The line beyond which ice counts as rock

The solar system has a shape that asks to be explained. Four small rocky planets inside two and a half astronomical units; a belt of rubble; then four large planets made mostly of hydrogen and ice. The transition is not gradual. Mars is a tenth of an Earth mass and Jupiter is three hundred, and between them there is a gap containing about a thousandth of an Earth mass of debris.

Something happened at about that radius, in the disc the planets were assembled from, and it was not a boundary anybody had to put there. It is where water froze.

The claim is not that ice is special. It is that ice is abundant, and that a phase change in an abundant species is a step change in how much solid material a growing body has within reach. Everything else follows from a race between the time it takes to assemble a large solid core and the time the disc’s gas takes to disperse.

Why a threshold in temperature is a threshold in mass #

A protoplanetary disc is mostly hydrogen and helium, and hydrogen and helium do not condense anywhere in it. What can condense is everything else, and everything else is a hierarchy: refractory materials — silicates, iron, aluminium oxides — condense at fifteen hundred kelvin and are solid throughout the disc; the volatiles condense much colder, and there is far more of them.

Water is the pivot because of abundance. After hydrogen and helium the commonest elements are oxygen and carbon, and oxygen’s dominant molecule in a hydrogen-rich gas is water. So the amount of solid material available to build planets from is roughly one thing inside the water condensation temperature and roughly four times that outside it — a step, not a slope, because condensation of a species happens over a narrow band of temperature.

The narrowness is worth a sentence, since it is what makes a line rather than a gradient. A species condenses when its partial pressure exceeds its vapour pressure, and vapour pressure is exponential in the inverse temperature. Over a ten per cent change in temperature the vapour pressure of water changes by a factor of several, so the transition from almost all vapour to almost all ice happens over a radius interval far narrower than the radius itself. The same exponential sharpness is why the ionisation of a gas is a step rather than a slope.

That step in solid surface density is what the whole essay is about, and the reason it matters is that planet formation is a race.

The temperature the line is drawn at #

The disc’s temperature comes from starlight, which it absorbs and re-radiates. Written out, $T(r) = 280\,\mathrm{K}\,(L/L_\odot)^{1/4}(r/\mathrm{AU})^{-1/2}$, and setting that to the condensation temperature gives the line. For the Sun’s present luminosity and 170 kelvin the answer is 2.7 astronomical units, which lands neatly in the middle of the asteroid belt, and it would be satisfying to stop there.

It is not that simple, and the reason is worth stating because it changes the conclusion. A young star is brighter than a main-sequence one, and a disc that is still accreting heats itself: material spiralling inwards releases gravitational energy, and in the first million years that internal heating exceeds the starlight over the inner few astronomical units. So the line starts much further out — five astronomical units or more — and sweeps inwards as the disc drains, passing through the region where the terrestrial planets are being built.

The accretion heating is not a small correction. A disc processing even a modest amount of mass through its inner regions liberates gravitational energy at a rate comparable with the star’s own output, delivered exactly where it matters, and the process by which it does so is the one any accretion disc has to solve: material cannot fall inwards without something carrying its angular momentum outwards, and whatever does the carrying dissipates energy doing it.

That the line moves is the single most important complication in the picture, and it is why the phrase “the snow line” is a simplification of a history rather than a location. A body assembled at three astronomical units early, when the line was at five, formed dry; the same radius later, when the line was at two, was outside it and could accrete ice.

What the step does to the outcome #

Core accretion is a two-stage process, and the step decides whether the second stage happens at all.

A solid core grows by sweeping up planetesimals, faster where there is more material. Once it passes roughly ten Earth masses, the hydrogen envelope it has been holding onto can no longer support itself against its own weight, 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 in a hundred thousand years. The instability there is the same one that governs a star held up by its own weight read backwards: a self-gravitating envelope in hydrostatic balance has a maximum mass it can hold up at a given core mass, and past it there is no equilibrium to sit in.

The whole of that depends on reaching ten Earth masses while the gas is still there, and discs disperse in a few million years. Inside the snow line the solids are too sparse to build a ten-mass core in the time available. Outside it, they are not. That mismatch is a large part of why the first exoplanet results were so surprising. A hot Jupiter at 0.05 astronomical units is inside the snow line by two orders of magnitude, and it cannot have been assembled there — the isolation mass at that radius is a fraction of an Earth. Its existence is therefore not a fact about formation but a fact about migration, and it took a while for that to be the obvious reading rather than a rescue.

The same correction has to be applied to any statistic drawn from the census. Every survey draws a different sky, and a survey that finds giant planets preferentially at short periods is not evidence that giant planets prefer short periods. It is evidence about the survey — which is why the part of an occurrence rate that is a definition has to be settled before the rate says anything about formation.

The evidence in the solar system #

The line’s fingerprints are all over the local system, and the asteroid belt carries the clearest.

Asteroids inside about 2.7 astronomical units are predominantly S-type: stony, anhydrous, related to the ordinary chondrite meteorites. Outside it they are predominantly C-type: dark, carbonaceous, and carrying water bound into hydrated minerals — water that was ice when the parent body formed and was liberated by mild heating afterwards. The compositional transition sits where the arithmetic above puts the line, and it was noticed as a pattern in reflectance spectra long before it was explained.

The belt also records that the compositional structure is primordial rather than acquired. Families produced by later collisions preserve their parent bodies’ composition, so a family’s spectral class is a statement about where its parent formed — and the dating of those families puts the collisions long after the disc dispersed.

The satellite systems say the same thing on a smaller scale. Jupiter’s inner large moon is rocky and its outer three are half ice by mass, in a sequence that looks exactly like a miniature version of the same condensation gradient — because Jupiter’s own accretion disc had its own temperature profile, with the planet in place of the star.

The Earth’s water #

If the Earth formed inside the snow line, it formed dry, and it is not dry. The oceans are only a two-thousandth of its mass, which is a small number and not zero, and something delivered them.

The candidates are the carbonaceous asteroids from just outside the line, scattered inwards during the chaotic late stages of terrestrial assembly, and the comets from very much further out. The deciding measurement is isotopic: the ratio of deuterium to ordinary hydrogen, which is a strong function of the temperature at which the water condensed and which is not changed much by anything afterwards. Ocean water matches carbonaceous chondrites well, and most comets measured poorly, though the comet measurements scatter enough that the case is a preponderance rather than a proof.

The transport itself is not gentle. Delivering material from just outside the line to a terrestrial planet requires scattering it inwards, which means close encounters with a growing giant planet, which means a period in the inner system that is anything but orderly. Chaotic systems have a horizon past which the trajectory cannot be followed, and the assembly of the terrestrial planets is firmly on the far side of it: what can be computed is a distribution of outcomes rather than the outcome.

The general shape of that argument is worth extracting. The snow line does not merely decide where giant planets form. It decides where the volatiles are, and therefore what has to be transported and how far, for any inner planet to have an atmosphere or an ocean at all. A planetary system’s habitability is downstream of a condensation temperature.

How the line is seen directly #

Everything above is inference from the outcome. The line itself has now been observed, or something close to it has, and the method is worth describing because it is the reverse of the usual one.

A disc’s midplane is cold and dense and opaque, and its surface is neither. Molecules that are frozen onto grains in the midplane are in the gas phase in the warm surface layer above it, so a molecular line drawn from the disc traces the region where that species happens to be gaseous — and the inner boundary of an emission ring is where the species is destroyed or hidden, while the outer boundary is where it freezes out. Mapping a line therefore maps a condensation front, provided the front is resolved.

Water is the hard case, because the Earth’s atmosphere is full of it and because the emitting region is small. The fronts that have actually been mapped are those of carbon monoxide and its isotopologues, which condense at around twenty kelvin and therefore sit tens of astronomical units out, where a millimetre interferometer resolves them comfortably. What is measured is not the water line but a line further down the same sequence, and the water line is then placed by the temperature profile that fits the one that was seen.

The indirect route is better than it sounds, because it also delivers a check. A disc’s temperature profile inferred from one molecule’s front should predict the front of the next, and in the best-observed discs it does.

Where the picture stops #

There is more than one line. Water is the most abundant condensable, but carbon monoxide, carbon dioxide, methane and ammonia each have their own condensation front further out, and the sequence of them is what sets the composition of a body assembled at any radius. The ratio of carbon to oxygen in a giant planet’s atmosphere is now measurable and is, in principle, a record of which of those lines the planet’s material came from — which turns a composition into a formation location.

The disc is not passive and it is not static. Real discs have dead zones, pressure bumps, and gaps carved by planets already formed, and each of those traps drifting solids. A pressure maximum near the snow line will concentrate material further, and the rings observed in nearby discs are almost certainly this happening — which means the step in solid density is sharper than the smooth argument above suggests.

And the drift is the hardest part. Metre-sized bodies spiral into the star in a few hundred years under gas drag, faster than they can grow. Everything above assumes planetesimals exist; making them is a separate and unresolved problem, and the snow line is one of the places where the conditions for solving it look most favourable, since ice is stickier than rock and vapour diffusing outwards across the line condenses onto whatever is already there.

The prediction that a long survey can test #

The argument makes one sharp population-level prediction: giant planets should be commonest just outside the line, because that is where cores can be assembled fastest and where the material is.

Testing it requires measuring the occurrence rate of giant planets against orbital distance out to several astronomical units, which means a radial-velocity survey running for longer than the orbital periods involved — two decades or more.

Those surveys exist, and the answer they give is that the occurrence rate of giant planets rises with distance out to about three astronomical units and then flattens or falls. The rise is steep: a giant is several times more likely between one and three astronomical units than inside one.

That is the predicted shape, and the agreement is worth two qualifications.

The turnover at the outer end is where the surveys run out rather than where the planets do. A survey of twenty years cannot establish the rate beyond about seven astronomical units, so the falling part of the distribution is a statement about sensitivity.

And the peak’s position depends on the host star. The line’s radius scales as the square root of the luminosity, so a more luminous star has its line further out — and the observed giant-planet occurrence does rise with stellar mass, which is consistent and is also what a more massive disc would produce independently.

The prediction is confirmed in its qualitative shape and is not yet sharp enough to distinguish the snow line from the disc mass, and separating those two requires the rate as a function of both stellar mass and distance, which needs a longer baseline than anybody has.

The line that lands inside the zone #

The hero figure draws two curves against stellar mass, and they converge. Following that convergence gives a result that changes what a habitable planet around a small star is likely to be made of.

For a solar-type star the snow line sits at a few astronomical units and the habitable zone at one, so a planet in the habitable zone formed well inside the line and formed dry.

For a star of a tenth of a solar mass both curves move inwards, and they do not move by the same factor. The habitable zone’s radius goes as the square root of the luminosity and so does the snow line’s, so in the simplest treatment their ratio is fixed — but the disc’s own accretion heating does not scale that way, and around a very low-mass star the disc is cooler at a given radius than the starlight alone implies.

The consequence is that around the smallest stars the snow line can fall at or inside the habitable zone, so a planet that ends up in the temperate band assembled from material that included ice.

Such a planet is not a drier Earth; it is a wetter one, by a large factor. A body assembled from material a few tens of per cent ice by mass and then warmed into the habitable zone has an ocean hundreds of kilometres deep and no exposed land at all.

That is a different kind of world from the Earth in ways that matter for what could be measured about it: no continental weathering means no carbonate–silicate cycle, and therefore no thermostat regulating the atmosphere’s carbon dioxide over geological time.

The same threshold that decides where giant planets form decides whether a temperate planet around a small star is a rock with an ocean or an ocean with a rock in it, and the answer depends on a disc temperature profile in the first million years.

The line’s position and the zone it sits outside are both worth reading at a second setting, since the essay’s whole point is that the two are set by unrelated physics.

Where this ladder goes next #

Later rungs on this anchor: the condensation sequence in full, and reading a giant planet’s carbon-to-oxygen ratio as a formation radius; the movement of the line over the disc’s lifetime, and what a body’s formation time therefore means; pressure traps and the observed rings in nearby discs; the delivery of volatiles inwards, which is the isotopic argument above done properly; and the connection to the census of planet occurrence, where the prediction that giant planets should be commoner just outside the line meets a survey that barely reaches it.