Concept

The disc–halo degeneracy — where it appears

The freedom to reproduce one rotation curve with a heavy stellar disc and a light halo or the reverse, because only their sum is measured. It is broken by measurements of the disc's own surface density, and it dominates every claim about a halo's inner profile.

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

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.

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

Three initial mass functions, normalised to the same light. The number of stars per logarithmic interval of mass, on a logarithmic axis, for three initial mass functions — Kroupa, Salpeter, bottom-heavy — each normalised to the same number of stars above one solar mass, so that the stars which make an old population's light are the same in each. They differ only below a solar mass. In a 10 billion year old population the stars below half a solar mass hold 62 per cent, 78 per cent, 91 per cent of the living stellar mass, and they give 15.8, 24.2, 41.7 per cent of the light of the main sequence, which is itself a third of the population's. The difference between them is almost entirely mass that emits almost nothing.

Stars that weigh and do not shine

Most of the mass in an old stellar population is in stars too faint to contribute to its light. How many of them there are is set by the mass function stars are born with, and the mass-to-light ratio every dark-matter measurement divides by can be wrong by a factor of two without a single colour changing.

galaxies · Dark matter

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.

galaxies · Rotation curves

Named alongside it

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

Mass-to-light ratio Maximum-disc Stellar population synthesis The baryonic Tully–Fisher relation Degeneracy Initial mass function Rotation curve Surface density Velocity dispersion Vertical velocity dispersion The core–cusp problem Dark matter halo

All concepts