Astronomy & Celestial Mechanics
the sky, computed
Concept

Lindblad resonance — where it appears

A radius in a disc where a particle's epicyclic frequency beats against a rotating pattern at the rate that makes the forcing coherent. Waves are launched there, and in a planetary ring the wave's chirp rate measures the ring's own surface density.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A wave whose crests count out 35 kilograms per square metre 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 5.6 km and has shortened to 1.13 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 kilograms per square metre 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.

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.

gravitation · Planetary rings Where a 2-armed pattern is allowed to exist in the Milky Way. Angular speed against radius, with the two combinations that matter for a pattern of 2 arms. A wave turning at 25 km/s per kpc corotates with the stars at 8.7 kpc, has an inner Lindblad resonance at 1.9 kpc and an outer one at 13.8 kpc. Between the Lindblad radii the wave propagates; at them it is absorbed. Every one of these radii is found by bisection on the curves as drawn, not quoted — and none of them is a property of the pattern alone, because κ comes from the rotation curve, which comes from the mass.

Where a pattern is allowed to turn

A spiral wave has one pattern speed, and that single number decides where in the disc it can exist at all. The boundaries are set by the frequency at which a star rocks radially about its circular orbit — a frequency the rotation curve already contains.

galaxies · Spiral structure 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 An arm that moves angular momentum outwards, and is spent doing it. Where a 2-armed spiral pattern of 25 km/s per kiloparsec takes angular momentum from the disc and where it gives it back, against galactocentric radius, for the Milky Way's rotation curve. The resonances are found on that curve rather than placed: the inner Lindblad resonance at 1.91 kiloparsecs, corotation at 8.67, the outer Lindblad resonance at 13.80. The lower curve is the angular momentum removed per unit radius and the upper one what the wave delivers; between them runs the flux the wave carries, which is flat across the whole region where nothing is resonant, because a wave that is not interacting with anything simply travels. The two deposits are required to cancel to a part in a million — the pattern is a conveyor and keeps nothing — so the disc's total angular momentum is unchanged while its distribution is not. That is the sense in which a spiral is a machine: it moves mass inwards by moving angular momentum outwards, which is the same operation an accretion disc performs and the same one a protostellar disc must perform to make a star. The cost is paid in random motion. Every exchange heats the stellar disc, a hotter disc supports a weaker wave, and the pattern that did the work is the thing the work destroys. What the figure cannot show is the pattern's own lifetime, which is the unsettled part of the subject.

An arm that is undone by the work it does

A spiral pattern takes angular momentum from the inner disc and delivers it to the outer, which lets mass move inwards without violating anything. It is paid for in random motion, and a disc with too much random motion cannot carry a wave — so the pattern destroys the conditions it needs to exist.

galaxies · Spiral structure

Named alongside it

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

Angular momentumAngular momentum transportDensity waveEpicyclic frequencyPattern speedResonanceSurface densityBar instabilityCorotationCorotation resonanceDisc heatingDispersion relation

All concepts