There is not one line, there is a staircase

The line beyond which ice counts as rock treats one phase change as a boundary and derives most of a planetary system from it. That is the right place to start and it hides a structure.

Water is not the only condensable species and its freezing point is not the only threshold. Silicates and iron are solid everywhere in a disc; carbon dioxide freezes far outside water; methane, ammonia and carbon monoxide further out still. Each has a temperature at which its partial pressure meets its vapour pressure, and a disc whose temperature falls with distance turns each of those temperatures into a radius.

What a body is made of is therefore not decided by a line but by an interval — which pair of fronts it assembled between — and the composition of every rock, moon and comet in the solar system is a reading of that — which is the same inference a bulk density cannot make on its own.

Why a temperature becomes a radius #

The conversion is the same one throughout and it is worth stating once.

A disc heated by the star it surrounds reaches a temperature set by the balance between the flux it absorbs and the flux it re-radiates. Absorbed flux falls as the inverse square of the distance and re-radiated flux goes as the fourth power of the temperature, so

$$T(r) = 280\ \mathrm{K}\,\left(\frac{L}{L_\odot}\right)^{1/4}\left(\frac{r}{\mathrm{AU}}\right)^{-1/2},$$

and inverting it at a condensation temperature gives

$$r_{\rm cond} = \left(\frac{280}{T_{\rm cond}}\right)^2\sqrt{\frac{L}{L_\odot}}\ \mathrm{AU}.$$

The inverse square in that expression is what spreads the fronts out. Water at 170 kelvin lands at 2.7 astronomical units; carbon monoxide at 20 kelvin is colder by a factor of eight and a half and lands at 196, a factor of seventy-two further out. The outer disc is therefore a place where the chemistry changes slowly with radius and the inner disc a place where it changes fast, and that asymmetry is entirely a consequence of the exponent.

The sharpness of each front is a separate matter and it comes from the other direction. A vapour pressure is exponential in the inverse temperature, so a ten per cent change in temperature changes it by a factor of several — which means the transition from almost all vapour to almost all ice occupies a radius interval far narrower than the radius itself. Each riser in the staircase is genuinely a step rather than a ramp.

Why water is the large step #

The heights of the risers are abundances, and the abundances are decided in stars rather than in discs.

After hydrogen and helium the commonest elements are oxygen and carbon, then nitrogen, then the rock-formers — silicon, magnesium and iron. Oxygen is about twice as abundant as carbon and eight times as abundant as silicon, and in a hydrogen-rich gas its dominant molecule is water.

So the amount of solid material a growing body has within reach is roughly one thing inside the water front and roughly four times that outside it, and no other front comes close to producing that ratio. Carbon dioxide’s step is a fifth of water’s; methane and ammonia together a seventh; carbon monoxide’s a tenth.

That is why one of these five steps gets the definite article. The others are real and they matter for composition; only one of them matters for how fast a core can be assembled, which is the quantity that decides where giant planets form.

Reading it backwards in the solar system #

The sequence’s clearest fingerprints are in the meteorites, and the correspondence is close enough to be a test rather than a story.

The enstatite chondrites are the most reduced meteorites known: their iron is metallic rather than oxidised, their silicates contain almost no oxidised iron, and they carry essentially no water. Their oxygen isotopes place them close to the Earth’s, which puts their formation inside about one astronomical unit — inside every front but the refractory one.

The ordinary chondrites are partly oxidised and contain a fraction of a per cent of water in hydrated minerals. They formed between roughly two and three astronomical units, which brackets the water front.

The carbonaceous chondrites are dark, oxidised, and contain up to twenty per cent water by mass in hydrated silicates, plus organic compounds. They formed outside the water front, and their subclasses differ in how far outside.

And the comets contain not just water but carbon dioxide, methane, ammonia and carbon monoxide as ices in the proportions the outer sequence predicts, which is the only direct sample anyone has of material from beyond the third and fourth fronts, and which arrives on orbits that remember where it came from.

Four classes of object, four intervals of the staircase, with the ordering and the water content matching. The correspondence was noticed as a pattern in reflectance spectra decades before it was explained, and the explanation is the drawing above.

The first solids, and how the top riser is dated #

The highest step in the staircase has been sampled, and the sample is the oldest dated object in the solar system.

Calcium–aluminium-rich inclusions are millimetre- to centimetre-sized objects found in carbonaceous chondrites, made of the minerals an equilibrium calculation predicts condense first from a cooling gas of solar composition — corundum, hibonite, perovskite, melilite — at temperatures above about 1,400 kelvin. They were computed before they were recognised: the condensation sequence was worked out in 1972, and the predicted mineral assemblage matched inclusions that had been sitting in meteorite collections for decades.

Their age is 4,567.3 million years, measured by lead–lead dating — the zero point every other date in the solar system’s chronology is quoted against to a precision of a few hundred thousand years, and it defines the zero point of the solar system’s chronology. Everything else is dated relative to them.

So the top riser of the staircase is not a modelled boundary; it is a set of rocks with an age. That is worth holding onto, because it is the only step in the sequence with a direct sample whose formation conditions can be read off its mineralogy.

What the inclusions also show is that the sequence was not followed tidily. Many of them have been melted and recrystallised, some carry isotopic anomalies indicating they formed in several distinct events, and their formation region — hot enough for refractories to condense, which means the innermost disc — is not where they were found, which is in meteorites from the outer asteroid belt. Something transported them outward by a couple of astronomical units within the first few hundred thousand years.

That transport is now attributed to the disc’s own outward flow at high altitude, or to an early wind. Either way, the material that samples the innermost step of the sequence was delivered to the region between the second and third steps, and any reading of composition as a birthplace has to contend with it.

What the simple sequence leaves out #

The staircase drawn here is an equilibrium condensation sequence in a static disc, and three things complicate it in ways that are not corrections.

The species compete for the same elements. Carbon monoxide is bound by eleven electronvolts, so it takes whichever of carbon and oxygen is scarcer to near-completion before anything else forms. In a disc with more oxygen than carbon — which is nearly all of them — the leftover oxygen makes water and the silicates; with more carbon than oxygen, the leftover carbon makes graphite and carbides and there is no water front at all. The staircase in the figures assumes the solar ratio, and a system with a different one has a different set of steps.

The solids drift. A metre-sized body feels a headwind — the same drag that sorts a debris disc by size, because the gas is partly supported by its own pressure gradient and orbits slightly slower than the local Keplerian speed. The body loses angular momentum to the gas and spirals inward, fast — a metre-sized object at a few astronomical units migrates into the star in a few hundred years. So solids do not stay where they condensed, and the sequence describes where material forms rather than where it ends up.

And the fronts are where material piles up. Drifting icy bodies crossing a front inward sublimate, releasing vapour that diffuses back out and refreezes on whatever is already there. That recycling concentrates solids in a narrow annulus just outside each front, so the steps are not merely steps in composition but in surface density — sharper than the equilibrium argument alone gives, and probably where planetesimals are easiest to make.

The rings that are probably fronts #

The most striking recent evidence for the staircase is not a composition at all. It is a picture.

Millimetre interferometry resolves nearby protoplanetary discs into concentric bright and dark rings, in almost every disc large enough to resolve. The rings are in the dust continuum, so they are places where millimetre-sized solids are concentrated, and something is holding them there against the inward drift described above.

A pressure maximum does that: a local bump in the gas pressure reverses the headwind on its outer side, so drifting solids pile up at the bump instead of passing through. Several mechanisms make a pressure bump, and one of them is a condensation front — the change in the solids’ opacity and in the gas’s thermal structure across a front produces exactly such a feature.

The test is positional. If the rings are condensation fronts they should sit at the radii the sequence predicts, and they should be in the same order in every disc, scaled by the square root of the luminosity.

The answer is partly encouraging and not clean. Some discs have rings near the predicted carbon monoxide and nitrogen fronts, located by the emission of the corresponding gas-phase molecule. Many rings are not at any front, and the leading alternative — that they are gaps carved by planets already formed — predicts rings wherever planets are, which is to say anywhere.

So the rings are evidence that something concentrates solids at particular radii, which the staircase requires, and they are not yet evidence that the staircase is what does it.

What the staircase does not decide #

It is a composition, not an architecture. Which species are solid at a radius says what a body formed there is made of. It does not say a body formed there, how big it got, or whether it stayed — those are questions about surface density, growth timescales and migration, and the sequence is an input to them rather than an answer.

It assumes the midplane temperature and the fronts are observed in the surface. A disc’s midplane is cold, dense and opaque; its upper layers are warm and thin. A molecule frozen out at the midplane is in the gas phase above it, so the emission that maps a front traces the warm layer rather than the plane where planets form, and converting one to the other requires a vertical temperature structure that is itself modelled.

And the abundances are the star’s, once. Everything above uses one set of elemental ratios. A disc’s composition changes as solids drift inward and vapour diffuses outward, so the local ratios at a given radius are not the star’s at all after a million years — which is the mechanism the essay after this one turns into a clock.

An isotope that survives what a composition does not #

There is a second observable that records the same chemistry and is much harder to erase, and it is the one that carries the argument for the Earth’s water.

The ratio of deuterium to ordinary hydrogen in water depends on the temperature at which the water last equilibrated, because the exchange reactions that set it are mildly favoured toward deuterium in the cold. Water that formed in the cold outer disc is deuterium-rich; water that equilibrated with the hydrogen reservoir in the warm inner disc is not. The ratio is not much changed by melting, by aqueous alteration or by the long storage since.

Measured values sort themselves informatively. Interstellar water is enormously deuterium-rich. Comets span a wide range and average above the terrestrial value, with a few close to it. The carbonaceous chondrites cluster near the Earth’s oceans.

That is the isotopic argument for the Earth’s water having come from just outside the water front rather than from the far outer disc, and it is a preponderance rather than a proof — the comet measurements scatter enough, and are few enough, that a substantial cometary contribution is not excluded.

The general point is worth extracting: an isotope ratio is a thermometer that survives processes a composition does not. A body’s water content can be changed by heating, by impact, by aqueous alteration; its deuterium ratio largely cannot. Wherever a composition is being read as a formation condition, the isotopic version of the same reading is the more robust one.

The habit this illustrates #

There is a general structure here worth naming, because it recurs here in several forms.

A threshold in an intensive variable becomes a boundary in space whenever that variable has a gradient. The condensation temperature is a property of a molecule; the disc supplies a monotone temperature profile; the composition of the two is a radius. The same construction produces the habitable zone from a surface temperature, the Strömgren radius from an ionisation balance, and the photosphere of a star from an optical depth.

What makes the construction powerful is that the gradient converts an equilibrium statement into a geometric one, and geometry is observable. Nobody measures a condensation temperature in a disc; they measure where a molecule’s emission stops.

And what makes it fragile is that the conversion inherits the gradient’s uncertainty. The fronts are only as well located as the temperature profile, and a disc’s temperature profile depends on its opacity, its accretion rate and its flaring — none of which is measured directly. The sequence is a solid fact about chemistry projected through a shaky fact about structure, and the projection is where the error is.

The moons are a miniature of it #

There is a second place the sequence can be read, and it is the cleanest because the whole system is small enough to see at once.

Jupiter’s four large moons run from rock to ice with distance, and the innermost of them is the most volcanic body there is. Io is anhydrous silicate, with a density of 3.53 grams per cubic centimetre and no water at all. Europa is 3.01 and has a water layer over a rock interior. Ganymede is 1.94 and Callisto 1.83, both about half ice by mass.

That is the condensation sequence applied to Jupiter’s own accretion disc, with the planet in place of the star. The circumplanetary disc was heated by the forming planet, its temperature fell outward, and the water front fell between Io’s orbit and Europa’s.

The same pattern appears at Saturn, less cleanly, and the exceptions are informative: Titan is icy and so is Enceladus, and the innermost small moons are not obviously rockier. Saturn’s disc was cooler throughout, so its water front sat inside the whole system and the gradient it would have produced is not there to see.

A sequence that reproduces itself at a scale a thousand times smaller, with the same ordering and a front in the predicted place, is the strongest kind of evidence a scaling argument can have — it is the same physics with every number changed.

Still open: whether a composition can name a birthplace #

The whole value of the staircase, for anything outside the solar system, is the inverse problem: measure what a body is made of, and read off where it formed.

That inversion is not unique and it is worth being honest about why. A body’s composition reflects the solids it accreted and the gas it accreted, in proportions that depend on its mass and its history. Two bodies with the same measured ratio can have formed at different radii and swallowed different amounts of solid material, and nothing in the measurement distinguishes them.

The next essay takes the sharpest version of this — the carbon-to-oxygen ratio of a giant planet’s atmosphere — and asks how far the inversion can be pushed. The short answer is that it works when there is a second measurement, and that the second measurement is usually the metallicity, which carries its own ambiguity.