The speed a line width stands in for

The Tully–Fisher relation connects a galaxy’s mass or luminosity to how fast it rotates, and the derivation of its fourth power treats the rotation speed as a single number. Each galaxy is given one speed, and the relation is a line through the points.

A real galaxy does not offer one speed. It offers a rotation curve — speed against radius — and the curve refuses to fall at large radius, but it does plenty of other things closer in. It rises from zero at the centre, sometimes steeply and sometimes slowly, sometimes peaks and dips before settling, and in the smallest galaxies it may still be rising at the last point measured. Any single number read off it is a choice of where to read, and the relation that comes out depends on the choice.

There are five choices in common use. The outer flat speed, averaged over the part of the curve where it has stopped changing. The peak of the curve. The speed at 2.2 disc scale lengths, where a pure stellar disc’s own rotation would peak, and a favourite of optical studies whose measurements do not reach far out. And two widths of the integrated hydrogen line, measured at 50 and at 20 per cent of its peak height, which are what a single radio dish provides without resolving the galaxy at all.

A galaxy has a curve, not a speed #

The rotation curve is the sum, in quadrature, of what each mass component contributes: the stars, the gas and the dark halo. How the curve is shaped depends on how those contributions are arranged, and the arrangement changes systematically with the size of the galaxy.

The model galaxies are built from a stated recipe. Each has an exponential disc of stars, an exponential disc of gas twice as extended, and a dark halo with a flat core, and the rotation of each component is computed from its mass distribution — the stellar disc by Freeman’s closed form, the halo by the formula for a sphere whose density flattens at the centre. Smaller galaxies are given larger gas fractions, as real ones have, and disc sizes grow with mass.

What changes across the four curves is how much of the inner mass is in stars rather than dark matter. A massive disc galaxy has a dense stellar disc that dominates its inner few kiloparsecs, so its curve rises steeply, overshoots and settles as the halo takes over. A dwarf galaxy has a thin, diffuse disc and is dominated by its halo nearly everywhere, and its curve rises slowly across the whole disc. Which of those components carries how much of a given curve is itself hard to pin down from the curve alone, but the systematic trend with mass is not in doubt, and it is the trend that matters here.

The consequence for the relation is immediate. A speed read at 2.2 scale lengths is 73 per cent of the outer speed in the smallest galaxy and 106 per cent in the largest. Across three decades of mass it is a different fraction of the outer speed by a factor of nearly one and a half, and that factor enters the relation as a tilt.

What a single dish measures #

A radio telescope pointed at a galaxy that it does not resolve records the twenty-one centimetre emission of all the galaxy’s hydrogen at once, spread in frequency by the Doppler shift of each parcel of gas. The resulting profile is a histogram of line-of-sight velocities weighted by where the gas is.

The double-horned shape is the signature of a flat rotation curve. A ring of gas turning at speed $V$ puts its emission at line-of-sight velocities $V\cos\phi$, and a uniform spread in $\phi$ piles most of the emission near $\pm V$. When many rings share nearly the same speed, their horns stack. The width between the horns, measured at some fraction of the peak height, is then close to twice the flat speed.

Close to, and not equal. The profile’s edges are softened by the gas’s own random motions, a few to ten kilometres a second in the neutral hydrogen of a galactic disc, and the softening pushes the level at which the profile crosses half its peak outward. In the giant galaxy above, half the 50 per cent width overshoots the outer flat speed by about three per cent. At 20 per cent of the peak, lower down the softened edge, the overshoot is seven per cent.

For a galaxy that rotates at a few hundred kilometres a second, a few kilometres a second of turbulence is a small correction. For a dwarf it is not.

In a dwarf the two effects that separate a width from the outer speed pull in opposite directions. The rotation curve is still rising through much of the gas, so a large share of the emission comes from gas turning slower than the outer speed, which narrows the profile. The turbulence broadens it. At the 50 per cent level these nearly cancel in this galaxy; at the 20 per cent level the broadening wins. Neither width measures the outer speed; each measures a gas-weighted mixture of speeds, blurred by motions that are not rotation at all. Real surveys subtract an estimate of the turbulent broadening before using a width, and the size of that correction for the slowest galaxies is one of the larger uncertainties in the whole method.

The relation placed in one speed #

To see what each choice of speed does, the model population puts the relation in a known place and then reads it back. Ninety galaxies are built, spread evenly in logarithm from 3 × 10⁷ to 3 × 10¹¹ solar masses of stars and gas. Each galaxy’s halo is adjusted so that its outer flat speed lies on the baryonic relation $M = 47\,V^4$, with a small scatter of 0.08 dex in mass. Disc sizes and halo cores are then scattered at fixed mass, by 0.15 and 0.2 dex, as the sizes and concentrations of real galaxies are.

The construction says outright that the outer speed is the fundamental one. That is an assumption, and it is made deliberately, because what the population then tests is how much of a relation placed in the outer speed survives being read with a different speed. If a different choice turns out to give a tighter relation in real galaxies, the assumption is wrong; if every other choice degrades it in the way the model predicts, that is evidence the assumption is right.

The fit returns what was built, and the scattered disc sizes and halo cores do not show up at all. That is the point of the outer speed. Out at five to eight scale lengths the stellar disc’s contribution has fallen away and the gas and halo dominate, so how compact a particular galaxy’s stars happen to be leaves almost no mark on the speed there.

Reading closer in #

Optical rotation curves, measured from the emission of ionised gas along a slit, typically reach only a few disc scale lengths before the light runs out. A widely used convention for them is the speed at 2.2 scale lengths, the radius at which an exponential disc’s own contribution to rotation peaks.

The tilt is the trend in the curves figure turned into a slope. Dwarfs are still rising at 2.2 scale lengths, so their speeds there fall short of their outer speeds; giants have peaked by then, so theirs exceed them. Low-mass points slide left and high-mass points slide right, and the line through them rotates to a shallower exponent. The extra scatter is the scattered disc sizes and halo cores coming through: at 2.2 scale lengths a compact disc and a diffuse one of the same mass turn at noticeably different speeds.

Nothing in this is a defect of optical data. It is a statement about what a speed at 2.2 scale lengths measures, which is the combined mass inside the part of the galaxy where the stars matter most. The relation read that way is a real relation, but it mixes the baryonic relation with the relation between a galaxy’s mass and how concentrated its stars are, and it inherits the scatter of the second.

Reading the width #

The width sits in between, because the profile is weighted by where the gas is, and the gas extends further out than the stars. Much of the emission comes from the flat part of the curve, but not all of it, and in the smallest galaxies the rising inner curve and the turbulence both leave their mark.

The 20 per cent width is a cautionary case. Fitted on its own it gives an exponent of 4.05, the closest of the four alternatives to the value that was put in. It does so because two errors happen to cancel: the rising curves of dwarfs pull their widths down and flatten the relation, and turbulence pushes the same widths up by a larger fraction in the slowest galaxies and steepens it. Its scatter, 0.118 dex, is still half again that of the outer speed. An exponent near four is therefore no certificate that a speed is the right one; the scatter says more than the slope.

The peak, and what a bulge would do to it #

The peak of the rotation curve is the most tempting alternative to the outer speed, because it needs no decision about where a curve has become flat: it is simply the largest speed measured. In the model it gives an exponent of 3.86 and a scatter of 0.110 dex, closer to the outer speed than any other choice. That is because most of the model’s curves have no pronounced peak. In the smaller galaxies the largest speed is the last one measured, which is close to the outer speed by construction, and only in the most massive, most compact discs does the curve rise to a clear maximum and fall back, by up to eight per cent.

Real massive spirals are less forgiving, for a reason the model leaves out. Many have a central bulge, a concentrated spheroid of old stars that adds rotation at small radii. A bulge can lift the inner curve well above the outer speed, so that the peak is set by how much mass sits in the central kiloparsec rather than by the galaxy as a whole. The effect is largest in the most massive galaxies, which are the ones most likely to have prominent bulges, so it pushes their peak speeds up and flattens the relation at its high-mass end. A relation read with the peak speed therefore inherits one more property of galaxy structure, the bulge-to-disc ratio, on top of the concentration of the disc itself, and it is looser in real samples than in a population built without bulges.

Five relations side by side #

In the model the ordering is built in by the assumption that the relation lives in the outer speed. The interesting comparison is with real galaxies, and it has been made. A sample of about 175 nearby disc galaxies with resolved hydrogen rotation curves and near-infrared photometry allowed the same speeds to be read off the same objects, and the outer flat speed gave the tightest baryonic relation of all of them, with the speeds read closer in giving shallower and looser relations in the way the model describes.

That observation carries more weight than it first appears to. The outer speed is the one the stars and gas contribute least to. Out at the edge of the hydrogen, most of the mass inside the orbit is the mass that is not the light, and yet it is there — where the baryons matter least to the dynamics — that the baryonic mass predicts the speed best. The inner speeds, set largely by the stars themselves, predict it worse. Whatever makes the relation tight is not the stars’ own gravity; it is a connection between the total baryonic mass and the halo that holds it, and the choice of velocity definition is how that is known.

Consequences for a distance #

A distance indicator has to be cheap to apply to thousands of galaxies, and that favours the single-dish width over a resolved rotation curve, which is expensive. The model shows the price.

The first cost is scatter: a relation with 0.119 dex of scatter in mass instead of 0.080 gives distances about half as uncertain again for each galaxy. The second is worse, because it does not average away. The relation read with a width has a different slope from the relation read with the outer speed, and the difference is not uniform with mass. A calibration built on nearby giant spirals and applied to a distant sample weighted towards smaller galaxies, or the reverse, carries a systematic error that depends on what kind of galaxies the sample holds. The rule that follows is simple and easy to break: calibrate with exactly the definition, the same turbulence correction and the same range of galaxy masses that the distances will be measured with.

The inclination correction, the other large error term, does not help. It divides the width by the sine of the inclination, so it scales every speed definition by the same factor and cannot separate them. Correcting a galaxy’s orientation perfectly still leaves it with the wrong kind of speed.

There is also a lesson about galaxies that fall off the relation. A galaxy whose surface brightness is unusually low has a diffuse, halo-dominated disc, so its rotation curve rises slowly, and its speed at 2.2 scale lengths falls well short of its outer speed. Read with an inner speed it appears to lie off the relation, too slow for its mass. Read with its outer speed it lies on it. Part of the long history of low-surface-brightness galaxies seeming to be exceptions is a history of reading them at the wrong radius, and part of the evidence that they are not exceptions is that their curves scale onto the same relation once the outer speed is used.

What the model does not decide #

Three things in the construction are choices, and each would move the numbers without changing the ordering. The halo was given a flat core; a halo with a density cusp would add rotation at small radii and change the inner curve most in the smallest galaxies, where the argument about cores and cusps is concentrated. The gas disc was made twice as extended as the stars; real hydrogen discs extend further still, which would push the widths closer to the outer speed. And disc sizes were tied to mass by a single power law with a single scatter, while real galaxies of the same mass range from compact to extremely diffuse.

None of those choices affects the central point, which is that different velocity definitions sample different parts of galaxies whose structure changes with mass. The relation is not one relation measured in five ways but five relations, related to one another by galaxy structure. Which of them is tightest in real galaxies is an empirical question, and its answer — the outer speed — says the relation belongs to the potential at the edge of the galaxy, not to the disc of stars within it.

Still open: does the relation stay flat beyond the gas? #

The outer speed is measured only as far as the hydrogen extends, typically a few tens of kiloparsecs. Beyond that, a halo of the kind cosmological simulations produce should eventually show a declining rotation curve, and where it starts to decline should differ from galaxy to galaxy. Weak gravitational lensing of background galaxies can weigh the mass around a galaxy far beyond its gas, and recent analyses stacking many lens galaxies have reported implied circular speeds that remain flat, and still on the baryonic relation, out to hundreds of kiloparsecs — much further than such halos should allow. Stacking mixes galaxies of different masses and environments, the conversion from a lensing signal to a speed assumes a spherical mass distribution, and nearby neighbours add signal that has to be modelled away. Whether the flatness is real, and whether the relation that is tightest at the edge of the hydrogen is still the relation at ten times that radius, is not yet settled.