A scatter that points the way dust does

A Cepheid tells its distance by how slowly it blinks: the longer its pulsation period, the more luminous it is, so a period read from a light curve gives a luminosity, and a luminosity compared with the observed brightness gives a distance. The relation was found in 1912 and has been the backbone of extragalactic distances ever since.

It is not a perfect relation. Cepheids of one period differ in luminosity by about half a magnitude in visible light — a sixty per cent range in brightness — and two quite different things contribute. One is the dust between the star and the observer, which dims and reddens it by an amount that differs from star to star. The other is the star itself. The pulsation happens only in a band of temperatures, the instability strip, and the strip has a width. At a given period, a star near the hot edge of the strip is brighter than one near the cool edge.

Astronomers remove the dust with a construction called the Wesenheit index, a magnitude minus a multiple of a colour, chosen so that dust leaves it unchanged. This essay is about why the same construction removes most of the strip’s width too — an accident of the way temperature and dust both move a star in the colour–magnitude diagram — and about where the accident runs out.

A population built from three relations #

The figures use a synthetic population of Cepheids, constructed from physics rather than fitted to observations, so that every source of scatter can be switched on and off.

Three relations do it. Cepheids are massive stars crossing the instability strip after hydrogen burning, and their luminosity is tied to their mass by a mass–luminosity relation, steep, with luminosity rising as about the three-and-a-half power of the mass. Their period is tied to their mean density by the pulsation equation, $P\sqrt{\bar\rho} = Q$, with $Q$ about 0.04 days for the fundamental mode — a radial sound wave crossing the star, whose crossing time depends on the density. And their radius is tied to their luminosity and temperature by the Stefan–Boltzmann law. Choose a luminosity and a temperature, and the radius, the mass, the density and the period all follow.

The strip is drawn 0.07 wide in the logarithm of the temperature — from about 6,200 to 5,250 kelvin at the luminosity of a ten-day Cepheid — sloping slightly to cooler temperatures at higher luminosity, as the observed strip does. Stars are placed at random across its width and at random luminosities, weighted towards the fainter ones as the initial mass function makes them.

Why the period does not fix the luminosity #

At a fixed luminosity, a hotter star is smaller. A smaller star of the same mass is denser, and a denser star pulsates faster. So across the strip at fixed luminosity, the period falls from the cool edge to the hot one. Turn it round: at fixed period, a star at the hot edge must be more luminous than one at the cool edge, to be large enough to have the same density.

The size of the effect is worth a number. Across the strip at ten days, the hot edge is about 0.7 magnitudes brighter in V than the cool edge — a factor of nearly two in luminosity, from stars with identical periods. In bolometric light the difference is a little smaller, because the hotter stars put more of their light into the visible band; in the infrared it is smaller still. A Cepheid of known period but unknown position in the strip has a luminosity uncertain by tens of per cent, and that is before any dust.

That is the intrinsic scatter of the period–luminosity relation: the strip’s width, projected onto the luminosity axis along the lines of constant period. It is about two-tenths of a magnitude in V in the model, which matches the scatter measured for Cepheids in the Magellanic Clouds, where all the stars are at nearly the same distance and the dust is modest.

The same picture explains the period–luminosity–colour relation that was recognised in the 1950s. If the colour of each star is known, its position across the strip is known, and the luminosity at a given period can be corrected for it: a relation in period and colour together is much tighter than one in period alone. The trouble is that the colour is exactly what dust changes.

The dust’s arrow #

Dust between a star and the observer absorbs and scatters light, more at short wavelengths than at long. It makes a star fainter and redder, and for a given kind of dust the ratio of the two is fixed. The ratio is set by the sizes of the grains: grains much smaller than the wavelength scatter blue light far more than red, while larger grains dim all wavelengths more evenly, so dust with more large grains is greyer and has a larger ratio of extinction to reddening. For the average dust of the Milky Way’s diffuse interstellar medium the numbers are these: in the visual band the extinction is about three times the reddening in B − V, about two and a half times the reddening in V − I. Dust makes everything look further away, and the correction requires knowing how much there is, which for an individual star in another galaxy is not known.

The Wesenheit index — the word is German for “essence” — is the construction that avoids needing it. Subtract from the magnitude a multiple of the colour, the multiple chosen to be the extinction-to-reddening ratio: $W = V - R\,(V-I)$. Dust adds $A$ to V and $A/R$ to V − I, so it adds $A - R \cdot A/R = 0$ to W. The index is reddening-free by construction, for dust with that $R$.

What it also does follows from the figure above. The index projects every star along the dust’s arrow. A star moved along the arrow — by dust — keeps its index. A star moved along a line of constant period — by being hotter or cooler within the strip — keeps its index too, to the extent that the line is parallel to the arrow. Since the two are close to parallel, most of the strip’s width collapses along with the dust.

The Wesenheit relation is not just reddening-free; it is tighter than the period–luminosity relation would be in V with no dust at all. That was the point of the index when it was introduced, and it is why modern distance work uses Wesenheit magnitudes almost exclusively.

Why not measure the dust instead #

The obvious alternative is to measure each star’s reddening and correct for it. For a Cepheid in the Milky Way that can be done: its spectrum gives its temperature, its temperature gives its intrinsic colour, and the difference between intrinsic and observed colour is the reddening. For a Cepheid in another galaxy, twenty million light-years away, there is no spectrum good enough, only photometry in a few bands, and the colour that would measure the reddening is the same colour that also measures where the star is in the strip. The two cannot be separated with one colour, and with several colours they separate only as well as the stars’ intrinsic colours are known — which brings back the colour–temperature relation, the metal content and the bolometric corrections.

The Wesenheit index sidesteps the separation. It never decides how much of a star’s redness is dust and how much is temperature; it uses a combination that does not care. That is its strength, and the reason it is used where a reddening correction is impossible, and it is also why it inherits a dependence on the dust law that a direct reddening measurement would expose.

The index pays off most where the dust is worst.

How much each band leaves #

The cancellation is not equally good in every combination of bands.

The infrared helps for two reasons that are the same reason. A hotter star’s extra light is mostly at short wavelengths, so in the infrared the difference between the hot and cool edges of the strip is smaller; and dust absorbs short wavelengths most, so in the infrared there is less extinction to scatter the relation. The K band at 2.2 microns gets most of the way to a Wesenheit index without any colour correction at all, which is one reason infrared period–luminosity relations became the standard once infrared detectors made them possible.

The Wesenheit indices differ from each other according to how nearly parallel the lines of constant period are to the dust’s arrow in that colour.

The near-agreement in B − V is the coincidence the index was first built on: the Wesenheit index was introduced in 1982 with B and V. The coincidence has a physical basis. Both slopes come from how fast a colour changes with something — the dust’s from how fast extinction changes with wavelength, the strip’s from how fast a star’s colour changes with temperature — and over the visible range the two happen to be similar. There is nothing that requires them to be, and in other colours they are not.

Modern work nonetheless prefers V and I, or the infrared, and accepts the larger leftover width. The reasons are outside the model: the B band is strongly affected by the metal lines in a star’s spectrum, so B − V depends on composition as well as temperature and introduces a scatter of its own between galaxies of different metal content; and the dust’s ratio of extinction to reddening varies more from one sight line to another in B − V than in V − I. A smaller scatter within a galaxy is worth less than a smaller systematic difference between galaxies, when the goal is a distance to another galaxy.

A dust-free magnitude for one kind of dust #

The index cancels the dust exactly only for dust with the assumed ratio of extinction to reddening. Dust in the Milky Way has, on average, a ratio of total to selective extinction near 3.1, with sight lines ranging from about 2.5 to over 5; dust in other galaxies, especially in the star-forming discs where Cepheids live, need not have the same average.

That residual is small, and it is not negligible. The two determinations of the Hubble constant disagree by eight per cent, and the local one rests on Cepheids in the host galaxies of type Ia supernovae, behind dust whose law is inferred rather than measured. A systematic difference between the dust in those galaxies and the dust in the calibrating galaxies, of the size the figure draws, would move the local value by a few per cent — not the whole disagreement, and enough that the dust law is one of the systematic errors every analysis of the tension has had to bound. The infrared Wesenheit indices, from the space telescopes that can measure Cepheids at those distances, reduce it by working where extinction is smallest.

From the Clouds to the supernova hosts #

The relation has to be calibrated somewhere, and the chain runs through a small number of anchors. Parallaxes give the distances of the nearest Milky Way Cepheids directly; eclipsing binaries give the distance of the Large Magellanic Cloud to about one per cent; masers orbiting the black hole in the galaxy NGC 4258 give its distance geometrically. Each of those fixes the zero point of the Wesenheit relation, and the relation then carries the distance to galaxies far enough away to host type Ia supernovae, which carry it further. Every distance in that chain is measured with the one before it, and an error in how the Cepheids are corrected for dust or for the strip enters every step after it.

The Wesenheit index enters at two places in that chain with different requirements. In the calibrating galaxies, the Cepheids are near and well studied and the index’s job is to give a tight relation. In the supernova hosts, the Cepheids are distant, crowded and behind more dust, and the index’s job is to give a relation with the same zero point as in the calibrating galaxies. The first needs the strip’s width removed; the second needs the dust law to be the same. A Wesenheit index that does the first superbly but carries a dust-law dependence is the wrong tool for the second, which is why the infrared indices, with their small leak, have become the ones used for the Hubble constant.

Four independent methods give the distance to the same galaxy in the cases where they can all be applied, and their agreement is the test of whether the Cepheid step, with its Wesenheit correction, has been done right. The cross-checks between Cepheids and the tip of the red giant branch, a method with no pulsation and almost no dust, are where a systematic error of the size the leak figure draws would show.

What the scatter says about the stars #

The width the Wesenheit index removes is not noise. It is information about the stars, and the index throws it away.

A Cepheid’s position across the strip says where it is in its evolution: the strip is crossed more than once, first on the way from the main sequence towards the red giants and then, for most of the time spent in it, during core helium burning, in a blue loop that carries the star back and forth. The crossing number shows in how fast the period is changing — lengthening on some crossings, shortening on others — and in where in the strip the star sits. The masses of Cepheids inferred from their position and luminosity disagree with their masses from pulsation theory by ten to twenty per cent, the long-standing Cepheid mass discrepancy, usually attributed to extra mixing in the cores of the stars before they became Cepheids.

So the colour that the index uses to cancel the strip’s width is also the measurement of where in the strip each star is, and of the evolutionary physics that put it there. A distance measurement wants that information gone; a stellar-evolution study wants it kept.

Where the model stops #

Empirical colours, not model atmospheres. The colours of the model stars come from an empirical relation between B − V and temperature, with V − I and V − K as linear functions of B − V matched to the Sun and a cool giant, and a small bolometric correction. That is enough for slopes across a strip a few hundred kelvin wide and not enough for the zero points of the relations, which are not drawn.

A fixed pulsation constant. The pulsation constant varies slightly with mass and temperature in real Cepheids, and the first-overtone pulsators have a different one; the model uses one value for fundamental-mode pulsators.

One mass–luminosity relation. The relation depends on metallicity, rotation and how much the stellar core was mixed — the very quantities behind the mass discrepancy — and the slope of the period–luminosity relation inherits that dependence. The model uses one relation and therefore one slope.

A uniform strip. Real Cepheids are not spread uniformly across the strip; they cluster where the blue loops spend most of their time, which depends on mass. A non-uniform filling changes the scatter and can tilt the relation slightly.

One dust law with a varying normalisation. The leak figure varies the ratio of total to selective extinction and adjusts the infrared extinction weakly with it. Real extinction curves vary in more complicated ways, especially in the ultraviolet and blue.

The surprising connection #

The Wesenheit index works for the same reason that a colour is a thermometer: in the visible, both a star’s colour and dust’s reddening are driven by a gradient across the spectrum, and over the narrow range of wavelengths and temperatures involved, a hotter star and a less-reddened star look similar. In most of astronomy this similarity is a nuisance — it is why reddened stars are mistaken for cool ones, and why the temperature read from a colour needs the reddening first. For Cepheids it is a gift. The degeneracy that makes temperature and dust hard to tell apart is exactly what lets one combination of magnitudes ignore both.

Still open: whether the relation depends on metal content #

The tightness of the Wesenheit relation makes a smaller effect visible: whether Cepheids of the same period are brighter or fainter in galaxies with more metals. The effect enters through the mass–luminosity relation, the colours and the bolometric corrections all at once, and measurements of it have disagreed in sign as well as size for decades. Parallaxes from the Gaia satellite for Milky Way Cepheids, of metal content close to the Sun’s, and precise distances to the Magellanic Clouds, with less, now constrain it to a few tenths of a magnitude per factor of ten in metal abundance — enough to matter at the per cent level for the Hubble constant. Whether the dependence is the same in V, I and the infrared Wesenheit indices, so that a correction made in one band carries to the others, is the question the next generation of Cepheid distance work has to answer before the Hubble constant from Cepheids can be trusted to a per cent.