Generator

The spiral-pattern generator

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.

4 essays call spiral-pattern. 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 Milky Way measured from inside it — the tangent-point construction. Left: the disc in plan, with the Sun at 8.2 kpc from the centre and four lines of sight at galactic longitudes 20°, 40°, 60°, 80°. Each one grazes a circle of radius R₀ sin l, and at that tangent point the whole circular velocity lies along the line of sight, so the largest velocity in the spectrum belongs to a radius the geometry fixes and no distance has to be measured. Right: the four radii and speeds that yields, on the model curve they are read against. The construction reaches only radii inside the Sun's, which is why the outer curve — the half of it that carries the argument — needs distances after all.

Galaxies

A galaxy measured from inside it

The Milky Way is the one galaxy nobody can photograph, and the only one whose rotation can be measured without knowing a single distance. A line of sight at a chosen longitude grazes a circle of known radius, and the fastest gas along it is orbiting there.

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.

A field strength read off the scatter of directions. The Davis–Chandrasekhar–Fermi estimate: field strength against the dispersion of polarisation angles across a cloud, at a density of 10⁴ molecules per cubic centimetre and a turbulent velocity of 0.9 kilometres a second. The polarisation directions themselves contain no field strength — an aligned grain reports which way the field points and nothing about how hard it is pulling. The strength is in the disorder. Turbulence of a given energy bends a stiff field less than a weak one, so the angular scatter is the ratio of the turbulent velocity to the Alfvén speed, and inverting it gives the field: 220 µG at a 9-degree dispersion. The relation is exactly inverse, so the estimate is most trustworthy where the field is most ordered and worst where it matters most. The factor of 0.5 in front is not theory — it is what simulations of a known field, analysed this way, turn out to need, and the honest statement is that this method is calibrated rather than derived.

Galaxies

A direction measured by something with no strength in it

An interstellar grain spins with its long axis across the magnetic field, so starlight through a cloud is polarised along the field and the cloud's own emission across it. The observable contains no field strength whatever — and the strength is recovered anyway, from how much the directions disagree.

Two geometries, one bit of data, and the bit decides. Rotation measure against galactic longitude for two field geometries that look identical in every image ever taken. An axisymmetric field runs the same way round the disc at every azimuth, so the component along the line of sight changes sign exactly 2 times as the longitude goes round — once where the field turns toward the observer and once where it turns away. A bisymmetric field reverses its own direction from one side of the galaxy to the other, and that doubles the count to 4. Nothing about the brightness, the arms or the colour distinguishes the two; the sign of a rotation measure does, and a sign is one bit. The measurement is made against several hundred background sources, each contributing one rotation measure through the whole disc — the sources are not the object of study, they are the illumination. The Milky Way's own answer is untidy: broadly axisymmetric with at least one reversal inside the solar circle, which is neither clean case, and is why the field's origin is still argued about.

Galaxies

Two geometries and one bit to choose between them

A galaxy's large-scale magnetic field either runs the same way round the disc everywhere or reverses from one side to the other. No image distinguishes the two. The sign of a rotation measure does — two reversals around the sky for one geometry and four for the other — and a sign is one bit of information.

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