths, are representative of published fits, which range from nearly flat to about a doubling across this interval depending on orbital period. Exoplanets

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

 Small planets and giant planets per star, against the mass of the star. Planets per star against the host star's mass, both on logarithmic axes, for small planets (one to four Earth radii, orbital periods under 200 days) and giant planets (above a third of Jupiter's mass, within 2.5 astronomical units). The small-planet rate is drawn as 1.1 per star at one solar mass rising as M^−0.6; the giant rate as 7 per cent at one solar mass falling towards small stars as M^1. Both are representative of published fits, not fits of their own. At 0.3 solar masses a star has 2.3 small planets and a 2.1 per cent chance of a giant; at one solar mass 1.1 and 7 per cent. The ratio of small planets to giants changes by a factor of 7 between the two, so a statement about how common planets are depends on which star it is about as much as on which planet. Exoplanets

More small planets round smaller stars

A red dwarf has about twice as many small planets as the Sun's kind of star and a third as many giants. The two populations move in opposite directions along the same axis, and the small planets were built from discs several times poorer.

 The chance that a candidate is a planet, against how strong its signal is. The reliability of a transit candidate — the fraction of candidates at a given detection statistic that are planets rather than false alarms — against the multiple-event statistic, the combined signal-to-noise of all its transits, for orbital periods of 10, 100, 350 days. Below the 7.1 threshold nothing is a candidate. At a statistic of 9 the reliability is 100 per cent, 80 per cent, 23 per cent; at 12 it is 100 per cent, 95 per cent, 44 per cent. The true planets and the false alarms are modelled, not fitted: the planets as a falling power law above the survey's completeness curve, the false alarms as an exponential tail above the threshold that grows longer and taller with period, because a long-period signal is two or three transits and two or three transits average away little of the non-Gaussian noise that makes false alarms. At short periods a candidate near threshold is nearly always real; at a year, a candidate near threshold is usually not. Exoplanets

At a year, most candidates are noise

A transit survey's threshold is set so that Gaussian noise yields less than one false planet in the whole catalogue. The noise is not Gaussian, and at year-long periods the candidates just above the threshold are more often artefacts than planets — so the rate of Earth analogues is partly a measurement of the noise.

 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. Exoplanets

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.

 A peak that follows the snow line inwards. Giant planets per star per unit of ln a against orbital distance, for stars of 0.3, 0.6, 1, 1.5 solar masses, if the peak's position scales as the 1 power of the star's mass — snow line, young star's luminosity. The shape is the broken power law measured round Sun-like stars, rising as a⁰·⁸ and falling as a⁻¹, with its height scaled in proportion to the star's mass as the giants' overall rate is. The peaks sit at 0.9 AU, 1.8 AU, 3.0 AU, 4.5 AU: a 0.3-solar-mass star would have its giants piled up 3.3 times closer in than the Sun's. The dots mark each peak. Exoplanets

A peak that should move with the star

Round Sun-like stars, giant planets are commonest at about three astronomical units, close to where water first froze in the disc. If the snow line put them there, the peak should sit closer in round smaller stars — three times closer round a star of a third of the Sun's mass. Every hypothesis agrees about the Sun, which is why the Sun-like stars cannot test them; and one of them makes a Jupiter at the peak wobble its star by the same angle whatever the star's mass.

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