A gap in eccentricity a density can only just see

A planet’s transit, fitted as though its orbit were circular, implies a density for its star, and if the orbit is eccentric the implied density is wrong by a factor that depends on the eccentricity and on where the periastron lies. A star with several transiting planets has several such measurements of one number, and their agreement says the orbits are nearly circular. A star whose density is known independently — from the comb of frequencies at which it oscillates — turns each planet’s mismatch into a measurement of that planet’s eccentricity, or rather of one combination of its eccentricity and orientation.

Done for hundreds of planets round oscillating stars, that comparison found a difference by company. Planets with transiting siblings have eccentricities of a few hundredths. Planets that transit alone are, on average, more eccentric, and the excess was fitted as a mixture: a cold majority like the planets in multi-planet systems, and a hot minority with eccentricities of a few tenths. The physical reading was that most singles are members of multi-planet systems whose siblings happen not to transit, while a minority are something else — survivors of instabilities that scattered or removed their neighbours.

The same data could also be fitted by a continuum, a spread of dynamical temperatures with no gap, and whether the hot planets are a separate population is a question about how planetary systems are disturbed. A separate population means a distinct event — an instability that some systems suffer and others escape. A continuum means a range of processes acting by degrees. The question here is what it would take to tell, given that what is measured is never an eccentricity.

Two distributions, and the closest pair #

A comparison between a specific mixture and an arbitrary continuum would prove nothing; a continuum chosen badly can always be rejected. The comparison that means something sets the mixture against the continuum that resembles it most, in the quantity actually measured. The continuum used here is a natural family: each planet’s eccentricities follow a Rayleigh distribution, as orbits excited by many small random perturbations do, and the scale of that distribution varies from planet to planet evenly in its logarithm between two limits. With the limits free, it can be broad or narrow, centred on cold orbits or warm ones. The closest member to the mixture has its scales spread from 0.015 to 0.34.

Drawn as eccentricity distributions the two are plainly different. The mixture has a sharp peak at a few hundredths, falls steeply, and then flattens into a long shoulder carried by its hot population; between them there is a dip. The continuum falls smoothly from its peak with no dip at all. If eccentricities could be measured one planet at a time with small errors, a few dozen planets in the range 0.1 to 0.15 would settle the matter.

They cannot. A single transit with an independently known stellar density gives one number, the ratio of implied to true density, and that ratio is the cube of $(1 + e\sin\omega)/\sqrt{1 - e^2}$. An orbit with eccentricity 0.3 seen with its periastron perpendicular to the line of sight gives a ratio of 1.15, hard to tell from a circle. The same orbit with its periastron towards the observer gives a ratio of 2.5, and away from the observer 0.4. Every eccentric planet is spread by its random orientation across a range of ratios, and the spreads of all the planets overlap.

The ratios they predict #

The prediction of each hypothesis for the measured ratios comes from integrating over eccentricity and orientation together: for each eccentricity the orientation is uniform, and each hypothesis weights the eccentricities its own way.

The two curves are nearly the same. The core is the cold planets, whose ratios cluster within a few per cent of one; the wings are the eccentric planets, seen at every orientation. The mixture’s dip in eccentricity appears in the ratios only as a slightly narrower core and a slightly more abrupt shoulder where the core meets the wings. Everything else — the height of the wings, their slope, their reach to factors of five — is matched by the continuum, because those are the features the continuum’s two free limits were chosen to reproduce.

How different two distributions are, for the purpose of telling which one a sample was drawn from, is measured by the Kullback–Leibler divergence: the average, over planets drawn from one hypothesis, of the logarithm of the ratio of the two hypotheses’ probabilities. It is the evidence each planet contributes, on average, in favour of the true hypothesis. Here it is 0.031 nats. Evidence adds over independent planets, and a likelihood-ratio test reaches the conventional three standard deviations when twice the summed evidence reaches nine — about 150 planets at this precision.

That number is a floor in several ways. It assumes the two hypotheses are the only ones; it assumes each planet’s error is known; and it assumes a density error of one per cent, which is not what the measurements have.

Evidence counted in nats #

The divergence is worth a moment, because it turns a vague question — can these be told apart? — into a number with units. Each planet drawn from the true distribution has a probability under each hypothesis, and the logarithm of their ratio is the evidence that planet supplies. Some planets supply evidence the wrong way, by happening to land where the false hypothesis is more probable. On average, the evidence is positive, and its average is the divergence. Because it is an average of logarithms, it adds over independent planets, and the total grows in proportion to their number while its scatter grows only as the square root.

A divergence of 0.031 nats is small. It means that an individual planet shifts the odds between the two hypotheses by about three per cent, on average, and that it takes about thirty planets to shift them by a factor of e. The three-standard-deviation threshold corresponds to odds of about ninety to one, which needs about 4.5 nats, or 150 planets. Counting only the planets that land in the dip, by contrast, would throw away the evidence in the lobes on either side of it and need several times as many.

A real analysis also has to fit each hypothesis’s parameters from the same data, and the mixture has one more free parameter than the continuum. A model with more freedom fits any sample at least as well, and the standard corrections charge it about half a nat for the extra parameter. At 150 planets that is a tenth of the evidence. It moves the numbers in the figures a little upward and changes nothing about where the evidence comes from.

Where the evidence lives #

The difference between the two predictions is worth drawing on its own, because it shows exactly which planets carry the evidence and so what precision is needed to see it.

The evidence is concentrated in a narrow band of ratios between about 0.6 and 1.7, and within that band it alternates: the mixture puts more planets just off a perfect match, at ratios near 0.9 and 1.15, and fewer exactly at one and in the shoulders beyond. Those are the planets with eccentricities around a tenth, the ones in and near the dip, seen through the cube of their orientation factor. Each lobe is a few hundredths wide in the logarithm of the ratio — five to ten per cent in the ratio itself.

An error larger than the lobes’ width averages them away. At a five per cent error on each ratio the pattern survives at three-quarters of its size; at fifteen per cent it is reduced by a factor of ten and smeared across its own width. The wings, where the two hypotheses agree, carry no evidence at any precision. A sample of strongly eccentric planets says nothing about the gap; only the planets near the gap do, and only if their densities are good enough to place them on one lobe or the next.

The transit, not the star #

The precision of a density ratio has two parts. One is the star’s own density, measured from its oscillations: the spacing between the frequencies of successive overtones is proportional to the square root of the mean density, and for a bright Sun-like star the comb of frequencies gives the density to two or three per cent. The other is the transit’s implied density, which comes from the duration and the shape of the ingress and egress — the four contact points — and depends on how well the impact parameter is determined. For a small planet the ingress lasts minutes and is buried in noise, the impact parameter is uncertain, and the implied density carries an error of ten or twenty per cent.

So the error on the ratio is set by the transit, and the sample needed grows elevenfold between the best transits and typical ones. The asteroseismic densities, which made the comparison possible in the first place, are no longer the limit; the planets’ own light curves are. That changes what a better sample means. More oscillating stars help only in proportion to the square root of their number. Better transits — more of them per planet, observed with larger telescopes or over longer baselines so that ingress and egress are resolved — help through the steep part of the curve.

It also says which planets to count. The planets with the best-determined implied densities are the large ones on central transits round bright stars. Restricting a sample to them buys precision at the cost of numbers, and on this curve the trade is favourable: a few hundred planets with five per cent ratios settle what a thousand with twenty per cent ratios cannot.

The planets that are more likely to transit #

There is a selection effect hidden in the sample, and it acts on exactly the ratios that matter. The chance that a planet transits at all is the star’s radius divided by the planet’s distance at the moment of conjunction, and an eccentric planet whose periastron lies towards the observer passes the star closer and faster than a circular one. Its transit probability is raised by the factor $(1 + e\sin\omega)/(1 - e^2)$, and the same orientation raises its implied density. Planets with ratios above one are over-represented in any transit sample, in proportion to how eccentric they are.

The effect is modest — for an eccentricity of 0.3 seen from the most favourable side the transit probability is about forty per cent higher — but it is not symmetric, and the evidence for a gap is a pattern of a few per cent in probability. A distribution of ratios that is not corrected for it is tilted towards high ratios, and the tilt is larger for the hotter hypothesis. Both hypotheses therefore have to be passed through the same selection before they are compared with a sample, and the comparison in the figures, which treats every orientation as equally likely to be observed, is the comparison after that correction has been made.

The correction is known exactly from geometry, which is what makes it harmless. Corrections that are not known exactly — the probability that a planet was detected at all, which depends on its size, period and the star’s noise, and the probability that its star was bright enough to oscillate detectably — select on properties correlated with eccentricity only weakly, and are handled the same way.

How cold the cold population is #

The mixture used so far puts its cold majority at a Rayleigh scale of 0.04, close to what the planets in multi-planet systems show. The answer depends strongly on that number, because the cold population’s width sets how sharp the core is and how deep the dip between it and the hot population.

At a cold scale of a tenth the two populations overlap so much that there is no dip left to find, and the mixture is itself very nearly a continuum. At two hundredths the cold planets are almost perfectly circular, their core in the ratios is as sharp as the measurement allows, and a few dozen planets suffice. The question “one population or two” therefore has no single answer independent of the populations: the more distinct the cold population is from the hot one, the easier the gap is to see, and a gap that is hard to see is a gap that is barely there.

The multi-planet systems are the natural reference. If the singles’ cold majority really is the same population as the planets with transiting siblings — the same systems, seen from an angle at which only one planet transits — its eccentricities should match theirs, and the left of the figure applies. That prediction can be tested independently, because the multis’ eccentricity scale is measured from the agreement of their own densities, with no reference to the singles at all.

How many are hot #

The other free number is the fraction of singles in the hot population.

The hardest case is near an even split. Two populations of comparable weight, a factor of six apart in scale, are what a spread of scales even in its logarithm most resembles: the continuum puts equal weight in every factor of two of scale, and a half-and-half mixture of two scales is the crudest version of that. A small hot minority is the easiest to distinguish. A continuum wide enough to make its wings puts too many planets in between, and one narrow enough to keep the core makes no wings. The published fits to the measured singles put a minority of them, of the order of a quarter, in the hot population, which is in the moderately hard part of the curve.

The asymmetry is a general property of population inference rather than of eccentricities. Two hypotheses about a population are easiest to tell apart where they make the most different claims about the commonest objects, and hardest where they disagree only about the rare ones or about the shape of a smooth distribution. The same structure appears when the spread of stellar spins about aligned planetary orbits is read from projected angles alone, and when a survey measures the prior it then imposes on its own weakest signals.

What a gap would mean #

A clean separation into two populations would say that the eccentric singles were made by an event, not a process. Planets in compact systems interact constantly, and their mutual gravity keeps their orbits nearly circular and nearly coplanar; a system that goes unstable has its planets collide, merge or scatter, and the survivors end on fewer, wider, more eccentric orbits. That is a discrete outcome. Systems either went through it or did not, and the survivors should form a distinct hot population whose number reflects how often compact systems break.

A continuum would point to processes that act by degrees. A distant companion — a giant planet or a star — drives slow cycles in a planet’s eccentricity and inclination whose amplitude depends on its distance and mass, and a population of such companions at a range of distances produces a range of eccentricities with no gap. Tides from the star then round the closest orbits over billions of years, by an amount that depends on period and age and blurs any gap that once existed. Both mechanisms certainly act on some planets; the question is which dominates among the singles.

The density test is not the only one. Eccentricities measured from the star’s reflex velocity are available for some planets, and transit timing variations constrain the eccentricities of planets in near-resonant pairs. Both give the eccentricity itself rather than a combination with the orientation, and both are available for far fewer, and differently selected, planets. And the classification of a planet as single is itself a statement about geometry and detection rather than about the system, so the singles are a mixture of genuinely lone planets and planets whose siblings are inclined or too small to see — a mixture that has to be modelled before the eccentricities can be.

What the calculation leaves out #

Both hypotheses here are idealised. The cold and hot populations are Rayleigh distributions, the continuum is a particular family, and real eccentricity distributions need not be any of these; a different continuum family would change the numbers somewhat and the conclusion — that the evidence lives in a narrow range of ratios near one — not at all. The density errors are Gaussian, known and the same for every planet, when in practice they are asymmetric, correlated with the impact parameter and different for each. Orientations are taken as uniform, which is right for single planets but not for the transiting members of systems with aligned periastra. And the eccentricities are capped at 0.9, ignoring the planets whose periastra would bring them into their stars. None of these affects the shape of the argument: the gap is a feature of a few hundred planets’ worth of evidence, located just off a perfect density match, and erased by the errors typical of small planets’ transits.

Still open: whether the cold singles are the multis seen from the side #

The cleanest version of the test does not compare a mixture with a continuum at all. It asks whether the cold singles have the same eccentricities as the planets in multi-planet systems — a prediction with no free parameters if the singles are mostly multis whose siblings do not transit. The multis’ eccentricity scale is measured from the agreement of their own densities; the singles’ cold scale comes from the core of their density ratios. If the two agree, most single planets are siblings seen from an unlucky angle, and the hot remainder is whatever is left. If the singles’ core is warmer, even the cold singles have been disturbed, and the line between a population and a continuum is not where the mixture model drew it. The space missions now recording oscillations for tens of thousands of bright planet hosts will give the stellar densities; whether the transits will be good enough to use them depends on how many hours of each planet’s ingress and egress can be gathered.