Astronomy & Celestial Mechanics
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Series

The snow line — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The ice line at 2.7 AU, and the 3.4-fold jump in solid material across it. Two temperature thresholds turned into radii, against stellar mass, both axes logarithmic. The shaded band is the habitable zone, where water can be liquid on a planet's surface. The heavy line is the snow line of the disc the planets formed in — the distance at which a passively heated disc, whose temperature falls as the inverse square root of radius, reaches 170 K and water freezes. For a solar-luminosity star it sits at 2.71 AU, just outside the asteroid belt, and it is 1.6 times further out than the outer edge of the habitable zone; around a 0.15 solar-mass star both have moved inwards and the ratio is 5.5. What makes the line matter is what happens as it is crossed. Water is by far the most abundant condensable material after hydrogen and helium, so freezing it raises the surface density of solids by roughly 3.4 times in one step. Everything about the architecture of a planetary system follows from that step: a core massive enough to capture gas can be assembled outside the line and not inside it, which is why the solar system has small rocky planets in and giant ones out, and why a giant planet found at 0.05 AU is a statement about migration rather than about formation. The line is drawn where a mature disc puts it; a young, accreting disc is hotter and its line is several times further out, sweeping inwards as the disc drains.

    The line beyond which ice counts as rock

    A disc of gas around a young star gets colder outwards, and at about a hundred and seventy kelvin water stops being vapour and becomes a building material. Crossing that one line multiplies the solid mass available by roughly three and a half, in a single step, and the architecture of every planetary system is downstream of it.

    part 1 · exoplanets
  2. The solid mass available, as a staircase of 5 fronts. The share of the condensable material that is solid, against distance from a 1 solar-mass star, on a logarithmic radius axis. Each riser is one species freezing out, at the radius where the disc's temperature — falling as the inverse square root of the distance — reaches that species' condensation point. silicates and iron at 1400 K and 0.04 AU; water ice at 170 K and 2.71 AU; carbon dioxide at 70 K and 16.00 AU; methane and ammonia at 30 K and 87.11 AU; carbon monoxide at 20 K and 196.00 AU. Inside every front the solid surface density is 22 per cent of what is available, which is the refractories alone; water alone contributes 53 per cent, more than twice everything else combined, which is why one of these steps is called the snow line and the others are not. A body's composition is decided by which pair of risers it formed between, and the steps are narrow because a vapour pressure is exponential in the inverse temperature.

    There is not one line, there is a staircase

    Water freezing is the biggest step in a protoplanetary disc and it is one of five. Each condensable species has its own temperature and therefore its own radius, and what a body is made of is decided not by which side of a line it formed on but by which pair of risers it formed between.

    part 2 · exoplanets
  3. The ice line sweeps from 6.4 to 2.7 AU while the disc drains. The radius at which a disc around a 1 solar-mass star reaches 170 K, against the disc's age, both axes logarithmic. The temperature is the fourth root of the sum of two fluxes: starlight, which does not change, and the disc's own accretion, which releases gravitational energy at a rate set by how fast material is flowing inward. The accretion rate decays as the disc drains — taken here as 5e-5 solar masses a year falling off as t^(−3/2) beyond 0.1 million years — so the viscous term fades and the line sweeps in. It starts at 6.38 AU and ends at 2.73, the passive value the closed form gives. A body at three astronomical units formed dry if it formed early and icy if it formed late, so a composition dates a formation rather than locating one, and the dating is only as good as the accretion history assumed.

    A composition that dates a formation rather than placing it

    The ice line in a young disc starts six astronomical units out and sweeps inward to under three as the disc drains. A body at four astronomical units therefore formed dry or wet depending only on when — so what it is made of is a clock, and reading it as a map is the mistake the moving line makes easy.

    part 3 · exoplanets
  4. A carbon-to-oxygen ratio that steps at every front. The carbon-to-oxygen ratio of the gas and of the solids against distance from a 1 solar-mass star, each in units of the star's own ratio. Crossing a condensation front moves one element or both out of the gas and into the solids, so the two curves step in opposite directions at water at 2.7 AU, carbon dioxide at 16.0 AU, carbon monoxide at 196.0 AU. Water takes oxygen and no carbon, so beyond it the gas is carbon-rich — 3.20 times the stellar ratio — and the solids are oxygen-rich. Carbon monoxide takes both, in a ratio of one to one, so beyond that front the gas ratio rises again. A giant planet's atmosphere is made mostly of gas it accreted, so measuring its ratio and inverting this staircase gives a formation radius — and the inversion is not unique, because more than one interval returns the same value once the solids a planet also swallowed are allowed for.

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

    Carbon and oxygen freeze out at different places, so the gas between the fronts is carbon-rich and the solids are oxygen-rich. A giant planet is made mostly of gas it accreted, so measuring the ratio in its atmosphere and inverting the staircase should give the radius it formed at — and the inversion turns out not to be unique.

    part 4 · exoplanets
  5. The deuterium in the solar system's water, body by body. The ratio of deuterium to ordinary hydrogen in water, on a logarithmic axis, for the Earth's oceans and for the bodies it has been measured in, with the protosolar hydrogen and the water around young protostars as the two ends of the range. The oceans, 1.6·10⁻⁴, are 7.8 times the protosolar value; the carbonaceous and enstatite chondrites sit within ten per cent of them, as do two comets from the Jupiter family. The comets from the Oort cloud, and 67P from the Jupiter family, are two to three and a half times richer. Water made in a cold molecular cloud is richer still, near 0.001. The ratio is fixed when the water forms or last exchanges with hydrogen gas, and is not changed by anything that happens afterwards except escape, so it is a label that each body's water carries from wherever it was made.

    Water that remembers how cold it was

    Every water molecule in the solar system carries a label: the share of its hydrogen that is deuterium, fixed when the water last exchanged with hydrogen gas. Water made in the cold cloud the Sun formed from is rich in it; water that passed through the hot inner disc is poor. The Earth's oceans sit within ten per cent of the carbonaceous meteorites and twice below most comets — a record of where the water was assembled, and of how much of it had ever been hot.

    part 5 · exoplanets
  6. The mass at which a growing core cuts off the pebbles drifting past it. The pebble-isolation mass — the mass at which a core perturbs the gas around it enough to raise a pressure maximum outside its orbit, where drifting solids stop — against distance from the Sun, for a passively heated disc whose thickness grows as the distance to the 2/7, with α = 10⁻³. The mass goes as the cube of the disc's aspect ratio, because a planet has to push the gas by a fraction of its own pressure, and so it grows outward: 7 Earth masses at 1 AU, 30 at 5 AU where Jupiter's core formed, 138 at 30. A core reaching this mass stops growing from pebbles, starts accreting gas, and becomes a dam: everything drifting inward from outside its orbit is held there.

    A core that shut the door on the pebbles

    Solids in a gas disc drift inward, and a growing planet eats them as they pass until it is heavy enough to bend the gas around it. Then a pressure maximum forms outside its orbit, the drift stops there, and everything beyond is cut off from everything within. Jupiter's core reached that mass early, and the solar system's meteorites still fall into two families that never mixed.

    part 6 · exoplanets

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