A composition that dates a formation rather than placing it

Every drawing of a condensation front so far here is of a disc heated by starlight alone. That is right for a mature disc and badly wrong for a young one, and the difference is not a correction.

A disc that is accreting is releasing gravitational energy. Material spiralling inward gives up potential energy at a rate set by how fast it is flowing, and the energy comes out as heat where it is released — in the disc, at the radius concerned, rather than at the star. For the first hundred thousand years or so that internal heating exceeds the starlight over the inner several astronomical units, and the ice line sits where the sum of the two fluxes puts it.

The accretion rate decays as the disc drains. So the heating fades, the line sweeps inward, and it passes through the region where the terrestrial planets are being assembled while they are being assembled.

The two fluxes and the two exponents #

The arithmetic is short and the exponents are what matter.

Irradiation gives a temperature falling as $r^{-1/2}$, which follows from absorbed flux going as $r^{-2}$ and re-radiation as $T^4$. Viscous dissipation in a steady accretion disc releases energy per unit area at a rate

$$D(r) = \frac{3GM_\star\dot M}{8\pi r^3},$$

so setting that to $\sigma T^4$ gives a temperature falling as $r^{-3/4}$ — steeper.

Two profiles with different slopes cross once. Inside the crossing the disc heats itself; outside it the star heats the disc. The crossing moves inward as $\dot M$ falls, because the viscous curve drops while the irradiation curve does not, and the ice line follows.

That is the whole mechanism, and it has a consequence the figure makes plain: the sweep is large early and small late. Between thirty thousand and a hundred thousand years the line moves from 6.4 to 5.3 astronomical units; between one million and ten it moves from 3.2 to 2.7. Most of the motion happens while the disc is embedded and unobservable.

What makes a composition a clock #

A body that assembles at a fixed radius sees the line pass over it, once, inward, and where it ends up is a separate question again. Before the passage its surroundings are above the condensation temperature and it accretes dry material; afterwards they are below and it accretes ice.

So the water content of a body is not a statement about where it is. It is a statement about how much of its growth happened after the line went past — which is a statement about the ratio of two timescales, the growth time and the sweep time.

That inverts the usual reading. The single-line picture says a composition locates a birthplace; the moving-line picture says it dates one, given a place. Neither is available without the other, and a measurement of composition alone constrains the product rather than either factor.

There is a case where the two readings give opposite answers, and it is the one that matters for the solar system. The asteroid belt’s compositional transition — stony inside about 2.7 astronomical units, carbonaceous outside — is usually read as the fossil of the ice line’s final position. On the moving-line picture the transition is a fossil of where the line was when the parent bodies of the asteroids finished accreting, which was earlier and therefore further out. The two differ by a factor that the figures here put between one and a half and two.

Why the inner disc is the awkward region #

The sweep passes through the terrestrial region, and that is where the argument does the most work and is least secure.

Water delivered to the inner disc has to come from outside the line, and there are two routes. One is dynamical: bodies from beyond the line are scattered inward by a growing giant planet, arriving as whole objects on paths no integration can follow for long. The other is the sweep itself: material that was outside the line becomes inside it without moving, and its ice sublimates in place, releasing vapour into the inner disc.

The second route has a consequence that is easy to miss. Vapour released at the line diffuses both ways, and the part that diffuses outward recondenses on whatever solids are there — so the annulus just outside the line is enriched in water beyond the equilibrium value, by a factor that depends on how fast the vapour diffuses against how fast the solids drift.

That enhancement is a candidate solution to a hard problem. Planetesimal formation requires solids to concentrate enough for a gravitational or streaming instability to act — the same threshold argument a molecular cloud’s own collapse turns on, and the ice line supplies both a density enhancement and a change in the stickiness of the particles, since ice sticks better than rock at low speeds. The line is not only a compositional boundary; it may be the place planetesimals are made.

Whether it is depends on numbers nobody has measured — the turbulent diffusivity of the gas, the drift speed of the solids, and the sticking behaviour of ice-coated grains at the relevant temperatures — and the calculations disagree by more than an order of magnitude.

What is actually measured #

Nothing in the first figure is observed directly, and it is worth separating what is measured from what is modelled, because the modelled part is most of it.

The accretion rate is measured, for stars old enough to be visible. It comes from the excess ultraviolet continuum produced where the accretion flow strikes the stellar surface, calibrated against the emission lines it also produces. The measurements span four orders of magnitude at any given age, and the accretion they trace is the flow a disc can only sustain by exporting angular momentum, with a median falling roughly as the inverse of the age — which is where the decay law in the figures comes from.

The disc temperature is measured, indirectly and in the surface layers. Molecular line ratios give excitation temperatures at the height where the lines form, and the continuum spectral energy distribution constrains the radial temperature profile of the emitting surface. Getting the midplane, which is where planets form, requires a vertical structure model.

The line’s position has been measured once, arguably. The water snow line is almost impossible to observe: water in the Earth’s atmosphere blocks the relevant transitions, and the emitting region is small. One outburst offered a way round — a young star that brightened by two orders of magnitude pushed its snow line out by a factor of several, into a region where a millimetre interferometer could resolve it, and the change was detected as a shift in the emission of a molecule that is destroyed by water vapour.

That is one system, in an unusual state, using a proxy molecule. The sweep in the first figure is otherwise entirely a calculation, resting on an accretion history that is measured statistically and on a viscous model that is assumed.

The outburst that moved a line far enough to see #

The steady decay in the first figure is an average of something much less orderly, and the departures are large enough to be the main event.

A young star’s accretion is episodic. Material accumulates in a region of the disc where the magnetic coupling is poor — a dead zone, too cold for thermal ionisation and too shielded for cosmic rays — until the pile-up becomes unstable, and then it drains onto the star in an outburst lasting decades to a century. The accretion rate rises by two to four orders of magnitude.

The luminosity rises with it, and the disc’s thermal structure follows within months. An outburst pushes every condensation front outward by a factor of a few, holds it there for the duration, and then lets it return as the disc cools.

That has two consequences worth separating.

For the chemistry, an outburst sublimates the ice on every solid inside the new front position, and the ice does not simply reform where it was — the vapour diffuses and recondenses, redistributing volatiles over a region much wider than the front’s own excursion. A body that has been through several outbursts has a composition reflecting the outbursts rather than its radius.

For the observation, the outburst is the one chance to see the front move. A star that brightened by two orders of magnitude pushed its water front out from an unresolvable fraction of an astronomical unit to several, and the shift was detected — not in water, which the Earth’s atmosphere blocks and which emits from too small a region, but in a molecule whose abundance collapses in the presence of water vapour. The front’s position was read off where that molecule’s emission stopped.

One measurement, in one unusual object, using a proxy, is the entire direct observational basis for a moving condensation front. Everything else in this essay is calculation.

Where the model stops #

A steady accretion disc is not a disc. The temperature profile used here assumes the accretion rate is the same at every radius, which is true only if the disc has settled. Real discs are not steady: they have dead zones where the magnetic coupling fails and material piles up, and they undergo outbursts when the pile-up releases. During an outburst the line jumps outward by a large factor and returns over decades, so the history is a staircase of excursions rather than a smooth decay.

The viscosity is a parameter. The disc has to throw angular momentum outwards for anything to accrete, and how it does so is not settled — magnetised winds remove angular momentum without dissipating it locally, which changes the heating at fixed accretion rate by a large factor. A wind-driven disc is cooler at the same $\dot M$, so its line is further in throughout.

And the condensation temperature is a function of the pressure. The 170 kelvin used in every figure is for a particular midplane pressure; the published range runs from about 145 to 190, and the radius goes as the inverse square of the temperature. That alone is a factor of nearly two, which is comparable to the whole sweep being computed.

The excursion and the composition it writes are therefore one measurement made twice — where the fronts went, and what the solids there ended up made of. Narrowing the drawn interval to the ages at which discs are actually observed shows how little of that excursion is on the record.

What a formation time even means #

There is an ambiguity in the phrase this essay’s title depends on, and it is worth confronting because it decides how sharp the clock is.

A body does not form at an instant. A planetesimal assembles over some interval, a core grows over a longer one, and a planet’s final composition is a weighted average over its whole accretion history with the late material weighted by how much of it there was. So “when a body formed” is a distribution rather than a date, and the composition records the convolution of that distribution with the line’s position over the same interval.

Two limits make the ambiguity concrete.

If the growth is fast compared with the sweep, the body samples the line at one position and its composition is a snapshot. The clock is sharp and the reading is a date.

If the growth is slow compared with the sweep, the body accretes material from both sides of the line and its composition is an average. The clock is blunt, and two bodies with different histories can end with the same water content.

Which limit applies depends on the growth timescale, which depends on the surface density, which is the quantity the line’s passage changes. The problem is therefore not separable, and the honest calculations are the ones that integrate the growth and the thermal evolution together.

The sharp version of the argument requires that planetesimals form fast, which is what the streaming instability provides and what the classical collisional growth picture does not — so the usefulness of the composition as a clock is contingent on an unsettled question about how planetesimals are made.

The general shape #

The argument has a structure that recurs wherever a boundary moves through material that is itself changing.

A stationary boundary sorts material by position. A moving one sorts it by time of arrival, and the record it leaves is a convolution of the boundary’s history with the material’s growth history. Recovering either from the record requires knowing the other.

The same structure governs the reionisation of the intergalactic medium, where an ionisation front sweeps through gas that is also collapsing, and the observable is the residue of both. It governs the freeze-out of any species in a cooling gas. And it governs crater counting, where a surface’s age is read from an accumulated population under a flux that was itself declining.

In every case the tempting reading is the stationary one, because it involves one fewer unknown, and in every case it is biased in a known direction: it reports the final state of the boundary as though it had always been there.

The spread in accretion rates is the spread in histories #

The decay law in the figures is a median through a distribution, and the distribution is wide enough that “the” history of a disc is a fiction worth naming as one.

Surveys of young clusters measure accretion rates for hundreds of stars of known age and mass. At any given age and mass the rates span about two orders of magnitude — not a measurement error, since the individual determinations are good to a factor of two or three, but real scatter between objects.

Two readings of that scatter are current and they have different consequences here.

On the first, the scatter is in the initial conditions: discs are born with different masses and different angular momenta, so they start at different rates and decay along parallel tracks. Then the ice line’s starting position varies between systems by a factor of a few, and the compositional gradient in a planetary system is a birth property.

On the second, the scatter is in time: every disc is episodic, and a snapshot of many discs catches them at random phases of their own variability. Then every system’s line has swept back and forth many times and the composition records an average.

The observations that would separate them are repeat measurements of the same stars over years, and those show variability of tens of per cent on short timescales — large, but far smaller than the population scatter. That favours the first reading and does not establish it, because the long-timescale variability an episodic model needs is by construction not visible in a decade.

The uncertainty on a single disc’s history is therefore not the measurement error on its accretion rate; it is the width of a population, and it is two orders of magnitude wide.

Still open: whether the belt remembers the line at all #

The asteroid belt’s compositional gradient is the best evidence anybody has for a snow line in the solar system, and there is a serious argument that it is not evidence of a snow line’s position.

Dynamical models in which the giant planets migrated substantially — inward and then outward — scatter bodies across the belt from a wide range of original radii, mixing material that formed at two astronomical units with material that formed at ten. On that account the belt’s gradient is a residue of a scattering history rather than a fossil of a thermal boundary, and the radius at which the stony and carbonaceous populations change over is set by dynamics.

The two readings make different predictions about the width of the transition and about whether the two populations overlap in composition at the boundary. The observed transition is broad and the populations do overlap, which favours the mixing picture; and a moving line also broadens the transition, which is the reading this essay has been developing.

Separating a boundary that moved from a population that was stirred requires a tracer that records one and not the other. Isotopic dichotomies between meteorite groups are the current candidate, and the ages of the asteroid families themselves bound how much later the stirring happened, and what they show — two reservoirs that stayed separated for several million years — is a constraint on the mixing rather than on the line.