A target list that counts pairs twice

Every survey draws a different sky: the planets a method finds are a measurement of what the method can see as much as of what is there. Most of the selection effects that essay and its successors described act on the planets — on their sizes, their orbits, their orientations. This one acts on the stars, before a single planet has been looked for.

A survey has to choose its targets, and it chooses them by brightness. A transit survey needs enough photons to see a dip of a part in ten thousand; a velocity survey needs enough to measure a shift of a metre a second. So the target list is all the stars brighter than some limit, in some region of sky. That sounds like a fair sample of stars, and it is not, because an unresolved binary — two stars too close to be separated in the survey’s images — is brighter than either of its components, and a magnitude limit reaches it from further away.

The previous essay on dilution took the binaries on a target list as given and asked what their companions’ light did to the planets found round them. This one asks how many of them there are, and which ones. The answer is: more than in the Galaxy, and preferentially the ones that dilute most.

Brighter, and seen from further #

A companion adds its light to the primary’s. On the main sequence, luminosity rises steeply with mass — roughly as the fourth power — so a companion of half the primary’s mass adds only a sixteenth of its light, and a companion of equal mass doubles it.

The volume factor follows from a simple argument. A star of a given brightness is seen out to a distance that grows as the square root of its luminosity, so doubling the light extends the reach by $\sqrt2$. If stars are spread uniformly in space, the number within reach grows as the cube of the distance, and the twin is seen in $2\sqrt2 = 2.83$ times as many places as either of its stars alone. This is the Malmquist bias in its most literal form: the brighter members of a population are over-represented in any sample limited by brightness, because they are seen from further away.

The Malmquist bias is usually discussed for standard candles, where it makes a sample of supposedly identical objects brighter on average than the population they came from. For binaries the brightening is not a scatter about a mean but a definite, mass-ratio-dependent boost, and the over-representation is of a particular kind of system rather than of a tail.

The same boost corrupts the stellar parameters a survey assigns. A star’s radius, which every transiting planet’s radius is scaled from, is usually derived from its brightness and its distance; an unresolved twin, treated as one star, is assigned a luminosity twice its primary’s and therefore a radius $\sqrt2$ too large. That error partly offsets the dilution of the planet’s transit, which made the planet’s radius $\sqrt2$ too small relative to the star — so for a planet round the primary of an equal pair the two errors cancel, and for one round a fainter star they do not. Parallaxes from the Gaia satellite made this effect visible, because they gave distances independent of brightness, and stars that were too luminous for their colours stood out as likely unresolved pairs.

More pairs on the list than in the sky #

About half of solar-type stars have a stellar companion. The companions have masses spread fairly evenly from a tenth of the primary’s to equal, with an excess of near-twins, and a large fraction of them are close enough to be unresolved at the distances transit surveys look.

The average excess is modest — a few percentage points — because most companions are faint and add little light. That modesty is misleading, because the excess is not spread evenly across the binaries.

So a magnitude-limited target list is enriched in twins. Those are the binaries in which a planet’s transit is diluted the most: a planet round either star of an equal pair has its transit depth halved by the other’s light, and its radius, inferred from the depth, underestimated by a factor of $\sqrt2$. The binaries that matter most for the planet statistics are the ones the target list over-represents most.

How deep the survey looks #

The volume factor depends on the geometry of the stars the survey reaches, and the Galaxy is not a uniform volume. Stars are concentrated in a thin disc a few hundred parsecs thick, and a survey that reaches beyond that thickness — looking out of the plane — runs out of stars before it runs out of sensitivity.

The surveys that found most transiting planets sit at different places on this curve. A survey of the brightest stars across the whole sky reaches a few hundred parsecs for Sun-like stars and sits near the shallow end, with the full factor of 2.8. A deep survey of a single field, staring for years at stars down to sixteenth magnitude, reached stars a kilometre or more away in a direction tilted out of the plane, and sat further along. The correction a survey needs is therefore not a universal number but depends on its depth and pointing — one more way in which every survey draws its own sky.

In the deep, disc-outreaching limit, the twins’ advantage is smaller but their place in the distribution is the same.

Planets per target, planets per star #

An occurrence rate is a number of planets divided by a number of stars, and part of any rate is a definition. Surveys usually divide by the number of targets, treating each as a single star. When some targets are pairs, two things go wrong, in opposite directions.

A binary target contains two stars, each of which can have planets; if planets are as common around stars in binaries as around single stars, a binary target offers twice the opportunity. But each planet’s transit is diluted by the other star’s light, and planets near the detection threshold are lost; planets around the fainter companion are diluted far more than planets around the primary, and are mostly lost entirely.

Neither number is the truth, and the gap between them is set by the binary fraction on the target list — which is the quantity the magnitude limit distorts. The occurrence rates most quoted are per target, and correct for dilution only partly; the corrections that do exist usually use the binary fraction of the Galaxy rather than that of the target list.

A complication in the other direction #

The picture so far assumes that planets are as common around stars in binaries as around single stars, and they are not. Close binaries — those with separations of tens of astronomical units or less — suppress planet formation: the companion stirs the protoplanetary disc, truncates it, and heats it. Surveys that have imaged the hosts of transiting planets at high resolution find far fewer close companions than among field stars of the same type, a deficit that sets in at separations below about a hundred astronomical units.

That deficit partly cancels the over-representation, in a way that depends on separation. The binaries a magnitude limit over-represents are all the unresolved ones, close and wide; the binaries that lack planets are the close ones. For a survey whose resolution leaves pairs closer than a few hundred astronomical units unresolved, the extra binaries on the list are disproportionately close, planet-poor pairs — which lowers the inferred occurrence per target, the opposite of the double-counting effect.

So the correction a survey needs is a product of three things: how many pairs the magnitude limit adds, which mass ratios they have, and how planet-poor they are at each separation. Each has been measured, with uncertainties that leave the net correction uncertain by several per cent of the occurrence rate — the same order as the statistical errors the largest surveys now quote.

The velocity surveys’ opposite bias #

Radial-velocity surveys choose their targets by brightness too, and they then do something transit surveys cannot: they remove binaries. A star whose spectrum shows two sets of lines, or whose velocity swings by kilometres a second, is a known binary and is dropped from a planet search, because its own motion would swamp any planet’s. Many velocity surveys also exclude stars with known visual companions within a few arcseconds, to keep the second star’s light out of the spectrograph.

So a velocity survey’s sample is biased the other way: it under-represents close binaries, and the planets it finds are planets of single stars and wide pairs. That is part of why occurrence rates from the two methods, for planets they can both detect, have to be compared with care. A transit survey’s sample contains more close pairs than the Galaxy and a velocity survey’s fewer, and planets are rarer in close pairs; the two surveys’ rates bracket the truth from different sides, and agreement between them is partly a coincidence of the two biases’ sizes.

Correcting it #

The corrections in current use come in two kinds. The first is direct: image as many hosts as possible at high resolution, find the companions, correct each planet’s radius for its dilution, and assign it to the star it most probably orbits. That removes the bias for the resolved pairs and leaves it for the unresolved ones.

The second is statistical: simulate the survey. A synthetic galaxy is populated with stars, binaries with a measured distribution of separations and mass ratios, and planets with a trial occurrence rate, and the synthetic sky is observed through the survey’s own selection function — its magnitude limit, its resolution, its detection efficiency. The trial occurrence rate is then adjusted until the synthetic detections match the real ones. That forward model includes the magnitude limit’s preference for pairs automatically, because the synthetic stars are selected by apparent brightness exactly as the real ones were.

Forward models of this kind have shifted the inferred occurrence of small planets by of order ten per cent relative to rates computed per target with no binary correction, and in the smallest radius bins by more. The direction depends on the assumptions about planets in close binaries, which is where the remaining uncertainty lives.

What follow-up imaging finds #

The cleanest check is to look. High-resolution imaging with adaptive optics or speckle interferometry resolves companions down to tenths of an arcsecond, and the hosts of transiting planet candidates have been imaged in large numbers precisely because a resolved companion changes the planet’s inferred radius.

What the images find is consistent with the bias. A substantial fraction of planet hosts from the deep transit fields — of order ten to thirty per cent, depending on the brightness and the resolution reached — have a companion within a couple of arcseconds, and those companions are skewed towards near-equal brightness relative to what a volume-limited sample would show. Imaging below a tenth of an arcsecond, which only the largest telescopes reach, finds more.

Imaging cannot find everything. A companion closer than the resolution limit, or much fainter than the primary, is missed, and the closest pairs — spectroscopic binaries, separated by less than an astronomical unit — are invisible to imaging entirely and are found only by velocity monitoring, which few transit hosts have had. The residual unresolved fraction, after imaging, is estimated statistically from the binary population, which brings the bias back in through the estimate.

Why it matters for the smallest planets #

The occurrence of the smallest planets — Earth-sized, and in the habitable zone — is the number transit surveys were built to measure, and it is the most sensitive to this bias. Small planets sit near the detection threshold; a dilution of a factor of two in depth moves many of them below it, and moves the survivors into smaller radius bins. The Earth-sized bin is where diluted planets land, and the magnitude limit decides how many of them there are to land.

The planet population’s most famous feature, the gap in the radius distribution between rocky and gas-enveloped planets, is affected the same way. Planets diluted by a factor of $\sqrt2$ in radius are moved across the gap from above, filling it partly; the gap’s measured depth and position depend on how many diluted planets the sample contains, and therefore on the binary fraction of the target list. Samples cleaned of known binaries show a sharper gap, which is itself evidence that the uncleaned samples contained enough diluted planets to blur it.

A brightness limit always chooses #

The binary bias is one member of a family. Any sample chosen by apparent brightness favours whatever makes an object look brighter at a given distance, and the favoured objects are over-represented by the ratio of the volumes from which they are seen.

A brightness is a distance only if something is known about the object’s luminosity, and the Malmquist argument is what follows when that knowledge is statistical: the sample is brighter than the population, by an amount set by the spread of luminosities and the geometry of the volume. For rapidly rotating stars, which are brighter seen pole-on than equator-on because their poles are hotter, a brightness limit prefers the pole-on stars — an orientation selected by a magnitude cut, with consequences for every statistic about stellar inclinations. For binaries, the brightening comes from a second star, and the sample prefers pairs.

What distinguishes the binary case is that the favoured property also changes the measurement the survey makes. A pole-on star is no worse a target for most purposes; a twin binary halves every transit depth on it. The selection and the measurement error are correlated, and that is what turns a modest excess in the binary fraction into a systematic in the planet statistics.

Where the model stops #

Luminosity as the fourth power of mass. The model uses $L \propto M^4$ throughout. On the lower main sequence the exponent is smaller, and in the red bands of most transit surveys low-mass companions contribute relatively more light than in bolometric terms. Both make faint companions slightly more important than drawn.

A flat mass-ratio distribution. The intrinsic distribution of companion masses is drawn as flat between a tenth and one. The measured distribution has an excess of twins, which makes the bias stronger, and depends on separation.

A smooth disc. The geometry figure uses an exponential disc of one scale height at one latitude. The real Galaxy has a thin and a thick disc and a halo, and the stars a survey reaches at different magnitudes sit in different components.

A planet population independent of binarity. The occurrence model gives every star the same planets. The suppression of planets in close binaries, discussed above, is not in the figure; its effect depends on the separation distribution of the unresolved pairs, which the model does not carry.

The surprising connection #

The same bias distorts the census of stars themselves. Counting stars brighter than a limit to build the luminosity function — how many stars of each brightness there are — over-counts the unresolved binaries as single stars more luminous than either component, which shifts stars from the faint end to the bright end and flattens the inferred initial mass function. The correction for that, worked out for star counts decades before exoplanets were found, is the correction transit surveys need: the same arithmetic of a $\sqrt2$ in distance and a $2\sqrt2$ in volume, applied first to the stars and now to the planets that orbit them.

Still open: the planets round the companions #

The companions on a target list are, by construction, the stars most likely to be ignored: fainter than the primary, often unresolved, and assumed when a planet is found to be the second most likely host. A planet round a twin companion is diluted by half and recorded as orbiting the primary; one round a faint companion is diluted almost to nothing and not recorded at all. Whether the companions in planet-hosting binaries have their own planets at the rate single stars of their mass do — which would mean that magnitude-limited surveys have systematically missed a population of planets round the commonest stars in the Galaxy — needs targets chosen without a magnitude limit, or companions resolved and searched separately, and neither has yet been done at the scale of the surveys that produced the occurrence rates.