A second peak the errors have to let through

A rocky planet is usually made of what its star was made of: iron, magnesium and silicon in the star’s proportions, which for most stars predict an iron core of about a third of the planet’s mass. A small number of planets have been measured well above that — core fractions of 0.6, 0.7, sometimes more — and the analyses that compared each planet with its own host star found that those planets did not look like the upper edge of the ordinary population. They looked like a separate cluster.

Whether they are is the question this essay is about, and it is worth being exact about what kind of question it is. It is not whether any individual planet is iron-rich; for a few planets with very precise masses and radii that is established. It is whether the dense planets form a distinct population, with its own peak in the distribution of core fractions, or whether they are the tail of one broad distribution that happens to have a long upper side. The two possibilities imply different histories — a separate route to forming planets, against one route with a wide spread of outcomes — and they are distinguished by the shape of a histogram rather than by the properties of any one point in it.

The error is as wide as the gap #

The first figure is the whole difficulty. The two populations are separated by about a third in core fraction. A planet’s core fraction is inferred from its mass and radius, and the uncertainty in that inference — set mostly by the radius, which enters almost four times as strongly as the mass — is typically a tenth or more for the planets in question. A measurement error a third of the gap is not small: it spreads each population into a curve whose width is comparable to the distance between them.

When two curves of that width overlap, the dip between them fills in. What is left is a single peak with an asymmetric upper side, and an asymmetric upper side is also what a single population produces if it has a skewed spread of compositions — if, say, most planets match their stars closely and a minority deviate upward by varying amounts. The data, at that precision, contain the information to tell these apart only statistically, by the precise shape of the tail, and a precise shape needs many points.

This is a general property of mixtures, and the same arithmetic governs every count of planets inferred from a survey’s detections rather than read off directly. A second component is visible by eye — as a separate hump — only when the separation between components exceeds about twice their combined widths. Below that, it has to be found by fitting, and the fit’s power to find it falls steeply as the widths grow.

A count, not a measurement #

Treating the question statistically turns it into a number: how many planets, measured to a given precision, are needed before a two-population model is preferred decisively over the best single population.

The calculation is a standard one. Suppose the truth is the two-population mixture. Each measured planet then contributes, on average, a small amount to the log-likelihood difference between the true mixture and the best single Gaussian that can be fitted to the same data — the Kullback–Leibler divergence between the two distributions. A sample of $N$ planets accumulates $N$ times that, and the two-population model has three more free parameters than the single one (the second peak’s position, width and share), so the margin it needs for a decisive preference is about seven in log-likelihood. The required sample is seven divided by the divergence per planet.

The curve rises steeply. Between an error of $\pm 0.05$ and $\pm 0.15$ — a factor of three in precision — the number of planets needed rises by more than a factor of fifteen. The divergence between two broad, overlapping curves collapses as the overlap grows, and at the precisions of current samples each halving of the error is worth about an order of magnitude in sample size. A survey that wants to answer the question has a choice between a few dozen planets measured very well and several hundred measured as well as current samples are, and the first is cheaper.

The published samples sit in the shaded box, below the line at their precision. On these assumptions they hold about a third as many planets as a decisive answer needs. That is consistent with what the analyses report: a dense group that is favoured, sometimes strongly, in particular samples, and not yet established in the sense of surviving any reasonable choice of sample and model.

Why comparing with the star helps, and by how much #

The comparison of small planets with their own host stars established that a planet’s composition follows its star’s, which turns each star’s spectrum into a prediction for its planets. Using that prediction here has a direct statistical effect.

If every star is assumed to have the Sun’s proportions, the main population’s width includes the real star-to-star variation in iron-to-rock ratio, which in core fraction is about 0.05 — wider than the planets’ own scatter about their stars. Normalising each planet by its own star’s prediction removes that variation from the main peak, narrowing it, and the narrower peak leaves more contrast for a second one. At the best precisions the improvement is a factor of two in sample size. At the precisions of current samples, where the measurement error dominates every width, it is a third.

The effect is largest where it is least needed. When measurement errors are small, the star’s own contribution is a large part of the main peak’s width, and removing it matters. When errors are large, the star’s contribution is a small part of a big number. So comparing planets with their stars is essential to the precise programmes of the coming decade and only modestly helpful to the samples available now.

There is a second, less statistical benefit. A planet whose core fraction is high relative to the Sun’s proportions might simply orbit an iron-rich star. A planet whose core fraction is high relative to its own star’s has been enriched in iron relative to its birth material, and that enrichment is what needs explaining. Without the star, a dense planet is a fact about the planet; with it, it is a fact about what happened to the planet.

Rare groups and common ones #

The cost depends also on how large the dense group is, and not monotonically.

The left side of the curve is intuitive: a group holding three per cent of planets contributes only three planets in every hundred, and a fit cannot find a peak defined by three points. The right side is less so. A group holding forty per cent of planets is not a minority with its own peak; it is half of a bimodal distribution whose two halves blur into one broad hump, and the fit has to distinguish “two equal peaks” from “one wide peak” — which, at the same errors, is about as hard as distinguishing “a small peak” from “a tail”.

That the cheapest share is near a fifth means the question is easiest to answer for a group of exactly the size first reported. It also means the answer is sensitive to the selection of the sample. A sample drawn from short-period planets, where stripping by the star could raise core fractions, might hold a larger share of dense planets than the population as a whole; a sample drawn to have precise masses might favour heavier, brighter-star systems. The share is a property of the sample, and the number needed is a property of the share.

The harder alternative #

All of the counts above compare a two-population mixture with the best symmetric single population. That is the weakest alternative, and it flatters the mixture.

A single population with a long upper tail is physically reasonable. Planet formation has many sources of scatter — the local chemistry of the disc where a planet formed, the order in which its building blocks arrived, collisions with neighbours — and most of them could push a planet’s iron fraction up by varying amounts without producing a distinct group. Such a population would have a skewed distribution, with most planets near their stars and a long tail of increasingly iron-rich ones.

Distinguishing that from two populations is the real test, and it is harder because the skewed population already puts planets where the second peak would be. What separates them is the shape in between — whether the density of planets dips between the main peak and the dense group, or declines smoothly — and the dip is exactly what measurement errors fill in. On these assumptions it costs three times as many planets as the symmetric comparison.

What a sample of forty can say #

This is roughly the situation of the published analyses, and the figure shows two things they share. The fitted group is too large. With few dense planets in the sample, the fit absorbs part of the main population’s upper side into the second component, so its share comes out larger and its centre lower than the truth. The improvement is larger than the expectation. The expected margin at this size and precision is under two; this draw gave almost four, because a fit with three extra parameters always finds some improvement in a finite sample even when it has nothing real to find. Both are the ordinary behaviour of mixture fits on small samples, and both push toward finding a group.

The same model with 150 planets measured to $\pm 0.08$ is unambiguous.

A valley that the errors had hidden #

There is a precedent for exactly this situation in the same field, and it ended in the structure being found.

The radii of small planets from the Kepler mission were for years quoted with uncertainties of twenty-five per cent or more, because the stars’ radii came from colours and catalogues rather than spectra. The distribution of planet radii looked like one broad hump of planets between one and four Earth radii. In 2017 the California-Kepler Survey measured the host stars spectroscopically, bringing the planet radii to about ten per cent, and a gap appeared in the histogram — a valley near 1.8 Earth radii separating rocky planets from planets with hydrogen envelopes. Distances from the Gaia satellite brought the radii to five per cent, and the valley deepened.

The valley had been in the population all along. It was invisible in the earlier data for the reason the dense group may be invisible now: the measurement errors were comparable to the width of the gap, and they filled it in. When the errors shrank below about half the gap, it became visible without any statistical argument at all.

That precedent cuts both ways. It shows that a population structure hidden by errors can be revealed by better measurement of the stars, which is what the composition question needs. It also shows that the structure, once found, was found by looking at the histogram — and that the early statistical arguments for and against a gap, made from the imprecise radii, had been inconclusive in just the way the forty-planet sample above is inconclusive. A suggestive mixture fit on a few dozen points is where the radius valley was in 2015.

A companion that makes a planet look like iron #

A dense group can also be manufactured, and one route to manufacturing it runs through a systematic error that measurements do not average away.

When a star has an unresolved companion, the companion’s light dilutes the transit, and the planet’s radius is measured too small. A planet measured too small at the right mass is measured too dense, and a planet measured too dense is assigned too high a core fraction. A ten per cent underestimate of the radius raises the inferred density by a third — enough to move an Earth-like planet into the range of the dense group.

The error runs one way only. It cannot produce a deficit of dense planets; it can only add spurious ones, scattered upward from the main population by amounts set by each companion’s brightness. That is a tail, not a peak, but a small sample cannot distinguish a tail from a peak, and a sample whose dense members include a few diluted planets will fit a dense group more readily than the truth warrants. The defences are high-resolution imaging of every host star to find companions, and the radial-velocity data themselves, which show a second star’s lines if it is bright enough to matter.

What the individual planets say #

The population argument does not make individual iron-rich planets less real. A planet whose mass is measured to three per cent and radius to one and a half, against a star whose composition is measured, has a core fraction known to about $\pm 0.05$, and for a few planets that number lies well above anything their stars could supply.

The best-measured cases are informative precisely because they are individual. Kepler-107 c is about twice as dense as its inner neighbour Kepler-107 b, although the two are similar in size and orbit the same star; the inner planet should, if anything, be the denser one, since it is the more exposed to the star’s radiation. The most natural explanation is a giant impact that stripped the outer planet’s mantle. GJ 367 b, an Earth-sized planet on an eight-hour orbit, is measured as mostly iron. K2-229 b has a density that implies a core fraction near that of Mercury. Each of these has a plausible individual history, and none of them requires a second population.

The population question is not whether such planets exist but whether they cluster. If collisions produce them, the outcome of each collision depends on its speed, angle and mass ratio, and the resulting core fractions should spread smoothly upward from the stellar value — a tail. If they form from iron-rich solids in the hottest inner part of the disc, where silicates evaporate and iron condenses first, they should share a characteristic composition set by that chemistry — a peak. The shape of the histogram is the test between two formation stories, which is why it matters whether the second peak is there.

Why the radius dominates the error #

The precision that sets everything above is the precision of the core fraction, and it is dominated by the radius. That is worth explaining, because it determines where the effort should go.

A planet’s density goes as its mass over the cube of its radius — a mass from the star’s wobble, a radius from its shadow — and the core fraction is a steep function of density at fixed mass. A one per cent error in radius is a three per cent error in density; a one per cent error in mass is a one per cent error. So a planet whose mass is known to five per cent and radius to three has a density uncertainty dominated by the radius, and improving the mass alone does little.

The planet’s radius is measured relative to its star’s: the transit depth gives the ratio, and the star’s radius gives the scale. A star’s radius known to two per cent — from its distance, its brightness and its temperature — puts a floor of two per cent under the planet’s radius however deep and well-sampled the transit. The precision programmes that would answer the population question are therefore as much about stars as about planets: distances from astrometry, temperatures from interferometric angular diameters, and models of stellar atmospheres good enough to turn a colour into a temperature to a per cent.

The mass has its own floor, set by the star’s surface: activity that mimics a planet’s signal in radial-velocity data at the level of a metre per second, which is the signal of a few Earth masses on a short orbit. For the planets in question the mass precision is achievable with enough observations; the radius precision depends on knowledge of the star that improves more slowly.

What the figures assume #

The populations are Gaussian. Both the main peak and the dense group are drawn as Gaussians in core fraction, and the alternative as a Gaussian or a log-normal. Real formation channels need not produce any of these shapes, and a different pair of shapes would change every count. The counts measure how hard the problem is for shapes of this kind, not for the shapes nature uses.

The errors are equal and known. Every planet is given the same measurement error, and the fits treat the error as known. Real samples mix planets measured to $\pm 0.05$ with planets measured to $\pm 0.2$, and the error for each is itself estimated; a hierarchical analysis that propagates each planet’s own error is the correct method and generally needs fewer planets than the equal-error case for the best-measured subsample.

Core fraction is the only property. The analyses that suggest a dense group also use orbital period, planet mass and stellar composition, and a group defined by several properties at once is easier to establish than one defined by core fraction alone.

The sample is unselected. Every simulated planet is drawn from the population; real samples are drawn from planets bright enough, close enough and precisely enough measured to be included, and the selection can create or hide structure.

Still open: the hundred planets that would decide it #

The question is set up to be answered. The radial-velocity spectrographs now operating at the ten-centimetre-per-second level, combined with transit photometry of nearby bright stars, will produce masses to a few per cent for well over a hundred small planets in the coming decade; astrometry and interferometry will put their stars’ radii at a per cent; and spectroscopy of the same stars will give each one’s iron, magnesium and silicon. By the second figure, that sample is past the point where a dense group of a tenth of the planets, if it exists, cannot be missed.

If it does exist, the next question is what defines its members — orbital period, stellar luminosity, the presence of neighbours — and each answer points to a formation route. If it does not, the dense planets are a tail produced by many routes, and the interesting quantity becomes the tail’s shape: how often does a planet end up with twice its star’s iron, and how often with three times. The one-density degeneracy that has limited every composition argument will not go away, but with a hundred planets compared with their stars it stops being the limit on this one.