Astronomy & Celestial Mechanics
the sky, computed
Concept

Rotation curve — where it appears

The circular speed of a galaxy's material plotted against radius. Its failure to fall at large radii is the classic evidence for dark matter, and its inner part constrains a mass distribution only once the stars' contribution has been assumed.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not.

A rotation curve that refuses to fall

Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure.

galaxies · Rotation curves Three mass models, one rotation curve. Rotation curves for three decompositions of the same galaxy, from a disc contributing 30 per cent of the outer rotation to one contributing 95. Each is the quadrature sum of a stellar disc, whose shape is fixed by the light distribution and whose amplitude is an unknown mass-to-light ratio, and a dark halo with parameters of its own. All three reproduce the same flat outer rotation, because the halo's amplitude is adjusted to make up whatever the disc does not supply. They differ in the inner few kiloparsecs, by 27 kilometres a second here — which is more than the measurement error and less than the uncertainty in the disc's own contribution, since that depends on a mass-to-light ratio nobody measures directly. The maximum-disc assumption picks the largest disc consistent with the data, and it is a convention rather than a result.

Three mass models that fit the same curve

A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.

galaxies · Dark matter One velocity, two distances — 4.7 and 9.5 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 30 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 84 kilometres a second is at either 4.7 or 9.5 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy.

One velocity and two distances

Inside the Sun's orbit a line of sight crosses each galactocentric radius twice, and the two crossings have identical radial velocities. So a cloud's velocity gives two candidate distances, near and far, differing by a factor — and nothing about the velocity says which, though the choice decides whether the object is ordinary or extraordinary.

galaxies · Galactic structure Five speeds from the same galaxies, and the relation each one gives. The exponent and the scatter of the baryonic Tully–Fisher relation fitted to the same 90 model galaxies with five different speeds, fitted as speed on mass. outer speed: exponent 3.98, scatter 0.080 dex in mass; peak speed: exponent 3.86, scatter 0.110 dex in mass; at 2.2 scale lengths: exponent 3.29, scatter 0.167 dex in mass; W50 ÷ 2: exponent 3.73, scatter 0.119 dex in mass; W20 ÷ 2: exponent 4.05, scatter 0.118 dex in mass. The galaxies were built with the relation in their outer halo speed, and the other four speeds each lose some of it: the ones read from the inner curve inherit how concentrated each galaxy's stars are, and the line widths add the turbulence of the gas, which matters most in the slowest galaxies.

The speed a line width stands in for

A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.

galaxies · Tully–Fisher How the scatter depends on how much dispersion is added to rotation. The scatter of the baryonic Tully–Fisher relation, in dex of mass, when the same 80 turbulent model discs are measured with S = √(K Vᵣₒₜ² + σ²), against the weight K given to the rotation, for rotation measured at 1 and 2.2 disc scale lengths. Measured at 1 scale length the scatter is smallest, 0.076 dex, at K = 0.55, and is 0.076 dex at K = 0.5; measured at 2.2 scale lengths the scatter is smallest, 0.077 dex, at K = 0.23, and is 0.134 dex at K = 0.5. For an exponential disc with constant dispersion the pressure correction is exactly K = one over twice the radius in scale lengths, so the best weight depends on where the rotation is measured. The widely used K = 0.5 is the value that makes a pure rotator and a pure isothermal sphere of the same mass agree; it is also the pressure correction for rotation measured at one scale length, and not at any other.

A disc that turns slower than its mass requires

The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.

galaxies · Tully–Fisher

Named alongside it

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

The baryonic Tully–Fisher relationDark matterMass-to-light ratioVelocity dispersionDegeneracyTully fisher relationThe 21-centimetre lineAsymmetric driftCircular velocityDark haloDark matter haloThe disc–halo degeneracy

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