Two elements in a ratio, and a birthplace read off it

A hot Jupiter sits at a twentieth of an astronomical unit and cannot have formed there. Everything about the planet says it arrived from somewhere else, and nothing about its orbit says where.

There is a proposal for recovering the missing information, and it is the most ambitious use anybody makes of a condensation sequence. The idea is that the gas a planet accreted carries a compositional fingerprint of the radius it was accreted at, that the fingerprint survives the migration, and that it can be read off the planet’s atmosphere from here.

The fingerprint is the ratio of carbon to oxygen, and the reason it works at all is that the two elements freeze out at different places.

Why the two curves must be opposite #

The structure of the figure follows from one fact and it is worth stating because it makes the staircase a theorem rather than a model.

Carbon and oxygen are conserved. Every atom of each is either in the gas or in the solids, so the gas’s inventory and the solids’ add to the total, and any front that removes oxygen from the gas adds it to the solids. There is no third place for it to go.

The water front removes oxygen and no carbon, because water contains no carbon. So beyond it the gas has lost a large fraction of its oxygen and none of its carbon, and its ratio rises steeply; the solids have gained oxygen and no carbon, and theirs falls.

The carbon dioxide front removes one carbon for every two oxygens, which is more oxygen than carbon relative to the solar ratio, so the gas ratio rises again — less steeply, because there is less left to remove.

The carbon monoxide front removes them one for one, which is more carbon than oxygen relative to the solar ratio of about one to two, so beyond it the gas ratio rises sharply toward the point where there is no gas-phase oxygen left at all.

Three fronts, three steps, all in the same direction for the gas and all in the other for the solids. Nothing about the magnitudes is guaranteed — those depend on the apportionment of each element among the molecules — but the signs are forced.

Why a planet’s atmosphere is mostly the gas #

The inference needs one astrophysical premise: that a giant planet’s envelope reflects the gas it accreted rather than the solids.

That follows from the core-accretion picture. A solid core grows to about ten Earth masses, at which point its hydrogen envelope can no longer support itself and gas falls in at a rate limited only by how fast it can cool — the runaway that turns a ten-mass core into a three-hundred-mass planet, which is a hydrostatic balance running out of solutions. The envelope is therefore three hundred masses of disc gas over ten masses of solids, and the observable atmosphere is the outer part of that envelope.

If the core stays at the bottom and the envelope does not mix with it, the atmosphere’s ratio is the gas’s ratio at the radius where the runaway happened.

That premise is where most of the difficulty lives, and the rest of this essay is about it.

What the measurement is #

The atmospheric ratio is measured by spectroscopy of a transiting planet, and the chain has more steps than the phrase suggests.

During a transit some starlight passes through the planet’s upper atmosphere, and molecules there absorb at their own wavelengths, so the planet’s apparent radius is larger at the wavelength of a strong band than in the continuum between bands. Measuring the transit depth as a function of wavelength gives a transmission spectrum, whose features are of order a hundred parts per million on a transit depth of one per cent.

For a hot Jupiter the accessible carriers are water, carbon monoxide, carbon dioxide and methane. Their relative abundances depend on the carbon-to-oxygen ratio, on the metallicity, and on the temperature — because the chemistry that partitions carbon between carbon monoxide and methane is temperature-dependent in the same way every dissociation equilibrium is.

The analysis is a retrieval: a model atmosphere with free abundances, a temperature profile and a cloud treatment is fitted to the spectrum, and the posterior on the abundance ratios is the result. It is a fit with more parameters than the data obviously support, and the quoted uncertainties depend heavily on what was allowed to vary.

The state of the art gives ratios with uncertainties of a few tens of per cent for the best-observed planets, which is enough to distinguish one interval of the staircase from another if everything else is right.

Why the inversion is not unique #

The hero figure is a staircase, and a staircase is not invertible.

A measured gas ratio of 1.6 times the stellar value corresponds to a plateau between two fronts — and on the drawn curve there is exactly one such plateau, so in this idealisation the inversion works. Three things break it.

A planet accretes solids too. The ten-Earth-mass core is the obvious contribution and it stays at the bottom, but planetesimals continue to rain into the envelope during and after the runaway. Those solids have the complementary ratio — oxygen-rich where the gas is carbon-rich — so adding them pulls the atmosphere’s ratio back toward the stellar value. A planet that formed far out and swallowed many planetesimals looks like a planet that formed nearer in and swallowed few.

The planet migrates while it accretes. The runaway takes of order $10^5$ years and a migrating giant crosses a substantial range of radii in that time, so the gas it accretes is an average over an interval rather than a sample at a point.

And the fronts move. The whole staircase sweeps inward as the disc drains, so the radii the plateaux sit at depend on when the accretion happened.

Each of those turns a point into an interval, and together they turn the inversion into a constraint on a combination rather than a determination of a radius.

The second number that helps #

The standard escape is to measure the metallicity as well, and the reason it helps is that it responds to the solids and the ratio responds to the gas.

A planet’s atmospheric metallicity — the total abundance of everything heavier than helium, relative to the star’s — is raised by swallowing solids and is unchanged by accreting gas, because disc gas has the stellar metallicity by definition in the inner regions where the refractories are already solid. So the metallicity measures how much solid material got into the envelope, and the carbon-to-oxygen ratio measures the mixture.

Two measurements, two unknowns, and the degeneracy breaks — in principle.

In the solar system the test is available. Jupiter’s atmosphere is enriched in carbon by a factor of about four relative to the Sun, and in nitrogen, sulphur, argon, krypton and xenon by similar factors. That uniform enrichment says the planet swallowed solids carrying all of those species, and the noble gases are the informative part: argon condenses only below about thirty kelvin, so trapping it requires solids that formed very cold, far outside where Jupiter is.

Jupiter’s atmosphere therefore contains material from beyond the carbon monoxide front, which is either evidence that the planet formed much further out than it now sits or evidence that cold planetesimals were delivered inward to it. The oxygen abundance would distinguish the two, and it was the one number the Galileo probe could not measure, because it entered a dry region and its water measurement is a lower limit.

That gap was the motivation for a later mission’s microwave radiometer, which measures water at depth by its absorption of the planet’s own thermal emission. The result is a lower enrichment than the other species — a carbon-to-oxygen ratio above the solar value — and it is still being argued about, because the measurement is of the equatorial region and the planet is not uniform.

The four ways a giant can get its heavy elements #

It helps to enumerate the routes, because the inversion’s ambiguity is exactly the ambiguity between them and naming them makes the count clear.

Gas accretion delivers the disc’s gas at the local ratio, in enormous quantity and with the stellar metallicity. It sets the baseline and it is the signal the method wants.

Core dissolution mixes the original ten Earth masses of solids upward into the envelope, raising the metallicity and pulling the ratio toward the solid curve. How much of it happens depends on the temperature and composition at the core–envelope boundary, which interior models disagree about.

Planetesimal accretion during the runaway delivers solids from the feeding zone into the envelope while it is forming. The amount depends on how much solid material was left in the feeding zone by then, which depends on how efficiently the core swept it up first.

Late bombardment delivers solids after the disc has gone, in the dynamical rearrangements that also produce migration. That material comes from wherever the scattered bodies came from, which is generally further out than the planet formed.

The first raises neither the metallicity nor the deviation from the stellar ratio. The other three raise the metallicity and move the ratio toward the solid curve, by amounts that differ and in ways that are not separable from a single atmosphere.

So a planet with a solar ratio and a high metallicity is a planet that formed anywhere and swallowed a lot, and a planet with an extreme ratio and a low metallicity is one that accreted almost pure gas, at a radius the staircase names. Only the second is legible, and the second is the rarer case.

What has been measured, and what it did not settle #

Several dozen giant exoplanets now have retrieved carbon-to-oxygen ratios and the picture is not the tidy one the method promised.

The distribution is broad. Values from well below solar to well above are reported, sometimes for the same planet by different groups analysing the same data with different retrieval codes. The spread between analyses is comparable to the spread between planets, which means the population statistics are not yet a measurement of anything about formation.

The systematic that does most of the damage is clouds. A grey cloud deck truncates the transmission spectrum, which is the same degeneracy a measured transit depth carries against limb darkening, muting all the features by the same factor, which is degenerate with a lower overall abundance. Distinguishing a cloudy metal-rich atmosphere from a clear metal-poor one requires observing the wings of a strong feature where the cloud’s opacity is not grey, and that is at the edge of what is achievable.

The second is the temperature structure. The partitioning of carbon between carbon monoxide and methane, and of oxygen between water and carbon monoxide, depends steeply on temperature, so a retrieval that gets the profile wrong misallocates the elements.

Neither of those is a limitation of the idea; both are limitations of the spectra, and both are improving. Emission spectroscopy at high resolution, which resolves individual molecular lines — a Doppler shift is a speedometer at any distance — and is far less affected by clouds, is now delivering ratios for a handful of planets with much smaller systematics.

High resolution changes the character of the measurement #

There is a technique that removes most of the systematics above, and it is worth describing because it works by a completely different principle.

A transmission or emission spectrum at low resolution is a shape, and fitting a shape means fitting everything that changes shapes — clouds, temperature profiles, instrument systematics. At a resolving power of a hundred thousand the spectrum is not a shape but a forest of individual molecular lines, thousands of them, at positions the laboratory knows exactly.

The planet is moving. Over a few hours of a transit its orbital velocity changes by tens of kilometres a second, so its lines shift while the star’s and the Earth’s do not. Cross-correlating the observed spectrum with a template for one molecule, at each of a range of assumed planetary velocities, produces a peak at the planet’s actual velocity and nothing anywhere else.

That has three consequences. The detection is of a specific molecule, because the template is specific. It is immune to a grey cloud, because a grey cloud reduces every line by the same factor and the cross-correlation is a shape match rather than an amplitude. And it delivers the planet’s orbital velocity, which with the star’s gives the mass ratio without an inclination assumption.

What it does not deliver easily is an abundance, because the normalisation is exactly what the method throws away. Getting a carbon-to-oxygen ratio requires cross-correlating for several molecules and comparing their strengths, which reintroduces some of the modelling — and the recent measurements that do this give ratios with the smallest systematics anybody has.

A method that is insensitive to the nuisance parameter is worth more than a method that models it, and this one is insensitive by construction rather than by care.

Where the method stops #

It assumes the envelope does not mix with the core. If the core is eroded and its material dredged upward — which some interior models of Jupiter require to explain its diluted core — the atmosphere’s composition is a mixture of gas and core, and the fingerprint is diluted by an unknown amount.

It assumes disc gas at the stellar ratio to begin with. The gas’s carbon-to-oxygen ratio depends on how much carbon was locked into refractory grains, which is a chemistry with its own uncertainty and which differs between the interstellar medium and the solar system by a large factor.

And it assumes the planet’s spectrum represents its atmosphere. A transmission spectrum probes the terminator, at pressures of millibars, on a tidally locked planet with a day side and a night side that differ by a thousand kelvin. Whether that region’s composition is the bulk envelope’s is a separate question that a global circulation model has to answer.

The habit, and why it recurs #

The shape of this argument is one that recurs whenever a present-day composition is read as a history, and it is worth stating in the abstract.

A tracer records a history only if the process that set it is not the process that is being asked about. The carbon-to-oxygen ratio works as a formation-radius indicator precisely because the chemistry that sets it happened in the disc and nothing since has changed it. The moment mixing, dissolution or bombardment can change it, the tracer records a sum of the formation and everything afterwards, and the inversion needs to solve for both.

The escape is always a second tracer with a different sensitivity. Here it is the metallicity, which responds to solids and not to gas. In stellar spectroscopy it is a second ionisation stage, which responds to pressure and not to abundance. In the interiors of icy moons it is a moment of inertia, which weights the interior differently from a tidal response. In every case the useful second measurement is the one whose degeneracy runs the other way, and finding it is more valuable than improving the first.

And the honest output is a region rather than a number. The literature reports formation radii for individual planets, and what the measurements support is a statement about which interval of the staircase is consistent with the data given a set of assumptions about the solids. Those are different claims, and the second is the one the data carry.

Still open: what has not been examined #

Four essays have taken one condensation front apart: what it does to the solid supply, what the other four fronts add, how the whole staircase sweeps inward while planets grow, and whether a composition can be inverted for a birthplace.

What has not been examined is the delivery. Volatiles get from beyond the fronts to the inner system — the Earth’s oceans are the proof — and the isotopic argument that says where they came from was only sketched here. Doing it properly means the deuterium ratio as a thermometer, the scatter among the comets measured so far, and the dynamical mechanism that moves material inward without heating it. That is a different subject from the one these essays have been building and it uses all of it.