Astronomy & Celestial Mechanics
the sky, computed
Generator

The lindblad-resonances generator

Where a 2-armed pattern is allowed to exist in the Milky Way
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.

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.

3 essays call lindblad-resonances. 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.

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. Galaxies

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.

 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. Galaxies

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.

 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. Galaxies

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.

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