Astronomy & Celestial Mechanics
the sky, computed
Generator

The ring-profile generator

A wave whose crests count out 35 grams per square centimetre of ring
A wave whose crests count out 35 grams per square centimetre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring.

Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring.

6 essays call ring-profile. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

A wave whose crests count out 35 grams per square centimetre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring. Gravitation

A ring weighed by the wave crossing it

Saturn's rings are a few tens of metres thick and spread over an area larger than the Earth, made of pieces nobody can resolve, and nothing has ever landed on them. Their mass per unit area is nevertheless known to a few per cent — from the rate at which the crests of a wave crowd together as it travels outwards.

 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. Gravitation

An edge is a balance, not a boundary

A ring of particles spreads, because collisions move angular momentum outward. Something has to push back, and the something is a small moon whose torque falls as the inverse cube of the gap. A flat curve and an inverse cube cross once, and the crossing is the sharp edge — which is why an edge exists at all rather than a gradient.

 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. Exoplanets

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.

 A resonance with the planet's rotation, and it sorts by size. Above: the radii at which a charged grain's orbital frequency is commensurate with the planet's spin, in planetary radii, with the measured edges of Jupiter's halo ring drawn over them. The synchronous radius is 2.24; the 3:2 resonance falls at 1.71 and the 2:1 at 1.41. The halo's outer boundary is at 1.71 and its inner extent near 1.4 — the ring ends where the resonances are, and both numbers were measured by a spacecraft camera with no reference to this arithmetic. Below: which grains care. The charge on a grain is proportional to its radius and the mass to the cube, so the force per unit mass goes exactly as the inverse square of the size, and the resonance grips sub-micron dust while leaving anything larger on a Keplerian orbit. That is why the halo is a cloud of fine dust puffed a thousand kilometres out of the ring plane while the coarse material stays flat: the same field acting on the same orbit sorts the material by size, which no gravitational resonance can do. Orbits

A resonance with the planet itself

Every other resonance in celestial mechanics is a commensurability between two orbits. A charged dust grain has a third clock available — the planet's rotation, which sweeps its magnetic field past the grain — and the commensurability with that selects by charge-to-mass ratio, which means by grain size.

 Where a disc pushes a planet outward. The sign of the total torque on a low-mass planet, as a function of the power-law indices of the disc's surface density (Σ ∝ r^−α, horizontal) and temperature (T ∝ r^−β, vertical), from the fitting formulae of Paardekooper and collaborators for an adiabatic disc with γ = 1.4 and the horseshoe torque unsaturated. Above the line the total is positive and the planet migrates outward; below it the planet migrates inward. The line is straight because every part of the torque is linear in the two indices, and it is steep in α because a shallow density profile strengthens the vortensity part and weakens nothing. The minimum-mass nebula (α = 1.50, β = 0.50) sits at a total of −3.76 in units of Γ₀/γ; The viscously heated inner disc (α = 0.60, β = 0.90) sits at a total of +0.74 in units of Γ₀/γ; The irradiated outer disc (α = 1.07, β = 0.43) sits at a total of −2.65 in units of Γ₀/γ. The minimum-mass nebula every textbook starts from is on the inward side by a wide margin, and a disc heated by its own accretion is on the outward side: the same planet can migrate either way depending on how the gas is heated, which the Lindblad torque alone could never say. Exoplanets

The gas that turns round beside the planet

The two spiral wakes a planet raises nearly cancel, and what decides the direction it drifts is a third, smaller torque from the gas sharing its orbit. That gas makes a U-turn in front of the planet and behind it, and whether the turn pushes the planet outward depends on how the disc was heated and on a viscosity nobody has measured.

 One patch of ring, many optical depths. The normal optical depth an observer infers from the starlight transmitted through one patch of the A ring, treating it as a uniform sheet, against the direction of the sight line projected onto the ring plane — measured round from the radial direction — for sight lines 12°, 30°, 60° above the plane. The ring is the slab-and-gap model of the wakes: slabs 33 m wide and 8 m tall every 60 m, tilted 22° from the orbital direction, gaps of optical depth 0.12. The minima are where the sight line runs along the slabs, twice per circle; the maxima where it runs across them, where at low elevation the slabs' shadows close the gaps altogether (curves cut off at 3). At 12° the inferred value runs from 0.29 to opaque; at 60° from 0.81 to 0.97. The dashed line is the same particles spread into a uniform layer, 1.43: every observer along the slabs sees less ring than is there. Gravitation

A ring that is thinner from one side

Saturn's A ring is not a sheet of particles but a field of tilted slabs, each a few dozen metres across, pulled together by the ring's own gravity and sheared apart again within an orbit. Starlight passing through it threads the gaps or meets the slabs depending on its direction, so one patch of ring has a different optical depth for every observer — and a uniform model of it hides most of the particles it contains.

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