A threshold that makes iron matter only to giants

The first stars found to have planets were unusual in a way nobody had predicted. The Sun-like star 51 Pegasi, the first of them, has about one and a half times the Sun’s iron. So did most of the handful found in the next two years, and in 1997 it was pointed out that as a group they were metal-rich — richer than the average star in the solar neighbourhood by something like a factor of two. With a dozen stars the statement was suggestive. With a thousand it became the most secure correlation in the study of exoplanets: in a radial-velocity survey of 1,040 stars published in 2005, the fraction with a giant planet rose from 3 per cent at the Sun’s metallicity to more than 20 per cent at three times it, and fell below 1 per cent at a third of it.

That relation is steep, and it is not shared. The small planets the transit surveys later found in their thousands — the super-Earths and sub-Neptunes that are the commonest planets in the galaxy — depend on their star’s metals weakly or not at all. An occurrence rate is a count divided by a probability, and when both halves are done carefully for both kinds of planet, the answer is two laws of different slope on the same axis.

This essay is about why. The usual reading is that giant planets and small ones form differently and that only the giants need the metals. The argument here is that a single mechanism, with a threshold in it, produces both slopes — and that which slope a population shows is decided less by what the planets are than by how rare they are.

Metallicity, and what a slope of two means #

The iron abundance of a star is written

$$[\mathrm{Fe/H}] = \log_{10}\!\left(\frac{N_\mathrm{Fe}}{N_\mathrm{H}}\right)_{\star} - \log_{10}\!\left(\frac{N_\mathrm{Fe}}{N_\mathrm{H}}\right)_{\odot},$$

so zero is the Sun, $+0.3$ is twice its iron per hydrogen atom and $-0.5$ is a third. It is measured from the strengths of iron absorption lines in the star’s spectrum, a quantity read from what the atmosphere removes, and for Sun-like stars it is good to about 0.05. Iron stands in for all the heavy elements, which in the thin disc of the Galaxy rise and fall roughly together.

A slope of two means occurrence $\propto 10^{2[\mathrm{Fe/H}]} = (N_\mathrm{Fe}/N_\mathrm{H})^2$: the probability of a giant planet goes as the square of the star’s metal content. Doubling the iron quadruples the chance. That is not a small modulation of a rate that is otherwise set by something else. Across the metallicities present among nearby Sun-like stars, it is the largest single factor in whether a giant planet exists at all.

The measurement had to rule out a selection effect first, because a radial-velocity survey is more sensitive to some stars than others. A metal-rich star has more and deeper absorption lines, and more lines mean a more precise velocity, so a metal-rich star’s planets are slightly easier to find. The survey that measured the slope checked this directly — its detection threshold was nearly the same for metal-poor and metal-rich stars at the planet masses in question — and the effect is far too small to produce a factor of ten. Transit surveys, which measure a radius rather than a velocity, see the same steep dependence for their giant planets, through an entirely different selection.

A histogram that moved before a rate was measured #

The correlation was noticed long before it could be measured, and the form it was noticed in is worth understanding, because it is a different quantity from the rate.

The early statement was about hosts: the stars with planets had a metallicity distribution centred higher than the stars without. That is what a rate rising with metallicity does to a population, and for a Gaussian population it does it exactly. If nearby stars have metallicities distributed normally with width $\sigma$, and each has a planet with probability proportional to $10^{\beta[\mathrm{Fe/H}]}$, the hosts are distributed normally with the same width and a mean shifted by

$$\Delta = \beta\,\ln 10\;\sigma^2.$$

With $\sigma = 0.22$ and $\beta = 2$, that is 0.22 — the hosts of giant planets average $+0.17$ against the field’s $-0.05$, which is what the first few dozen hosts showed. A small planet’s slope of a quarter or a half moves the host distribution by a few hundredths, which is inside the error on a single star’s metallicity. So the fact that the transit surveys’ small-planet hosts looked like ordinary stars was, at first, taken as evidence that small planets do not care about metals, when a shallow slope and no slope at all give host distributions that cannot be told apart without thousands of stars.

The shifted histogram says only that the rate rises. Turning it into a rate needs the stars that were searched and have no planet, and a completeness correction for each — the denominator that holds everything difficult about an occurrence rate.

Born rich, or made rich #

Two explanations were put forward within a year of the correlation’s discovery, and they predict opposite things about stars that are otherwise alike.

The first is that the metals came first: a star forms from the same cloud as its disc, a metal-rich cloud makes a disc with more solid material, and more solids make giant planets easier to build. The second is that the planets came first: a star that formed giant planets had some of them, or the rocky debris they scattered, fall into it, and swallowed rock enriched its surface in iron. The star would be metal-rich because it had planets.

The two can be separated by where the swallowed iron would end up. A Sun-like star’s surface is the top of a convection zone that mixes it with the outer part of the star, and the depth of that zone depends on the star’s mass. A star of 1.3 solar masses has a convection zone containing a thousandth of a solar mass; the Sun’s contains a fortieth; a subgiant that has begun to swell after leaving the main sequence has deepened its zone to a large fraction of the star. The same few Earth masses of swallowed rock would raise the surface iron of the first star by a large factor, the Sun’s by a few per cent, and the subgiant’s by almost nothing.

The correlation shows none of that. Planet hosts with thin convection zones are no more metal-rich than those with thick ones, and subgiants hosting giant planets are as metal-rich as main-sequence hosts, which after their convection zones deepened they could not be if the enrichment were a surface veneer. The metals are in the whole star, which means they were in the cloud it formed from. Pollution happens — some stars show the chemical fingerprints of swallowed rock, and a quarter or more of white dwarfs show it unmistakably, their atmospheres carrying the metals of rock that fell in within the last few days to millions of years — but it is not what makes planet hosts metal-rich.

A core racing its own disc #

The accepted mechanism is core accretion. A giant planet begins as a solid core, built from the dust and ice in the disc by collisions, and when the core reaches roughly ten Earth masses its gravity can hold the surrounding gas faster than the gas can escape, and it gains a massive envelope in a runaway. The solids set how fast the core grows; the gas sets the deadline, because the gas disc lasts only a few million years before the star’s radiation and the disc’s own accretion remove it. Beyond the line where ice condenses there are more solids to build with, which is why giant planets form beyond it; the metallicity multiplies the solids everywhere.

The figure is a deliberately minimal version of that race. A core’s growth time is taken to be inversely proportional to the mass of solids in its disc — twice the solids, half the time — which is the simplest dependence and the weakest plausible one. Disc masses around stars of the same mass are observed to spread over more than an order of magnitude, and are drawn as a lognormal with a width of 0.4 dex. The gas disc’s lifetime is exponential with a mean of three million years, which is what infrared surveys of young clusters show: about half the stars in a three-million-year-old cluster still have discs, and almost none in a ten-million-year-old one. The one free choice is the growth time for a median disc at solar metallicity, set at thirty million years — ten times the typical disc’s life — so that giant planets are rare, as they are.

The result is that a factor of five in metallicity, from $-0.4$ to $+0.3$, moves the growth-time distribution by a factor of five and moves the probability of winning the race by a factor of thirty. The reason is visible in the figure. Only the fast tail of each distribution overlaps the surviving discs; the bulk of every curve is far to the right of the deadline. Sliding a distribution to earlier times changes its tail by far more than it changes its bulk, and it is only the tail that makes giant planets.

Steep when winning is rare, flat when it is common #

The same calculation can be run with the race made easier, and that is where the two slopes come from.

With a median growth time of thirty million years, winning is a 2-per-cent outcome and its slope against metallicity comes out at 2.1 — close to the measured slope for giant planets, from a dependence on metals that is only linear. With three million years, winning is a 38-per-cent outcome and the slope is 0.7. With three hundred thousand years, almost every disc wins and the slope is 0.1. Nothing changed but how hard the target was.

The general statement is about thresholds on broad distributions. The logarithmic slope of the fraction of a lognormal population beyond a fixed line grows the further into the tail the line sits: near the median, moving the population by a small amount moves the fraction by about the same amount, while far into the tail it moves it by a large multiple. A rare outcome of a threshold process is extremely sensitive to anything that shifts the population; a common one is insensitive to it. The same arithmetic is why the frequency of extreme floods is far more sensitive to a change in mean rainfall than the frequency of ordinary ones.

So the steep law for giants and the flat law for small planets need not be two mechanisms. A small planet is an outcome most discs achieve — a few Earth masses of rock is what is left over in almost any disc — and it sits on the flat side. A giant planet requires a core to finish before the gas is gone, which most discs fail, and it sits on the steep side. Sub-Neptunes, which need to keep a few per cent of their mass in hydrogen and so need a core to have grown while some gas remained, are drawn at a slope between the two, and published fits put them there. A planet of the right size and composition where one cannot have formed is the complementary puzzle — a planet that won a race it should have lost, and the rarity of those is the same tail read from the other end.

The toy model does not prove this is what happens, and its limits are stated below. What it shows is that no special physics is needed to produce a slope of two, and that the contrast between the giants and the small planets is the contrast any threshold would produce.

The star’s mass enters the same way #

If the solids are what matter, the metallicity is not the only thing that sets them. A more massive star has, on average, a more massive disc; disc masses measured at millimetre wavelengths scale roughly in proportion to the star’s mass. The prediction is that giant-planet occurrence should rise with stellar mass as well as with metallicity, and in the same currency.

It does. A survey that included M dwarfs at the low end and evolved stars of up to two solar masses at the high end — stars that were A-type on the main sequence, observed after they had cooled into giants with lines narrow enough for precise velocities — fitted the occurrence as a product: proportional to the stellar mass and to $10^{1.2[\mathrm{Fe/H}]}$. The contours in the figure are that fit. A metal-rich M dwarf has about the same chance of a giant planet as a metal-poor Sun, which is the trade the solid budget predicts: halve the disc and double its metals, and the solids are unchanged.

The slope on metallicity in the joint fit, 1.2, is shallower than the 2 of the single-variable fit, because some of the metallicity dependence in the Sun-like sample was really a mass dependence — metal-rich stars in a sample selected by colour are slightly more massive. That is a caution about every slope quoted here: each is a fit of a power law to a finite sample, and the power law is a description of the data rather than a law of nature.

The M dwarfs are where the two sides of the threshold are clearest. They host giant planets rarely — less than half as often as Sun-like stars — and small planets more often than Sun-like stars do, in the stars that make up most of the Galaxy. A smaller disc makes a hard outcome harder and leaves an easy one easy.

A rate set by the Galaxy’s own history #

The metallicity of a star is not a property it chose. It is the metallicity of the gas it formed from, and that was set by how many generations of stars had enriched that gas before it — a record the Galaxy writes as it ages. The old stars of the thick disc and the halo are metal-poor, typically $[\mathrm{Fe/H}]$ of $-0.5$ to $-2$, and by the steep law they should have almost no giant planets; the searches that have looked find very few. The inner Galaxy is more metal-rich than the solar neighbourhood, and the gradient outward is one the oldest stars have partly walked away from, carried by radial migration from where they were born.

The surprising consequence is that the frequency of giant planets is a function of cosmic time. In the first few billion years of the Galaxy, before the interstellar medium had reached anything like the Sun’s metallicity, giant planets would have been rare, and a planetary system with a Jupiter was a late development of chemical evolution. Small planets, by the flat law, would have been forming all along. Whether any giant planets formed around the most metal-poor stars at all is one of the open tests of the threshold picture: a race that is essentially never won at $[\mathrm{Fe/H}] = -1$ would leave that population with none.

What the figures do not establish #

The measured laws are fits of power laws to samples of a few hundred to a few thousand stars, with the slopes for small planets less certain than the figure’s clean lines suggest; the published values for them depend on orbital period — close-in super-Earths show more metallicity dependence than those on wider orbits — and are drawn here as representative values rather than a single fit.

The race is a model with one tuned number. Its growth time for a median disc was chosen to make giant planets rare, and the slope of 2.1 it produces follows from that choice together with the assumed spread of disc masses; a narrower spread would give a steeper slope for the same rarity, and a growth time that depended more than linearly on the solids would give a steeper slope still. What is robust is the ordering — rare outcomes steep, common ones flat — and not the particular numbers. Real core growth depends on the solids’ size distribution, on where in the disc the core forms, on pebbles drifting inward from further out, and on whether the core migrates while it grows; none of that is in the figure.

And metallicity is not the only variable that could matter. The ratios of carbon, oxygen and magnesium to iron vary between stars at the same iron abundance, and it is the refractory and icy solids, not the iron itself, that build cores. Iron is what is measured most easily.

Still open: whether the threshold is one threshold #

The steep law for giant planets and the shallow law for small ones are both measured, and a single race with a single deadline reproduces both from the rarity of the outcome alone. What would test it is the middle: planets of Neptune’s mass and Saturn’s, which by this reading should show slopes that rise smoothly with how rarely they form, and the most metal-poor stars, where the race should almost never be won. If a population of giant planets turns up around stars at a hundredth of the Sun’s iron, formed by some route that does not need a core to beat its disc — the gravitational collapse of a massive, unstable disc is the candidate — then the steep law is the sum of two mechanisms and not one. If the metal-poor stars stay empty of giants while their small planets remain as common as anywhere, the iron in a star is the clearest record anyone has of a race run in its disc four billion years ago.