Astronomy & Celestial Mechanics
the sky, computed
Concept

Pattern speed — where it appears

The angular rate at which a spiral or a bar rotates as a rigid figure, independent of how fast the material rotates. It is what makes a density wave persist, and where it equals the material rotation rate defines corotation.

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

The winding problem, drawn. A straight spoke of stars in the Milky Way from 2 to 15 kpc, and the same stars after 0.15, 0.3, 0.6 Gyr. Nothing has been done to them except let them orbit: the star at radius R has turned through Ω(R)t, and Ω falls as 1/R wherever the rotation curve is flat, so the inner end laps the outer one. By 0.6 Gyr the arm has a pitch angle of 3.4° at 8 kpc, which is tighter than any spiral galaxy that has ever been photographed, and the disc is thirteen gigayears old rather than 0.6.

An arm that cannot be made of stars

A galaxy's disc turns differentially, so any feature made of a fixed set of stars is wound to invisibility within a couple of rotations. Spiral arms are ten billion years old and still open, which means an arm is a place rather than a thing.

galaxies · Spiral structure 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 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 The orbit that turns across the bar. The radial displacement with which a nearly circular orbit answers a weak rotating bar, against radius in kiloparsecs, in linear theory, scaled to its largest value: positive means the orbit's long axis lies along the bar, negative across it. The disc's rotation curve rises within 0.35 kpc to 220 km/s, the bar ends at 3 kpc and turns with corotation at 3.6 kpc, and the disc then has two inner Lindblad resonances, at 0.37 and 0.78 kpc (vertical lines). Without dissipation (dashed) the response diverges at each resonance and changes sign across it: orbits between corotation and the outer resonance are elongated along the bar — the x1 family that builds the bar itself — and orbits between the two resonances are elongated across it — the x2 family. Gas, which dissipates (solid, damping 8 km/s/kpc), cannot follow the flip abruptly; its orbits turn through ninety degrees over a band round the resonance, where streams of gas on neighbouring orbits cross, collide and shock. That band is where a nuclear ring forms. Near corotation the linear theory itself fails, and the curves stop short of it.

The bar leaves its gas where the orbits turn

A rotating bar in a disc galaxy drives gas inwards from kiloparsecs out, and the gas does not fall to the centre. It stops in a ring a few hundred parsecs across and forms stars there at rates the rest of the galaxy cannot match. The ring sits at the bar's inner Lindblad resonance, where the orbits that support the bar give way to orbits turned across it — so its existence says there is mass concentrated at the centre, and its size says how fast the bar turns.

galaxies · Spiral structure

Named alongside it

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

Density waveAngular momentum transportCorotationEpicyclic frequencyLindblad resonanceStar formationSwing amplificationAngular momentumBar instabilityCorotation resonanceDifferential rotationDisc heating

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