Astronomy & Celestial Mechanics
the sky, computed
Concept

The snow line — where it appears

The distance from a star beyond which water condenses as ice in a protoplanetary disc. Solid material is several times more plentiful past it, which is why giant planets are expected to form there and why finding one much closer in requires migration.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

An impulse delivered inside 0.5 AU, and a comet 2050 hours early. Above: Marsden's outgassing law, the factor g(r) that scales a comet's non-gravitational acceleration, against distance from the Sun over one orbit of a comet with perihelion at 0.336 AU and aphelion at 4.09. It is close to an inverse square inside the water snow line and then falls off a cliff, because water ice that is not being heated does not sublimate. Half the whole revolution's impulse is delivered inside 0.55 AU — a few weeks out of a 3.3-year orbit — so the force is effectively a kick at perihelion rather than a perturbation spread around the path. Below: what a kick of that kind does to the timekeeping. A transverse component changes the semi-major axis and so the period, by 2.5 hours per revolution here, and a constant change in the period accumulates as the square of the number of revolutions rather than in proportion to it. After 40 returns the comet arrives 2050 hours — more than 85.4 days — before an orbit fitted without the term predicts, and doubling the number of returns multiplies the discrepancy by 3.90. That is why the effect was found in the eighteen-twenties from nothing but arrival times, and a century and a half before anyone photographed a jet.

A comet that arrives a day early

Encke's comet returned two and a half hours ahead of prediction, every revolution, for decades before anybody could say what was pushing it. The force is a rocket — a few tonnes a second of vapour leaving the sunward side of a rotating nucleus, delivered almost entirely in the few weeks around perihelion, and it accumulates in the arrival time as the square of the number of returns.

orbits · Non-gravitational forces Three orders of magnitude in astronomical units, and a factor of 9.6 in mutual Hill radii. Every adjacent pair of planets in three systems, plotted by the separation between them measured in their own mutual Hill radius, ((m₁+m₂)/3M⋆)^⅓ · (a₁+a₂)/2. In astronomical units the same 17 separations span a factor of 2554 — from 0.0043 AU between two TRAPPIST-1 planets to 10.9 between Uranus and Neptune — and carry no visible structure at all. In this unit they span 9.6, with a median of 11.8. The solid line at 2√3 = 3.46 is a theorem: two planets on circular coplanar orbits wider apart than that can never have a close encounter, whatever else happens, and every pair here is clear of it. The dashed line at 10 is a fit, to integrations of systems of several planets over billions of years, and it is the one that bites — a system packed tighter than about ten does not survive, which is why the observed distribution has a floor there and not at the theorem. A planetary system's spacing is not measured in kilometres. It is measured in a unit the planets define.

A feeding zone, and the spacing it forces

The radius that decides what a planet may keep also decides what it could reach while it was growing — and measuring the gaps between planets in that unit turns a distribution spanning three orders of magnitude in astronomical units into a band a factor of ten wide, with a floor that is partly a theorem and partly a fit.

gravitation · Hill sphere A magnification, and a spike inside it. The brightness of a background star as a foreground one passes in front of it. The smooth curve is exact: a point mass magnifies a point source by (u² + 2)/(u√(u² + 4)), which peaks at 6.7 for this track. The spike is the planet, of mass ratio 0.001, lensing one of the two images. Its duration is the Einstein time scaled by √q — about 23 hours against 30 days — so the whole planetary signal is a few hours in an event lasting a month, and it never repeats. The deviation is drawn in the approximation that the planet lenses the image in isolation; a real caustic crossing has structure this smooths over, and its true height is set by the source's size rather than by the geometry.

A star magnified by a planet nobody will see again

Microlensing weighs a planet by the way its gravity bends light around it. The measurement lasts a few hours, cannot be repeated, and is the only one that does not require the planet's star to be visible at all.

exoplanets · Microlensing Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.

A planet where one cannot form

A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in.

exoplanets · Planet migration 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.

exoplanets · The snow line An edge where two torques balance, 21 kilometres from the shepherd. Two torques on the edge of a ring, against distance from a shepherding moon, both axes logarithmic and both scaled by the same combination of surface density, radius and orbital rate so that only their shapes are being compared. The flat line is the viscous torque, which comes from collisions between ring particles and does not care how far away anything is; it is drawn for a kinematic viscosity of 12 square centimetres a second, within the range ring seismology gives. The falling line is the moon's, summed over the first-order resonances that crowd together as the gap narrows, which makes it an inverse cube. A flat curve and an inverse cube cross once, and the crossing is where an edge can sit: closer in the moon wins and pushes the material back, further out viscosity wins and the ring spreads. For Daphnis and the Keeler gap the balance lands 20.8 kilometres out against a measured half-width of 21, which is agreement to well inside the uncertainty on the viscosity — and it is the only handle anybody has on that viscosity, since the quantity being inferred is the collision rate among particles a metre across, a billion kilometres away.

A torque that nearly cancels

A planet embedded in a gas disc pulls on the material inside its orbit and outside it, and the two torques are almost equal and opposite. What survives the subtraction is a per cent of either, and it is still enough to carry a planet from where it formed to its star in less time than the disc lasts.

exoplanets · Planet migration Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.

Ninety-nine per cent of the mass and none of the spin

The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

orbits · Angular momentum 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.

exoplanets · The snow line 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.

exoplanets · The snow line 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.

exoplanets · The snow line

Named alongside it

The objects these essays reach for when they reach for this one.

Protoplanetary discPlanetesimalCondensation sequenceDisc evolutionEquilibrium temperaturePlanet migrationAngular momentumCore accretionDegeneracyHot jupiterIsolation massSolid surface density

All concepts