Astronomy & Celestial Mechanics
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Rotation curves — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · galaxies
  2. Three mass models, 2.3× apart in the disc, agreeing to 3.1 km/s everywhere. NGC 3198's rotation curve, decomposed three ways. The stellar disc has been scaled by 0.2, 0.65, 1.1 times its photometric mass, and for each scaling the halo's asymptotic speed and core radius have been fitted — not chosen — to reproduce the same total. The heavy curve and the marked points are that total: the three models agree with it to 3.08 km/s at every radius, well inside a measurement error of 4.5. The light curves below are the disc's own contribution, and at 17 kpc they differ by a factor of 2.3 — from 36 to 85 km/s — with the halo taking up exactly the slack, 126 down to 97. The curve is one function and the decomposition asks for two. The free parameter is the mass-to-light ratio of the stars, which the kinematics never measures, and it is why a "maximum disc" fit and a halo-dominated fit are both published for the same galaxy. The degeneracy does have one hard edge: scaling the disc to 1.35 times its photometric mass cannot be fitted by any halo in the family — the best leaves 5.6 km/s rms — because past the maximum-disc solution the stars alone already overshoot the curve and a halo cannot have negative mass. That is the one thing a rotation curve says about M/L on its own, and it is an upper limit. What separates the rest has to come from somewhere else: the vertical velocity dispersion of the disc, which weighs the stars alone; gas-rich dwarfs where there is scarcely a disc to argue about; or the baryonic Tully–Fisher relation, which ties the halo's speed to the baryons and would be a coincidence if the two were independent.

    The same curve, two galaxies

    A rotation curve is one function of radius. Decomposing it asks for two — how much of the speed is the stars and how much is the halo — and the mass-to-light ratio that trades one against the other is not measured by anything in the kinematics. A maximum disc and a halo-dominated fit run through identical points.

    part 2 · galaxies
  3. A disc weighed by how far its stars rise. The share of the circular speed at 2.2 disc scale lengths supplied by the disc, against the vertical velocity dispersion of its stars measured at 1.5 scale lengths, for a disc of scale length 3 kpc whose rotation curve reaches 200 km/s, with scale heights of 0.30, 0.41, 0.60 kpc. The share grows linearly with the dispersion, because the surface density goes as its square and the disc's speed as the root of the surface density. A maximal disc, supplying about 85 per cent of the speed (dashed), needs a dispersion of 44 km/s at the middle thickness. A measured 24 ± 3 km/s (shaded) gives 0.41–0.52: a sub-maximal disc, with the halo supplying most of the speed even at the disc's own peak.

    A disc weighed by how far its stars climb

    The rotation curve cannot say how much of a galaxy's speed belongs to its disc, because any split between disc and halo fits it equally well. The stars' motions perpendicular to the disc can, with no halo in the argument at all — and when that was done for thirty face-on spirals the discs came out light. But the vertical weighing needs the disc's thickness, which a face-on galaxy hides, and it needs the dispersion of the stars that hold the disc up, which the light measured may not belong to.

    part 3 · galaxies

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