A line width that is a distance

A distance indicator needs one thing: an observable that predicts an intrinsic luminosity without needing to know how far away anything is. A Cepheid’s pulsation period is such an observable. So is the width of a spiral galaxy’s twenty-one centimetre line.

The relation between that width and the galaxy’s luminosity was published by Brent Tully and Richard Fisher in 1977, and it is the workhorse of extragalactic distance measurement beyond the range where individual stars can be resolved. What makes it more than an empirical convenience is that its exponent can be derived, from two assumptions and no fitting.

Where the fourth power comes from #

Two assumptions do all the work, and both are observations rather than hypotheses.

Discs have roughly the same central surface brightness. This is Freeman’s law, established in 1970: the extrapolated central surface brightness of a spiral disc clusters around 21.65 blue magnitudes per square arcsecond across a wide range of galaxy size. So if $L$ is the luminosity and $R$ the disc’s scale length, $L \propto \Sigma R^2$ with $\Sigma$ roughly fixed, which gives $R \propto \sqrt L$.

The mass-to-light ratio inside the measured radius is roughly constant. So $M \propto L$, with the constant of proportionality including whatever dark matter is enclosed.

Then the circular speed at the measured radius satisfies

$$v^2 = \frac{GM}{R} \propto \frac{L}{\sqrt L} = \sqrt L,$$

so $L \propto v^4$. In magnitudes — a scale that runs backwards, so that a brighter galaxy has a more negative number — that is a slope of $-10$ magnitudes per decade of rotation speed.

Nothing about galaxy formation enters. The relation is a consequence of two regularities that were themselves discovered empirically, and it works because those regularities hold — which means the scatter in Tully–Fisher is a direct measurement of how well they hold.

The observable the width actually is #

What is measured is not a rotation speed. It is the width of a spectral line integrated over a whole galaxy.

Two corrections stand between that width and $v$.

Inclination. The width measures $2v\sin i$, so a face-on galaxy has a narrow line and an edge-on one a wide line at the same rotation speed. The inclination is estimated from the axis ratio of the isophotes, assuming the disc is circular and has some intrinsic thickness. For a nearly face-on galaxy the correction is a large division by a small number, and such galaxies are simply excluded from the samples.

Turbulence. The gas has an internal velocity spread of eight to ten kilometres per second, which broadens the profile independently of rotation. It is subtracted in quadrature, and for a slowly rotating dwarf it is not a small correction.

The bias in the fitted slope #

The first figure records something that looks like a defect in the machinery and is in fact a property of the method: the fitted slope is not the true one, and the two regressions bracket it.

The construction makes $L \propto v^4$ exactly, so the true slope is $-10.00$. Ordinary least squares of magnitude on log speed returns $-8.45$. The reverse regression, of log speed on magnitude, inverted, returns $-9.15$.

The reason is that the scatter in this family is scatter in the speed at fixed luminosity: surface brightness and mass-to-light ratio both move $v$ and neither moves $L$. Error in the abscissa attenuates a fitted slope towards zero, always, and by an amount that depends on how large the scatter is compared with the range of the data.

That is not a curiosity. A distance indicator is used by inverting the relation — measuring a width, predicting a luminosity — and the slope used for that inversion is the one being estimated here. Using the wrong regression introduces a distance error that is systematic and that varies with the galaxy’s rotation speed, so it bends the whole distance scale rather than shifting it.

The Tully–Fisher literature is consequently careful about which variable is regressed on which, and about the fact that the answer depends on how the sample was selected. In this figure the true value is known by construction, which is the only situation in which a bias of this kind can be seen rather than argued about.

The other bias, which is about the sample #

There is a second systematic and it is a general hazard of distance work.

A survey of galaxies is limited by apparent brightness. At a large distance, only the intrinsically bright members of a population are detected, so the average luminosity of the detected sample rises with distance. Predicting a luminosity from a relation calibrated on nearby galaxies and applying it to a distant sample therefore underestimates their luminosity and their distance.

This is Malmquist bias, and it is the reason distance-scale work is done with volume-limited or carefully modelled samples rather than with everything that turns up. Its correction depends on the intrinsic scatter of the relation, which is why the scatter is quoted as prominently as the slope.

A distance, worked through #

It is worth doing one, because the chain is short and every step is a number.

Take a spiral with a twenty per cent line width of 400 kilometres per second, inclined at seventy degrees, with an apparent magnitude of 11.2 after correction for extinction.

The projected rotation speed is half the width: 200 km/s. Dividing by $\sin 70° = 0.94$ gives 213 km/s, and $\log_{10} 213 = 2.33$.

A calibrated relation of the form $M = -10.0\,(\log v - 2.3) - 21.0$ then gives an absolute magnitude of $-21.3$.

The distance modulus is the difference between the apparent and absolute magnitudes: $11.2 - (-21.3) = 32.5$. And a modulus is a distance through $d = 10^{(m-M+5)/5}$ parsecs, which gives 32 megaparsecs. Every step there is arithmetic except the calibration constant, and that constant is the entire content of the rung below.

Why the rung exists at all #

There is a range of distance in which nothing else works well, and Tully–Fisher owns it.

Cepheids are the gold standard, and they run out. Resolving an individual variable star in another galaxy requires the galaxy to be close enough that the star can be separated from its neighbours; even with a telescope above the atmosphere that reaches some thirty megaparsecs, which contains a few dozen suitable spirals. Type Ia supernovae reach far further, and there is a catch: they are events, not objects, so a given galaxy supplies one about once a century and cannot be measured on demand.

A line width can be measured for any spiral bright enough to detect in hydrogen, at any time, in an afternoon. That is why Tully–Fisher distances exist for thousands of galaxies while Cepheid distances exist for dozens — and why the relation is the backbone of every map of the local peculiar-velocity field, which needs distances for a large sample rather than superb distances for a few.

The observation behind the number #

Tully and Fisher’s original sample was small — a couple of dozen galaxies in nearby groups — and the relation was immediately controversial, partly because the scatter looked too small to be believable.

Two things sharpened it since. The first was moving to the near infrared. The mass-to-light ratio of a stellar population is much more nearly constant in the infrared than in blue light, because the infrared is dominated by old low-mass stars while the blue is dominated by whatever has formed recently. The second assumption above therefore holds better, the scatter falls, and the measured slope steepens towards the predicted $-10$.

The second was replacing the single-dish line width with resolved rotation curves, so that a genuine flat rotation speed is measured rather than a profile width. This removes the turbulence correction and much of the inclination uncertainty, and it turned up something better still.

What is really being measured #

Replacing luminosity with baryonic mass — stars plus gas, since gas is a large fraction in the smallest galaxies — produces a relation that is tighter and straighter than the luminous one, over five decades of mass:

$$M_{\text{baryonic}} \propto v_{\text{flat}}^4.$$

This is the baryonic Tully–Fisher relation, and it is one of the most striking empirical regularities in extragalactic astronomy. Its tightness is a genuine puzzle for galaxy formation, because the visible mass is a small and variable fraction of the total and there is no obvious reason why it should track the halo’s rotation speed so exactly.

Two corrections that share a variable #

The inclination correction was described above as the largest contributor to the scatter. There is a reason it is worse than a single division by a sine, and it is worth setting out because it is the kind of error that does not average away.

A spiral galaxy is not transparent. Its own dust absorbs and scatters light, and how much depends on how much of the disc the light has to traverse — which depends on the inclination. A face-on galaxy is seen through one scale height of dust; an edge-on one through many, and the internal extinction can exceed a magnitude in the blue.

So the apparent magnitude fed into the relation needs a correction that is a function of inclination, at the same time as the line width needs a correction that is a function of inclination. The two corrections use the same estimated angle, so an error in that angle moves both — and it moves them in the same direction, because an overestimated inclination simultaneously reduces the deprojected rotation speed and over-corrects the extinction, making the galaxy appear both slower and brighter than it should.

An error that moves a point along the relation rather than across it is far less damaging than one that moves it perpendicular, and this one does neither cleanly: it moves points along a direction set by the ratio of the two correction slopes, which is not the relation’s own slope. The result is a contribution to the scatter that is correlated between the two axes, so the ordinary treatment of errors in a fit — independent uncertainties on each coordinate — understates it.

The response has been to move to the infrared for the photometry, which reduces the internal extinction correction by a factor of several and thereby decouples it from the width correction. That is a second and independent reason for the near-infrared calibration described above, alongside the mass-to-light argument, and the two reinforce each other: the same change of band makes one assumption hold better and makes one systematic smaller.

Where the picture stops #

The relation is for spirals with regular rotation. Ellipticals do not obey it and are not expected to; they have their own relation, involving dispersion rather than rotation, and it is a plane rather than a line.

Low-surface-brightness galaxies deviate. A galaxy with a central surface brightness well below Freeman’s value violates the first assumption directly, and such galaxies fall off the luminous relation. They lie on the baryonic one, which is one of the arguments for regarding that version as the more fundamental.

The figure draws a model family and not observations. The points are constructed from the two assumptions with stated scatter; they are what the assumptions predict, not what a telescope found. What can be checked against reality is the slope, which the near-infrared measurements put close to $-10$, and the scatter, which they put at a few tenths of a magnitude — tighter than this figure’s, because the model’s assumed scatter is deliberately generous.

And the whole thing is calibrated, not absolute. Tully–Fisher gives a luminosity in units of whatever the calibrating sample’s luminosities were. A systematic error in the Cepheid scale propagates through unchanged.

What it is used for besides distances #

A distance indicator is more useful than it sounds, because a distance and a redshift together give a peculiar velocity — the part of a galaxy’s motion that is not the general expansion.

The recipe is subtraction. The redshift gives the total recession velocity. The Tully–Fisher distance, multiplied by the Hubble constant, gives the part expected from expansion alone. What is left over is the galaxy’s own motion relative to the smooth flow, and that motion is caused by the gravity of everything around it.

Maps of those peculiar velocities are how the mass distribution of the nearby universe is charted, and they produced one of the more startling results of the 1980s: the Local Group is moving at some six hundred kilometres per second relative to the frame in which the microwave background looks isotropic, and the direction points towards a concentration of mass behind the plane of the Milky Way.

The precision required is worth noticing. A peculiar velocity of 600 km/s at a distance of 50 megaparsecs is a difference between a measured distance and an expected one of about fifteen per cent — which is about the accuracy of a single Tully–Fisher distance. The entire field therefore works by averaging over many galaxies, and its results are statements about flows rather than about individual objects. That is a recurring shape in this subject: a measurement too noisy to trust once becomes a map when it is made a thousand times.

What the tightness is being read as #

The baryonic relation’s tightness was called a puzzle above, and it is worth saying what the argument about it actually is, because the relation is the most-cited empirical result in a long-running dispute.

Stated carefully, the observation is this. The flat rotation speed of a disc galaxy is set by its total mass, most of which is dark; the baryonic mass is a small and variable fraction of the total, running from a few per cent in dwarfs to perhaps a fifth in bright spirals. So a relation between the baryonic mass and the rotation speed is a relation between a quantity and something it is not proportional to — and it holds over five decades, with a scatter consistent with the measurement errors alone.

In the standard picture that has to be a conspiracy, and a fairly delicate one: the efficiency with which a halo turns baryons into a visible galaxy must vary with halo mass in exactly the way that cancels the halo’s own mass–velocity relation. Simulations do produce something like the relation, and they produce it because their feedback prescriptions were tuned to reproduce galaxy properties — so the agreement is a demonstration that the picture can accommodate the relation rather than a prediction of it. The scatter is the harder part: models generically produce more of it than is observed, because a halo’s concentration varies at fixed mass and that variation should show up in the rotation speed.

The alternative reading is that the relation is fundamental — that the acceleration scale appearing in it is a property of dynamics rather than an accident of galaxy formation, and that the dark matter’s apparent contribution is a restatement of a modified force law. On that reading the exponent is exactly four by construction, the intrinsic scatter is exactly zero, and the observed scatter is entirely measurement error.

The honest position is that the observation is not in dispute and its interpretation is. What can be said from this rung is narrower and still worth having: the relation’s exponent and its tightness are measurements, and both were derived here from Freeman’s law and a constant mass-to-light ratio without any dark matter being mentioned. That derivation works because those two regularities happen to hold, and why they hold is the question the argument is really about.

Three of the relation’s inputs are assumptions rather than measurements, and each of them is worth moving to see which part of the relation it damages.

The generalisation #

The structure of the method is worth naming: find an observable that is independent of distance, and a physical argument connecting it to one that is not.

A Cepheid’s period is such an observable, and the physical argument is that the pulsation period is set by the star’s mean density while its luminosity is set by its radius and temperature. A Type Ia supernova’s light-curve shape is such an observable, and the argument is about the mass of nickel synthesised. A galaxy’s line width is such an observable, and the argument is the one above.

In each case the derivability of the connection is what makes the rung trustworthy, because an empirical correlation with no mechanism behind it might not extend to the population it is being applied to. That is the standing worry about every rung of the distance ladder, and it is the reason a two-line derivation from Freeman’s law is worth more here than a better fit would be.

And the case where the assumed surface density itself is wrong, which is the systematic the whole rung is exposed to.

Where the ladder goes next #

The immediate next rung is the baryonic relation and what its tightness constrains — because a relation between the visible matter and the halo’s rotation speed, holding over five decades with almost no scatter, is a fact any account of galaxy formation has to reproduce and none reproduces easily.

Later rungs on this anchor: the near-infrared calibration and why the slope steepens; the inclination correction as the dominant error term; Tully–Fisher applied to the Hubble constant, and what its systematics contribute there; the fundamental plane as the elliptical galaxies’ analogue; the relation at high redshift, where discs are turbulent and the width means something different; and the low-surface-brightness deviants, which are the most informative galaxies in the sample precisely because they break the first assumption.