A transit that says which star it crossed

About half of the Sun-like stars in a transit survey’s target list are pairs, and more of them are pairs than the sky’s share would suggest because a pair is brighter than a single star and a magnitude limit draws it from further away. Most of those pairs are too close for the survey’s camera to separate. When a planet transits one of the two, the catalogue records a dip in the combined light, assigns the planet to the brighter star, and computes its radius from that star’s size.

If the planet in fact orbits the companion, both steps are wrong. The dip has been diluted by the primary’s light, so the planet is larger than it looks; and the radius has been scaled by the wrong star, which is usually larger than the true host, so the error partly cancels and partly does not. The essay on where diluted planets land followed the consequences: planets round companions are mismeasured by factors of two or more, the faint companions’ planets are mostly never found, and the ones that are found are filed in the catalogue’s smallest bins.

That essay treated the host as unknowable from the light curve and asked what a population of misassigned planets would look like. It is not entirely unknowable. A transit carries two pieces of information about the star it crossed, independent of the catalogue’s assumption: how dense that star is, and where it is. This essay works out when each one is enough to name the host.

What naming the host changes #

The stakes for a single planet are set by the size of the correction, and the correction is large. Suppose a transit in the blended light of a Sun-like primary and a half-solar-mass companion implies a planet of 1.3 Earth radii when assigned to the primary. Assigned to the companion instead, two things change. The companion supplies only about four and a half per cent of the pair’s light in a red-optical band, so its own dip must have been more than twenty times deeper than the one observed; and the companion’s radius is about half the Sun’s. The planet’s radius is the observed one multiplied by the companion’s radius and by the square root of one plus the inverse flux ratio — a factor of about 2.6 here. The rocky planet of 1.3 Earth radii becomes a sub-Neptune of 3.4.

For a 0.8-solar-mass companion the factor is about 1.6; for a 0.4-solar-mass companion, 3.3. A planet assigned to the primary when it orbits the primary is wrong only by the dilution itself, the square root of one plus the flux ratio, which for a half-solar-mass companion is two per cent. So the catalogue’s default is a good bet for any one planet — most planets in binaries do orbit the brighter star, because the brighter star’s planets are easier to find — and a bad one for the population, because the planets for which it is wrong are wrong by a factor of two or three, all in the same direction, all towards the small end of the catalogue.

That asymmetry is why naming the host matters even when it can be done for only some of the planets. Each planet moved from the primary to the companion leaves the bin of Earth-sized planets and enters the bin of sub-Neptunes, and the bins are where occurrence rates, compositions and the radius valley are all read.

A star weighed by the shape of a shadow #

A transit’s light curve has a depth, a total duration, and the duration of its ingress and egress — the time the planet’s disc takes to cross the star’s limb. Four contact points fix the geometry: from the ratio of the ingress to the total duration and the depth, the impact parameter and the ratio of the orbit’s size to the star’s radius, $a/R_\star$, follow. Kepler’s third law then turns $a/R_\star$ and the period into the star’s mean density,

$$\rho_\star = \frac{3\pi}{GP^2}\left(\frac{a}{R_\star}\right)^3,$$

with no reference to the star’s mass or radius separately and no model of its interior. The density belongs to whichever star the planet crossed. Dilution reduces the depth but, for a small planet, barely changes the shape of the ingress, so the density measured is the host’s.

Main-sequence stars become denser as they become smaller. The Sun’s mean density is 1.4 grams per cubic centimetre; a star of half the Sun’s mass has about three times that, and one of a third of the Sun’s mass about eight times.

So a planet assigned to a Sun-like primary whose transit implies three times the Sun’s density did not orbit the primary. It orbited something smaller, and in a known binary that is the companion. Twins give no signal, because their densities are the same; the test strengthens as the companion becomes fainter, which is also how its planets become harder to detect at all. The range in which it is useful is bounded on both sides — above by dilution, which hides the faintest companions’ planets entirely, and below by similarity — and between them it covers companions from about 0.8 to about 0.45 solar masses, which is most of the companions round Sun-like stars.

An orbit that changes the clock #

The density test has a well-known confusion. The ingress and total durations measure the planet’s speed across the star, and the formula above assumes the planet moves at the speed of a circular orbit. On an eccentric orbit the planet crosses faster near periastron and slower near apoastron, and the density inferred from a circular fit is wrong by a factor

$$\psi = \left(\frac{1+e\sin\omega}{\sqrt{1-e^2}}\right)^3,$$

where ω is the angle of periastron. That is the duration that measures an eccentricity: with the host’s density known independently, the mismatch is a measurement of e. Here the host’s density is the unknown, and the eccentricity is a nuisance that can mimic a different host.

How much it matters depends on how eccentric the planets are.

For planets in compact multi-planet systems, whose eccentricities are a few hundredths, the factor ψ is within twenty per cent of one and the two hosts’ densities, a factor of three apart, do not overlap. The test is clean. That is the regime in which every planet of a star must agree on its density — and the agreement among siblings is itself a check that they share a host. If two planets in one light curve imply densities a factor of three apart, they orbit different stars.

For single transiting planets, whose eccentricities are much larger — a measured scale of about a third, with a cold core and a hot tail — the factor ψ spreads over a decade. A planet on the primary at an unlucky orientation implies a density three times too high; one on the companion at the opposite orientation implies the primary’s. A third of the primary’s planets would be assigned to the companion and a fifth of the companion’s to the primary. The test then says something about the population — an excess of high densities in a sample of binaries, beyond what the single stars show, is a count of planets on companions — but not about any one planet.

A transit strong enough to weigh its star #

The density is limited by the ingress. A small planet crosses the limb in a few minutes, and timing those minutes against the noise of a star’s light needs a strong signal.

Carter and collaborators worked out in 2008 how the uncertainties of the transit parameters scale with the signal: the ingress duration’s fractional error is roughly the inverse of the total signal-to-noise times the square root of six times the ratio of total to ingress duration, and the density inherits one and a half times that. For an Earth-sized planet on a Sun-like star, the ingress is about a hundredth of the transit, and separating a solar host from a half-solar-mass one at three standard deviations needs a total signal-to-noise of about a hundred. For a sub-Neptune, sixty; for a Jupiter, thirty.

The planets that matter most for the diluted population are exactly the ones that fail this. A survey detects a transit when its signal-to-noise exceeds seven or so, and most of its small planets sit between seven and fifteen. At that strength a transit’s density is uncertain by a factor of two or worse, and the factor of three between hosts is not resolved. The density test names the host for the bright, deep, well-sampled transits — and those are the ones least likely to be on a faint companion.

A centre of light that moves the wrong way #

The second piece of information is position. An unresolved pair’s image is the sum of two point spreads a separation s apart, and its centre of light sits between the two stars, weighted by their brightness. A transit dims one of them, and the centre moves.

The shift has a direction and a size, and both depend on which star was crossed. If the primary was dimmed, the centre moves towards the companion; if the companion was dimmed, it moves towards the primary. The size differs by the pair’s flux ratio. To produce the same blended depth, a faint companion has to lose a much larger fraction of its own light than the primary would, and so its transit moves the centroid further — ten times further for a companion ten times fainter. For a sub-Neptune’s thousand-part-per-million dip, a pair a tenth of an arcsecond apart with a half-solar-mass companion shifts by about a hundred microarcseconds if the companion was crossed and by four if the primary was.

Reading the direction needs the pair’s orientation on the sky, which an unresolved pair does not provide by itself; reading the size needs its separation and flux ratio. Both can come from imaging when the pair is resolved, and from the astrometric orbit when it is not — an astrometric survey sees an unresolved pair’s photocentre wobble round the centre of mass, and the wobble’s size and shape give the separation and the flux ratio. Given those, the centroid shift during a transit names the host whenever it exceeds the measurement’s precision, and its threshold separation falls as the companion gets fainter: the centroid test works best for exactly the planets the density test cannot reach, the faint companions’ weak transits, and fails for the near-twins whose shifts in the two cases differ least.

This is the same physics the surveys already use to reject background eclipsing binaries. A transit whose difference image is centred off the target star is a blend, and that check has removed thousands of false positives from candidate lists; it is one of the tests by which a planet is validated rather than confirmed. Applied to a bound pair rather than a chance alignment, it does not reject the planet; it moves it to the other star.

Where each test works #

Put the two tests on one plane — the pair’s separation across, its brightness difference up — and add the third route, imaging, which simply resolves the pair.

The top of the plane is empty of planets to assign: beyond four magnitudes a sub-Neptune on the companion is diluted below any survey’s threshold. The right of the plane belongs to imaging, which separates pairs down to a few hundredths of an arcsecond at small brightness differences and a few tenths at large ones. Between them, the centroid test reaches pairs a little closer than imaging can resolve, provided the pair’s geometry is known from its astrometric orbit. And the density test is a horizontal band, valid at any separation — including pairs far too close for any image — but only between a magnitude and four, and only for strong transits on near-circular orbits.

The corner they leave is the close, nearly equal pairs: separations under a few hundredths of an arcsecond, brightness differences under a magnitude. Those are pairs a few astronomical units apart at a few hundred parsecs, where their orbits are short enough to affect planet formation but long enough to leave room for planets on either star. For them, which star the planet orbits is undetermined by anything in the transit. The consolation is that it matters least there, because for twins the dilution error is only the square root of two whichever star is the host.

What the companions’ planets are #

The motive for naming hosts is not tidiness. It is the question left open by the dilution argument: whether companions have their own planets at the rate single stars of their mass do. The companions round Sun-like primaries are mostly red dwarfs, and red dwarfs, observed as single stars, carry more small planets than Sun-like stars do — two or more per star within a couple of hundred days. If the companions are like single red dwarfs, a quarter or more of the small planets in the binary systems of a transit survey orbit the companion, and most of them have never been found.

The density test turns that into a measurement for the planets that have been found. A sample of transiting planets in binaries known from astrometry, each with a density from its transit shape, should split into two groups — densities like the primary’s and densities like the companion’s — with the share in the second group measuring how many of the detected planets are the companion’s. Corrected for how much harder those are to detect, which the dilution figures quantify, that share becomes the companions’ occurrence rate. The planets in multi-planet systems give the cleanest split, because their eccentricities are small; the single planets give a blurred one, but in large numbers it can still be deconvolved.

There is a check built into the method. A star whose oscillations are measured — as tens of thousands of bright Sun-like stars’ now are — has a density known from the comb of its own pulsation frequencies to a few per cent. In a pair, only the primary is bright enough for that. A transit implying the primary’s seismic density is on the primary; one implying three times it is not.

What the tests assume #

Dilution leaves the shape alone. For a small planet the ingress and total durations are measured from the transit’s timing, and the depth enters the density only weakly. For a large planet on a faint companion the true depth is much larger than the observed one, the fitted impact parameter is biased, and the density is too. The test is cleanest for small planets, which is where it is needed.

The stars are on the main sequence. A companion that is a subgiant, or a white dwarf, breaks the mass–density relation drawn here. Evolved primaries are a larger problem: a primary that has begun to swell has a density lower than the Sun’s, and a companion’s planet then looks even more out of place.

The centroid precision is illustrative. The thirty microarcseconds drawn is not a property of any particular instrument. Ground-based and space-based photometers differ by orders of magnitude in what they can centroid, and the threshold separation scales directly with it.

The host’s density has a spread of its own. A main-sequence star’s density depends on its age and composition as well as its mass, by tens of per cent over the main-sequence lifetime of a Sun-like star. That spread is small compared with the factor of three between a solar primary and a half-solar-mass companion, but it is comparable with the factor of 1.5 for a 0.8-solar-mass companion, which is why the useful band begins near a magnitude rather than at zero.

Triples exist. About a tenth of multiple systems have three stars or more, and an unresolved third star adds a third host and a third density.

Still open: how many planets the companions have #

The planets on the faint companions of Sun-like stars are a population that magnitude-limited surveys systematically miss and misfile, and its size is not known. The light curves already in the archives carry two measurements of host identity for the planets that were found — a density from the transit’s shape and, where the pair’s orbit is known, a centroid shift — and neither has been applied systematically to the binaries that astrometry has now catalogued.

Applied, the density alone would sort the strongest transits in the multi-planet systems by host, and the share on companions would be the first direct measurement of how often a red dwarf with a brighter neighbour has planets of its own. If the share comes out near what single red dwarfs predict, the occurrence rates of small planets have been undercounting the Galaxy’s commonest planetary systems wherever a brighter star stood in front of them. If it comes out low, something about a companion — a truncated disc, a stirred one — suppresses the planets that would have formed round it, and the binary fraction of a survey’s targets becomes a correction to the planets’ formation as well as to their count.