Every planet of a star must agree on its density

A transit light curve carries more than the planet’s size. Its depth gives the ratio of the planet’s radius to the star’s; its four contact points fix how long the planet takes to cross the star and how long to cross its own diameter at the limb; and those durations, together with the orbital period, give the size of the orbit in units of the star’s radius. Kepler’s third law then turns the orbit’s size and period into the star’s mass divided by its radius cubed — its mean density — without any model of the star at all. It is one of the few places in astronomy where a star is weighed by a geometrical measurement alone.

The derivation assumes the orbit is circular. A planet on an eccentric orbit crosses the star at a speed that depends on where in its orbit the transit falls — faster near periastron, slower near apastron — and a transit fitted as circular then implies a density that is wrong by a factor set by the eccentricity and the argument of periastron,

$$\frac{\rho_{\rm circ}}{\rho_\star} = \left(\frac{1 + e\sin\omega}{\sqrt{1-e^2}}\right)^{3}.$$

An eccentricity of 0.2, with the transit near periastron, makes the implied density more than twice the true one. For a single planet this is a degeneracy with no escape from the light curve alone: the star’s density and the planet’s eccentricity are traded against each other, and one can be measured only if the other is known. When the star’s density is known independently — from its spectrum and a stellar model, or from its oscillations — the transit measures the eccentricity. When it is not, it measures neither.

Why a duration is a density #

It helps to see why a transit’s duration measures a density at all, because the reason is also why the eccentricity spoils it. A planet on a circular orbit of radius $a$ moves at a speed $2\pi a/P$, and it takes roughly the star’s diameter divided by that speed to cross the star. The duration therefore measures $R_\star/a$ for a given period; Kepler’s third law says $a^3 \propto M_\star P^2$; and so $(a/R_\star)^3 \propto M_\star P^2/R_\star^3$, which is the star’s mean density times the period squared. A planet with a ten-day period crosses a star like the Sun in about three and a half hours; the same planet crossing a star eight times denser, with half the radius and the same mass, takes about half as long.

An eccentric orbit changes the speed at the moment of transit without changing the period. At periastron the planet moves faster than a circular orbit of the same period would, the transit is shorter, and the star appears denser; at apastron the transit is longer and the star appears less dense. The factor in the density is the cube of the ratio of speeds, because the density goes as the cube of $a/R_\star$, and that cube is why modest eccentricities make large errors. The duration measures a speed, and it cannot tell whether the speed is set by the star’s gravity or by the orbit’s shape.

Four measurements of one number #

A star with several transiting planets breaks the degeneracy, because every planet orbits the same star. Each planet’s transit, fitted as circular, gives its own implied density, and each is the true density distorted by that planet’s own factor. The true density is common to all of them; the factors are not.

In the simulated system, three planets with eccentricities of a few hundredths agree, within their errors, on a density near the truth. The fourth, with an eccentricity of 0.2 and its transit falling near periastron, implies more than twice that density, five standard deviations from the others. No choice of star can reconcile the four; one planet is eccentric, and the system says which. A single planet with that light curve would have been assigned a star twice as dense as the true one, or an eccentricity nobody could check.

The logic has the structure of every over-determined measurement. Several planets give several estimates of one quantity with different, independent systematic errors, and their agreement both determines the quantity and tests the errors. The star’s density is measured to the precision of their agreement, and each planet’s disagreement with the consensus is a measurement of that planet’s eccentricity — or rather of the combination $e\sin\omega$ its transit is sensitive to.

The spread measures the eccentricities #

The planets of a single system usually disagree only slightly, and the size of the slight disagreements carries information even when no single planet stands out.

For small eccentricities the logarithm of the distortion factor is close to $3e\sin\omega$. If eccentricities follow a Rayleigh distribution — the natural distribution for small, randomly oriented eccentricity vectors — with scale $\sigma_e$, and the orientations are random, then $e\sin\omega$ is Gaussian with width $\sigma_e$, and the logarithms of the implied densities scatter with width $3\sigma_e$. The simulated scatter follows that rule for small dispersions and grows a little faster for larger ones. Measurement error adds in quadrature: with twenty per cent errors on each planet’s density, an eccentricity dispersion of a few hundredths hardly changes the scatter, and systems that circular are indistinguishable from exactly circular ones.

The measurement uses nothing about the star. It asks only how much the planets disagree with each other, not with any model, and so it is free of the systematic errors in stellar models that afflict any comparison of transit densities with spectroscopic ones. Applied to the Kepler mission’s hundreds of systems with several transiting planets, it gave the first measurement of how eccentric small planets’ orbits are in such systems: a few hundredths, typically — nearly circular, as the Solar System’s planets are.

How well several planets weigh their star #

If the orbits are nearly circular, several planets measure their star’s density almost as well as if they were known to be circular. Four planets with fifteen per cent errors each give the density to eight per cent when the eccentricity dispersion is two hundredths; each planet’s unknown eccentricity adds a little noise, and averaging over planets reduces both. For more eccentric systems the eccentricities dominate and the density is known less well, and a single planet in an eccentric population gives a density with a factor-of-two uncertainty that no amount of photometry reduces.

That reverses the usual order of an exoplanet analysis. Normally the star is characterised first, from its spectrum and a stellar model, and the planets’ radii follow from the star’s — a planet’s radius is only ever as good as its star’s. For a compact system of several transiting planets, the planets characterise the star: their transits give its density directly, and that density, combined with its spectrum, constrains its radius and age better than the spectrum alone. The best-studied compact systems, with six or seven transiting planets round a small star, have their stellar densities measured from the transits to a few per cent.

The candidate that orbits the wrong star #

The same consistency is a test of whether a candidate belongs to the system at all.

Many transit candidates are not planets of the star they appear to orbit. The most common impostors are eclipsing binaries in the background, blended with the target in the same pixels, whose deep eclipses are diluted into shallow ones that look like planets, and planets orbiting a different star that happens to lie along the same line of sight. A candidate is almost never confirmed, only validated — shown to be far more probable as a planet than as any of the alternatives — and the probability calculation uses every clue the light curve offers.

A candidate in a system with other transiting planets offers a strong one. If it orbits the same star, its transit must imply the same density as theirs, within what a modest eccentricity allows. If it orbits a background giant — a star a hundred times less dense than a dwarf — its transit is too long for its period, and it implies a density twenty or more standard deviations from the system’s. If it orbits a bound companion star of similar type, the test is weak, because such a star’s density differs from the target’s by less than a factor of two. Density consistency is one of the reasons candidates in multiple systems are validated at much higher confidence than single ones: the impostors that dominate among single candidates are mostly the ones the density comparison catches.

Singles and siblings #

The strongest use of transit densities has been to compare them with densities measured independently, for stars bright enough to show their oscillations.

A star’s oscillation frequencies — the comb of modes an interior writes — give its mean density directly: the spacing between modes of successive order is proportional to the square root of the density, to about a per cent for Sun-like stars. For stars with both oscillations and transiting planets, the ratio of the transit density to the oscillation density is the distortion factor itself, one planet at a time, and a population of such ratios measures the population of eccentricities. The figure shows what the two kinds of population look like: a narrow peak for nearly circular orbits, a broad distribution with long wings for eccentric ones.

When that comparison was made for small planets with asteroseismic hosts, the answer depended on the planets’ company. Planets with transiting siblings produced the narrow peak, with eccentricities of a few hundredths. Planets transiting alone produced something broader: a majority still nearly circular, and a substantial minority with eccentricities of a few tenths. The difference is a clue to how the systems formed and evolved. A system whose planets are all still in nearly circular, nearly coplanar orbits has probably never been disturbed; a single transiting planet with an eccentric orbit may be the survivor of instabilities that scattered or removed its siblings, or that tilted them out of the transiting geometry — which is why how many planets a star appears to have is not itself a measurement.

The Solar System as a transiting system #

The size of the effect is easiest to feel for a system whose orbits are known. An observer far away, looking along the plane of the Solar System, would see the inner planets transit the Sun. Venus and the Earth, with eccentricities of under two hundredths, would imply the Sun’s density to within a few per cent each, and agree. Mercury, with an eccentricity of 0.21, would imply a density anywhere from about half to nearly twice the Sun’s depending on where its periastron happened to lie relative to the observer — for a typical orientation, a disagreement of a few tens of per cent, easily detected against the other two. The observer would conclude, correctly, that the innermost planet has a noticeably eccentric orbit and the others do not, without ever measuring an eccentricity by any other means.

The most dramatic real case is the compact system of seven Earth-sized planets round the small, cool star TRAPPIST-1. All seven transit, with periods from one and a half to nineteen days, and their transits together imply a stellar density of about fifty times the Sun’s, measured to a few per cent — far better than the star’s spectrum could give, since models of such small stars are uncertain. The agreement among the seven is also a statement that their orbits are very nearly circular, as the resonant chain linking them requires, and the density they share has since been used to pin down the star’s radius and mass, and through them the planets’ own sizes and densities.

Where the eccentric minority’s eccentricities come from is itself a population question. Planets can be pushed onto eccentric orbits by close encounters with their siblings, which tend to leave a system with fewer, more eccentric planets on more widely spaced orbits; by a distant stellar or planetary companion driving slow cycles of eccentricity and inclination, which act on a single inner planet much more readily than on a tightly coupled system, because the siblings’ mutual precession suppresses the cycles; or by interactions with the disc in which they formed. Each leaves a different signature in how eccentricity correlates with period, with the host star’s composition and with the presence of outer companions. And each is opposed by tides from the star, which circularise the closest orbits over the system’s lifetime, so that the youngest and widest singles should be the most eccentric. Transit densities compared with asteroseismic ones are one of the few ways to measure small planets’ eccentricities in large numbers, and so one of the few tests of which of these processes dominates.

What the transit density leaves out #

The implied density depends on more than the eccentricity. It depends on the impact parameter — how far from the star’s centre the planet crosses — which is fitted from the shape of the ingress and egress and is poorly determined for small planets whose ingress is short and noisy; and on the limb darkening, which shapes the transit’s edges and is partly degenerate with the impact parameter. A grazing transit, crossing near the limb, has a density nearly unconstrained, and systems with several planets often contain one. The figures take each planet’s density error as given, but in practice those errors are correlated with the impact parameter and are larger than a simple estimate suggests.

The rule that the scatter measures the eccentricities assumes random orientations of the orbits’ periastra. In tightly packed systems near resonances, the eccentricities and their orientations are coupled by the planets’ mutual gravity, and the periastra can be aligned; then the planets’ distortion factors are correlated, the scatter underestimates the eccentricities, and the consensus density itself is biased. Transit timing variations in such systems carry the same eccentricities in another form, and the two measurements are fitted together in the best analyses. And the stellar density assumed common to all planets is the density of the star they orbit — which is the assumption the blend test exploits, and which fails for a system whose planets orbit two different stars of a close binary.

One star, weighed by its planets #

It is worth stating plainly what the multi-planet measurement achieves. A star’s mean density is a fundamental quantity: combined with its temperature and luminosity, it fixes its radius and mass, and through them its age and evolutionary state. For most stars it is estimated from spectra through models of stellar structure, with uncertainties of ten or twenty per cent that are dominated by the models’ assumptions. Asteroseismology measures it directly for stars bright and large enough to oscillate detectably. The planets offer a third route, for stars too faint or too small for asteroseismology — the small, cool stars that host most compact multi-planet systems, and whose models are least certain. Their transiting planets weigh them.

That matters for the stars as much as for the planets. Small stars are observed, when their radii can be measured directly, to be several per cent larger than the standard models predict for their masses — a discrepancy attributed, without certainty, to magnetic fields suppressing convection near their surfaces. A density measured geometrically from a compact system of planets, for a star whose mass is estimated from its brightness, is a direct test of that discrepancy for a star whose radius cannot otherwise be measured, and each well-measured compact system adds one more point to a relation the models still fail to reproduce.

Still open: whether the singles’ eccentricities are one population or two #

The broad distribution for singly transiting planets can be fitted as a mixture: a circular majority like the multis, and an eccentric minority. The mixture suggests that most single transiting planets are members of multi-planet systems whose other members do not transit, and a minority are genuinely different — dynamically hot systems, or lone survivors. But the eccentric minority could also be a continuum, with no gap between the two, and the samples with asteroseismic densities number only a few hundred. The space missions now observing tens of thousands of bright stars will measure oscillations for many more planet hosts, and whether the singles’ eccentricities separate cleanly into two populations or spread continuously between them is the question the larger sample is expected to settle — and with it, how often planetary systems are disturbed after they form.