The Earth-sized bin is where diluted planets land

Every bias examined so far in this account of exoplanet surveys has been about whether a planet is found. The detection domains decide which planets a method can see at all; the threshold is a ramp rather than a step; eccentric orbits are favoured or penalised differently by each method; and the number of planets a system shows depends on an angle nobody can measure. Each changes which planets end up in a catalogue.

The bias here is different in kind. It changes what a planet that has been found appears to be. A transit measures a depth — the fraction of the light in the photometric aperture that the planet blocks — and the depth becomes a radius only through an assumption about whose light is in the aperture. If a second star is there too, unresolved, its light is in the denominator, the depth is shallower than the planet would make on its own star, and the radius computed from it is too small.

The essay on validation met this case as the blend that passes every test: a real planet around a companion, diluted by the target, measured too small by a factor of two or three. What follows takes the same physics to the population. How large is the error for each star of a pair, which star’s planets are actually found, and what does a catalogue in which three stars in ten have an unresolved companion do to the distribution of radii that most of exoplanet demographics is built on?

Two stars, two very different errors #

A transit of depth $\delta$ in front of a star of radius $R_\star$ means a planet of radius $R_p = R_\star\sqrt{\delta}$. Suppose the target is a Sun-like primary with a fainter bound companion in the same aperture, contributing a fraction $F_2/F_1$ of the primary’s light in the survey’s band.

If the planet transits the primary, the companion’s light dilutes the depth by $1/(1 + F_2/F_1)$, and the inferred radius is too small by

$$\frac{R_\text{true}}{R_\text{inferred}} = \sqrt{1 + F_2/F_1}.$$

If the planet transits the companion, two things go wrong at once. The primary’s light — usually much more than the companion’s — dilutes the depth by $F_2/(F_1 + F_2)$; and the radius is then computed with the primary’s radius, because the primary is the star whose properties were measured. The inferred radius is too small by

$$\frac{R_\text{true}}{R_\text{inferred}} = \frac{R_2}{R_1}\sqrt{1 + F_1/F_2}.$$

The two curves are not mirror images, and the asymmetry is the first thing the figure shows. A planet on the primary is diluted by a star that is fainter than its own, so the correction can never exceed $\sqrt2 = 1.41$, the case of an equal twin, and for most companions it is a few per cent. A companion of half a solar mass has a surface near 3,900 kelvin and, in the band from 430 to 890 nanometres where Kepler observed, gives about 4.5 per cent of the Sun’s light. A planet on the primary of such a pair is measured 2.2 per cent too small — less than the uncertainty in the star’s own radius.

A planet on that companion is measured 2.6 times too small. Its transit is deep against its own small star, and that depth is almost entirely drowned by the primary’s light; then the shallow remainder is converted to a radius using a star twice the size. For a companion of 0.3 solar masses the factor is 4.3. A sub-Neptune of three Earth radii around such a companion would appear in the catalogue at 0.7 — a planet smaller than the Earth.

The stars in the figure are a sketch of the main sequence — radii as a power of mass, a smoothed version of the standard broken luminosity law, blackbody spectra integrated over the band — and the flux ratios are uncertain at the level of tens of per cent. That changes the factors by ten or twenty per cent. It does not change the asymmetry, which comes from which star is brighter and which star’s radius is used, and not from the details of either.

A faint companion hides the planets it would mis-measure #

The second figure asks which star the planets in binaries actually orbit, once the survey’s own threshold has been applied.

The survey behind it is synthetic and its assumptions are stated. Planets are drawn from a radius distribution with two peaks, at 1.35 and 2.4 Earth radii, separated by a valley; three stars in ten have a companion with a mass ratio drawn uniformly between a tenth and one; each star of a pair is equally likely to have a planet; and a planet is detected if its diluted depth is at least 70 parts per million, a little less than an Earth crossing the Sun produces. Nothing else about the survey — periods, noise, the number of transits — is modelled, because the question is only how dilution and a depth threshold interact.

They interact strongly. A small companion’s planets are diluted below the threshold before they can be mis-measured: around companions of less than 0.4 solar masses almost none are found, and the large correction factors on the right of the first figure belong mostly to planets that never enter a catalogue. Above half a solar mass the companions’ planets start to be found, and for twins they are found almost as often as the primary’s. Across all mass ratios, a quarter of the detected planets in binaries orbit the companion.

That is the population the large errors apply to: planets around companions of more than about half a solar mass, measured too small by factors between 1.4 and 2.6. The same dilution that would have produced the most extreme errors hides the planets it would have produced them for, and the errors that survive are large but bounded. It is a selection effect acting on a measurement error, and it is why the effect is a bias in a distribution rather than an occasional disaster.

A distribution with a tail no planet populates #

The whole synthetic survey can now be histogrammed twice: once with the planets’ true radii, and once with the radii a pipeline would infer, assuming every planet orbits the target and the target is alone.

Two features of the inferred distribution are not in the true one, and both follow from the fact that dilution moves every planet in one direction.

The first is the tail at the small end. The true distribution has essentially no planets below one Earth radius; the inferred one has a shelf there, made of super-Earths around companions and of the lower edge of the super-Earth peak around primaries, all moved down. Nothing compensates for it, because nothing makes a planet look larger.

The second is in the valley. The radius valley between super-Earths and sub-Neptunes is a gap that says how planets are built — planets either kept a few per cent of their mass in hydrogen or lost all of it — and its depth is part of the evidence for what removed the hydrogen. Dilution moves sub-Neptunes downwards into it. In this survey the valley’s floor rises from 20 per cent of the peak height to 24 per cent: a modest change, because most of the planets that move are moved by only a few per cent, and the planets moved far are mostly on companions and mostly undetected. But it moves in one direction. A valley measured in a catalogue with unresolved companions is always shallower than the true one, by an amount that depends on the companion fraction, and the depth of the valley is one of the numbers theories of atmospheric loss are fitted to.

This is a different effect from the one that once hid the valley entirely. Before parallaxes, the stellar radii every planet radius is multiplied by were uncertain by a quarter, and that scatter smeared the valley symmetrically; better radii made it appear. Dilution is not symmetric and better stellar radii do not remove it. A perfectly known primary still gives the wrong planet radius if it is not alone.

The bin that receives and never gives #

The radius bin that dilution affects most is the smallest, and the reason is arithmetic rather than physics. A bin is contaminated by planets moved into it from above. The largest bin receives from nothing larger; every smaller bin receives from all the bins above it, and the smallest receives from every one.

In the synthetic survey the bin usually called Earth-sized, from 0.8 to 1.25 Earth radii, is clean when no star has a companion and is contaminated in proportion to the companion fraction when they do. At three stars in ten, a little over a quarter of the planets in the bin are larger planets measured small, and almost one in ten is a sub-Neptune — a planet with a thick hydrogen envelope, catalogued as rocky and the size of the Earth. At one star in two, the shares are 42 and 14 per cent.

The contamination matters most where it is largest. The bin is the one every estimate of the frequency of Earth-like planets starts from, and where that estimate’s box is drawn is already the largest source of disagreement between published values. Dilution adds a bias that is the same sign in every analysis, and it inflates the count of small planets at the expense of larger ones. Whether it inflates the rate of Earths depends on a second effect that runs the other way: a star with a companion is two stars counted as one, so the number of stars searched — the denominator of every occurrence rate — is undercounted, and the companions’ planets, most of them undetectable, are missing from the numerator too. The two corrections have to be made together. Each is a change to a count divided by a detection probability, one to the count and one to the probability, and corrected separately they double-count the planets that are both moved and missed — which is why the analyses that take dilution seriously now forward-model the whole survey, companions included, rather than correcting its output.

A density that turns a sub-Neptune into iron #

A radius is rarely the end of the calculation. When a planet’s mass is measured from the star’s reflex velocity, the two together give a density, and the density is what decides whether a planet is rock, iron, water or gas. The mass is not affected by dilution for a planet on the primary, because the velocity is measured in the primary’s own spectral lines. The radius is. Density goes as the inverse cube of the radius, so a radius error is tripled:

$$\frac{\rho_\text{inferred}}{\rho_\text{true}} = \left(1 + \frac{F_2}{F_1}\right)^{3/2}.$$

For the half-solar-mass companion that is a 7 per cent overestimate, within the errors of most measured densities. For a companion of 0.7 solar masses, giving a fifth of the primary’s light in a Kepler-like band, it is 31 per cent. For 0.8 solar masses it is 59 per cent, and for an equal twin a factor of 2.8. A sub-Neptune with a density of 2 grams per cubic centimetre, around one star of an unresolved twin, would be reported at 5.6 — the Earth’s density, and read as rock rather than gas. A rocky planet of 5.5 around the same star would be reported at about 15, which at a few Earth masses is as dense as a planet made entirely of iron.

That connects dilution to a puzzle it did not create but could imitate. A handful of small planets are denser than the iron their stars could have given them, and the explanations proposed — mantles stripped by giant impacts, iron concentrated by the chemistry of the inner disc — all assume the density is right. A twin companion inside the imaging limit would produce exactly that kind of planet out of an ordinary one. The super-Mercuries that have been imaged and found to be alone are not affected; the argument is a reason to image every one, and to treat an anomalously high density around an unimaged star as a question about the star before it is a question about the planet.

The colour of a diluted transit #

A companion of different temperature leaves a second fingerprint. A planet is opaque at every wavelength, so its undiluted transit has the same depth in every band. Dilution does not: it depends on the companion’s share of the light, and a cooler companion’s share grows towards the infrared. For a planet on a Sun-like primary with a half-solar-mass companion, the companion gives 4.5 per cent of the primary’s light at Kepler’s wavelengths and 14 per cent at 2.2 microns, so the transit is 8 per cent shallower in the infrared. For a planet on the companion the effect is reversed and much larger: the companion’s share of the total rises from 4 per cent to 12, and the transit is nearly three times deeper in the infrared than in visible light.

A transit whose depth changes with colour is therefore a transit through more than one star, and multi-band photometry is one of the standard tests for blends. But the test has a blind spot exactly where the planet-on-primary error is largest. An equal twin has the same temperature as the primary, contributes the same fraction of the light in every band, and produces a perfectly achromatic transit diluted by a factor of two in depth — the case that makes the radius $\sqrt2$ too small and the density 2.8 times too high. The colour test catches the cases that are already easy to catch, the cool companions whose planets are rarely found, and misses the twins whose planets are found as often as the primary’s.

What imaging can and cannot remove #

The obvious remedy is to look. High-resolution imaging, with adaptive optics or speckle interferometry from large telescopes, resolves companions that a transit survey’s pixels cannot, and since 2014 most of the Kepler planet hosts have been observed this way. A companion that is seen can be corrected for: its flux ratio is measured, the depth is undiluted, and the planet’s radius is computed for both possible hosts.

The figure shows what that leaves. Companions to Sun-like stars have separations spread over many decades, with a broad peak near fifty astronomical units. An imaging survey’s resolution is an angle, about 0.05 arcseconds at best, and a fixed angle is a separation that grows with distance: 15 au at 300 parsecs, 50 au at 1,000. Most Kepler stars are several hundred to a thousand parsecs away, so at their distances the angle hides between a third and a half of all companions outright. Further out, the imaging reaches a contrast of a few magnitudes near the resolution limit and seven or eight beyond an arcsecond, and the faintest companions are missed there too. Combining the two, a sketch of a good modern contrast curve misses 47 per cent of companions at 300 parsecs and 60 per cent at 1,000.

The missed companions are not a random half. They are the close ones, which are the ones most likely to share an aperture, and the faint ones, whose planets are mostly undetectable anyway. The imaging removes the cases that are easiest to correct and leaves a residual that is still a large fraction of the total. Astrometry adds a further check — a binary’s orbital motion disturbs the fit of a single star’s parallax, and the quality of that fit flags companions too close to image — and spectroscopy flags those bright enough to add their own lines. Every method has a corner of separation and contrast it cannot reach, and the corners overlap. The residual dilution is therefore something a catalogue corrects statistically, by assuming a companion population and its effect, not something it can remove star by star.

What the synthetic survey leaves out #

The figures use one Sun-like primary for every pair, a mass ratio drawn uniformly, a fixed companion fraction and a depth threshold standing in for a whole detection pipeline. Real primaries span a range of masses; real mass-ratio distributions have an excess of twins; the fraction of companions that are close enough to share an aperture depends on the survey’s pixel size and on each star’s distance; and a pipeline detects on signal-to-noise, which depends on period, duration and the star’s brightness as well as depth. Each of these changes the numbers, and the published estimates — made with all of them modelled — find typical corrections to the mean planet radius in Kepler of ten to fifty per cent depending on how planets are assigned between the stars.

The model also ignores companions that are not bound. A background star at a different distance dilutes the depth in the same way, and the analysis of a background star is the false-positive analysis already described, with the difference that a background star’s planet is measured against a star of unknown radius at an unknown distance. And it ignores the effect that the companion’s planets, if they exist, are missing from the catalogue entirely — which is a bias in the occurrence rate rather than in the radius, and the larger one for small companions.

Still open: what the companions’ own planets look like #

The calculation turns the companions from a nuisance into a population. A quarter of the planets detected in binaries orbit the companion, they are measured too small by factors that reach two and a half, and they are found only around the more massive companions. Around the less massive ones — M dwarfs half the Sun’s mass or less, which are the commonest stars in the Galaxy and carry more small planets than Sun-like stars do — their planets are almost all undetected, and every one of them would be in a bin of the catalogue’s smallest planets if it were. What a survey of the companions themselves would find, at their own brightness and with their own radii, is the measurement that would turn a statistical correction into an observed population. The next space photometry missions resolve stars more finely and observe brighter, nearer targets, for which imaging resolves companions down to a few au; for them the companion’s planets become planets in their own right, measured against their own star. Whether the Earth-sized bin then shrinks, and by how much, is the test of how much of it was larger planets seen through someone else’s light.