More iron than the star could give

The Earth’s iron core holds 32.5 per cent of its mass. That number comes from seismology and from the Earth’s moment of inertia, and it is one of the best-known facts about any planet. There is a second way to arrive at almost the same number that uses nothing about the Earth at all. Take the Sun’s composition, measured from its spectrum, and keep only the elements that condense into rock and metal at the temperatures of the inner solar system. Put all the iron into a core and all the magnesium and silicon into a rocky mantle of oxides. The core’s share of the mass comes out at 33.3 per cent.

That agreement is not a coincidence. The Earth was assembled from the solids of the disc around the young Sun, and those solids had the Sun’s proportions of the refractory elements, the ones that do not evaporate. If that is a general rule, a rocky planet’s composition is predictable from its star’s spectrum, and a planet’s mass and radius together are a test of the rule. Most small exoplanets pass the test within their uncertainties. A few fail it badly, in one direction. They are too dense, and the excess density implies iron cores of around seventy per cent of their mass — Mercury’s proportion, not the Earth’s. They are called super-Mercuries, and the question they pose is where the rock went.

A core fraction from a mass and a radius #

A mass and a radius alone do not fix a rocky planet’s composition, because one density is consistent with many interiors: more iron and more water can trade against each other at constant size. For a planet without water or a substantial atmosphere the trade disappears, and the mass and radius fix a single number, the core mass fraction. Interior models built from laboratory equations of state for iron and silicate rock, compressed to the pressures inside planets of several Earth masses, give a compact relation:

$$\frac{R}{R_\oplus} = (1.07 - 0.21\,{\rm CMF})\left(\frac{M}{M_\oplus}\right)^{1/3.7},$$

valid from about one to eight Earth masses. The exponent is less than one-third because a heavier planet compresses its own interior, so radius grows more slowly than mass to the one-third power. The Earth, with a core fraction of 0.325 and one Earth mass, lands at one Earth radius.

The curves are close together. Going from no core at all to a pure iron planet changes the radius by about forty per cent at fixed mass, and the interesting range — between the Earth’s proportions and Mercury’s — is less than ten per cent of the radius. The difference is real and measurable, but only if the radius is known to a few per cent, and the radius of a transiting planet is the radius of its star times a measured ratio. Everything that follows depends on how well the star is known.

What a star’s spectrum predicts #

The stellar side of the comparison is a piece of chemistry with very few assumptions. A star’s photosphere preserves the composition of the cloud it formed from, apart from the lightest elements, and that composition is measured from the strengths of its absorption lines. The refractory elements that make rocky planets — iron, magnesium, silicon, with smaller contributions from nickel, calcium and aluminium — all condense from the disc’s gas at temperatures above about 1,300 kelvin, so in the inner disc, where rocky planets form, they are all solid and in the star’s proportions.

A planet assembled from those solids has, to a first approximation, the star’s ratios of iron to magnesium to silicon. If all its iron sinks to the core, which is what happens in a molten planet, and its magnesium and silicon form the oxide minerals of the mantle, its core mass fraction is set by those ratios:

$${\rm CMF}_\star = \frac{55.85\,{\rm Fe}}{55.85\,{\rm Fe} + 40.30\,{\rm Mg} + 60.08\,{\rm Si}},$$

with the elements counted by number and weighted by the masses of iron, magnesium oxide and silicon dioxide. For the Sun’s proportions it is 0.333. Some iron in a real planet stays in the mantle as iron oxide, and some light elements dissolve into the core; the two corrections run in opposite directions and are a few hundredths each.

The stars that host planets do not vary much in these ratios. Their overall metallicities range over a factor of ten, but the ratio of iron to magnesium among them varies by only about forty per cent either way, because both elements are made in the same kinds of supernovae. The core fractions they predict for their planets therefore cluster tightly, between about 0.27 and 0.40. The figure shows what a core fraction of seventy per cent would require: a star with nearly five times the Sun’s iron for its magnesium. No such star is known among planet hosts. A planet with a 0.7 core did not get it by keeping its star’s proportions.

How sure a measurement can be #

Before asking where the iron came from, the measurement has to be good enough to say that it is there. A core fraction inferred from a mass and a radius inherits their uncertainties, and the radius enters with nearly four times the weight of the mass, because a few per cent in radius is a large change in volume.

With a radius measured to two per cent and a mass to ten, a planet with a 0.7 core is separated from the stellar allowance, but not by much: the inferred core fraction ranges over half the possible scale. A more demanding question is how well each quantity must be known to make the separation decisive.

Three standard deviations of separation need the mass to better than about seven per cent and the radius to one or two per cent. Such numbers have been reached for only a few dozen small planets. Masses at that level take hundreds of radial-velocity measurements with the most stable spectrographs, against stars whose own surface activity mimics a planet’s signal at the same level. Radii at that level need the star’s radius to one or two per cent, which in turn needs an accurate distance, an accurate temperature and a model of the star’s atmosphere. The super-Mercury question has been argued at the edge of what the measurements can support, and several planets once proposed as super-Mercuries have moved back towards Earth-like composition as their masses were remeasured.

The population that survives remeasurement is not large, but it is not empty. A few planets of a few Earth masses, orbiting within days of their stars, have core fractions measured at 0.6 to 0.8 with uncertainties that exclude the stellar band. Analyses comparing the planets’ compositions with their own host stars’ abundances, star by star, find that most small planets follow their stars, with a scatter consistent with the measurement errors, and that a separate group lies well above — dense planets in a cluster of their own rather than the tail of one distribution.

Where the rock went #

There are two ways to make a planet with more iron than its star could have given it, and they place the separation of rock from metal at opposite ends of the planet’s history.

The first is to make a normal planet and then remove its rock. A collision between two planets late in the formation of a system, a giant impact, can strip a large fraction of the target’s mantle, since the rock is on the outside and the iron core is shielded beneath it. Mercury is the standard example: its core is enormous, about seventy per cent of its mass, and simulations of a grazing impact on a larger proto-Mercury can remove enough mantle to explain it. The Moon, which the Earth could not have captured, is thought to have formed in a giant impact on it, and impacts of that scale are expected in the final phase of every rocky system’s assembly.

The arithmetic is unforgiving. To go from a third to seven-tenths, a planet has to lose four-fifths of its mantle, more than half its original mass, without losing its core. That is a collision of a very particular kind: energetic enough to remove most of the rock, gentle enough not to disrupt the planet, and followed by the stripped rock not falling back. For a planet in a short-period orbit, close to its star, the stripped material might be lost to the star’s gravity or ground away in further collisions rather than re-accreted. For Mercury itself the rock would have had to escape the inner solar system rather than return.

The second way is to separate metal from rock before the planet forms. In the hottest inner parts of a protoplanetary disc, the condensation temperatures of iron and of the magnesium silicates differ by a hundred kelvin or so, so a region exists where iron is solid and the silicates are partly vapour. Solids forming there are iron-rich. Iron grains are also denser and conduct heat differently, and processes in the disc that sort grains by density or by their response to the star’s radiation can concentrate them further. A planet assembled in such a region would be born with a large core. That route places super-Mercuries naturally at short orbital periods, where they are found. It also predicts that their iron enrichment should correlate with how hot the disc was where they formed, and not with any history of collisions.

The two routes leave different traces. A planet stripped by an impact should have lost its volatile elements along with its mantle and may have debris around it; one formed from iron-rich solids should have the composition of its formation zone throughout. Neither trace is observable in any super-Mercury yet. What can be compared is the population: a planet that formed iron-rich in a hot zone should have company in neighbouring orbits that is also iron-rich, while a planet stripped in an impact can sit beside planets with normal cores. The few multi-planet systems that contain a super-Mercury suggest that its neighbours are often ordinary, which favours collisions, though the sample is small.

The one that can be visited #

Mercury is the only super-Mercury whose surface has been examined, and what the MESSENGER spacecraft found there complicates both stories. Its core is as large as expected, a molten outer core and a solid inner one together making up most of the planet’s radius. Its surface, though, is unexpectedly rich in volatile elements — sulphur, sodium, potassium — at abundances comparable to the other rocky planets’. A giant impact energetic enough to strip four-fifths of a mantle would be expected to heat the remaining rock enough to drive those elements off, and formation from solids in the hottest part of the disc would be expected to start without them. Neither expectation is met.

Several resolutions have been proposed. The stripped rock may have been replaced by later accretion of cooler material; the impact may have been a sequence of smaller hit-and-run collisions, each less heating than one large one; or the iron enrichment may have happened in the disc by a process that sorts grains by density without heating them, such as the pressure of the young Sun’s light on dark, conductive iron grains. Each resolution has a cost elsewhere, and none is established. Mercury’s orbit has been measured to the arcsecond for centuries; its origin is less settled than that of planets around other stars that are known only as a mass and a radius, because only Mercury has supplied the extra constraint that rules out the simple answers.

For the exoplanetary super-Mercuries the volatile test will not be available for a long time. What is available is the comparison of their compositions with their stars’, which is a test of the starting material. It says that whatever produced these planets did not start with iron-rich stars, since there are none. The iron was concentrated by something that happened in the disc or after, and the solar-system example shows that the something is not yet understood even where it can be examined in detail. A planet’s bulk composition records where and when it formed only once the processes that can alter it are known.

What the stars say about the planets that follow them #

The other half of the result — that most small planets do match their stars — is the quieter discovery and the more important one. It means a rocky planet’s bulk composition is, to first order, predictable from its star’s spectrum, before the planet’s mass is ever measured. That is a powerful prior. An interior model of a small planet with a known mass and radius can be constrained further by requiring its iron-to-rock ratio to be its star’s, which breaks part of the degeneracy between iron and water that a mass and radius alone leave open.

It also says something about planet formation that the solar system alone could not. The Earth’s agreement with the Sun might have been a coincidence, or a property of one particular disc. Agreement across dozens of systems, for stars with a range of compositions, says that the disc’s refractory solids reach their planets without being sorted much on the way, except in the special cases where a planet ends up with too much iron. Those special cases are rare, and their rarity is itself a measurement: separating rock from metal on the scale of a whole planet is something planetary systems do occasionally, not as a rule.

The metallicity of a star has long been known to matter for whether it hosts giant planets, which need enough solid material to build a core before the gas disperses. For small planets the overall metallicity matters much less, and what matters is the proportions. The proportions are almost the same in every star, which is why most rocky planets are so nearly alike in composition.

Where water fits #

The comparison has assumed planets without water. Planets formed beyond the snow line, where water condenses, can carry tens of per cent of their mass in ice, and a water layer makes a planet larger at fixed mass, lowering its apparent core fraction. A water-rich planet with an iron-rich core could therefore masquerade as an Earth-like one. Such a planet would be counted as normal, so water makes super-Mercuries harder to find, not easier, and the planets identified as super-Mercuries are, if anything, more iron-rich than their apparent core fractions suggest.

The reverse confusion is possible for planets near the boundary of the radius valley, where a thin hydrogen envelope inflates the radius of a planet that is otherwise rock and metal. A planet with even a fraction of a per cent of its mass in hydrogen can appear several per cent larger, and its core fraction is then underestimated. The super-Mercuries are found at short periods where such envelopes are stripped by the star’s radiation, which removes that confusion for them while leaving it for their wider-orbit cousins.

What the figures assume #

The interior relation is a fit to models of two-layer planets — an iron core and a silicate mantle — at Earth-like temperatures, and it is valid only from about one to eight Earth masses; outside that range, and for planets with molten interiors or large amounts of light elements in the core, it would need replacing. The stellar prediction keeps only iron, magnesium and silicon and assumes all the iron is in the core; including nickel raises the predicted core fraction by a couple of hundredths, and iron left in the mantle lowers it by a similar amount. The measurement figure assumes Gaussian, independent errors in mass and radius, while real errors are correlated through the star’s own parameters. The stripping figure assumes the core survives intact and the stripped mantle is lost for good, the most favourable case for the impact route.

Still open: whether the dense planets are one population or two routes #

The super-Mercuries are few, and their core fractions are measured near the precision limit, so it is not settled whether they form a separate population at all or the dense tail of a broad distribution blurred by measurement errors. The analyses that find a distinct group compare each planet with its own star’s abundances. That comparison removes the star-to-star variation in the prediction and leaves the planets’ scatter from their stars, which is what a separate population would show as a second peak. With a few dozen planets the second peak is suggestive rather than secure.

If the group is real, the question of how it formed is open in a measurable way. Collisions predict that super-Mercuries should be scattered among normal neighbours and have no preference for particular stellar compositions. Formation from iron-rich solids in a hot inner disc predicts a correlation with the star’s luminosity during its early life, which set where in the disc the iron-rich zone lay, and with the planet’s orbital distance. Masses and radii precise enough to test either prediction across a hundred planets are the target of the next generation of radial-velocity instruments and of space photometry that can measure stellar radii to a per cent. Whether the rock went after the planets formed or before is a question the iron in the stars can now be asked to answer.