A distance that rests on one unmeasured number

A Cepheid’s period tells its luminosity, and its luminosity and apparent brightness together give its distance, but only once the relation between period and luminosity has been calibrated on Cepheids whose distances are known some other way. Every such calibration borrows a distance from somewhere else, and every distance is measured with the one before it. A Cepheid also offers something the calibration does not need. It pulsates, because a layer of partly ionised helium acts as a valve at the right depth, and the pulsation can be measured twice, in two different units.

The spectral lines of the star shift back and forth with the motion of its surface: a Doppler velocity, in kilometres a second, which integrated over time gives the change of radius in kilometres. And the star’s angular size, measured by an interferometer or inferred from its colour and brightness, swells and shrinks with the same rhythm: a change of angle, in milliarcseconds. A length divided by an angle is a distance. This is the Baade–Wesselink method, proposed by Walter Baade in 1926 and made practical by Adriaan Wesselink in 1946, and it is often described as a distance with no calibration at all.

An idea older than the means to use it #

When Baade proposed the method, the pulsation interpretation of Cepheids was barely a decade old and still contested: the alternative, that Cepheids were eclipsing or orbiting binaries, had been abandoned only when the velocity curves were shown to demand a single star changing size. Baade’s point was that if the star really pulsates, then its size is changing by a measurable number of kilometres and a measurable fraction of its angle, and the two together give a distance. Neither angle could be measured then. Wesselink’s contribution was to avoid measuring it: at two moments of the cycle when the star has the same colour, and so the same surface brightness, the ratio of its brightnesses is the square of the ratio of its radii, and the radius difference comes from the velocity curve. That gives the radius, and with it the luminosity, without any angle at all.

The version drawn here — measuring the angle directly and fitting it against the integrated velocity — became possible only when interferometers of a hundred metres or more could resolve a disc of a milliarcsecond and see it change by a tenth of that, in the first years of this century. By then the method had been used for seventy years with a projection factor taken from geometry and a model atmosphere, and the question of whether that factor was right had been waiting for an independent distance to answer it.

Two measurements of one motion #

The velocity curve of a Cepheid is not a sine wave. The star contracts slowly and expands fast — the curve drops steeply when the surface starts moving outward and climbs back gradually — and it oscillates not about zero but about the star’s own motion through space, here $-16.8$ kilometres a second. Subtract that, integrate the rest over time, and multiply by a factor $p$ explained below:

$$\Delta R(t) = -p\int_0^t \big(v_r - \gamma\big)\,dt'.$$

The minus sign is because a negative radial velocity, towards the observer, means the near side of the star is expanding. For the model star the result is a smooth swing of 4.9 solar radii, about 11 per cent of its mean radius, peaking half a cycle after the minimum.

The angular diameter follows the same shape:

$$\theta(t) = \frac{2\,\big(R_0 + \Delta R(t)\big)}{d}.$$

At 272 parsecs a star of 44 solar radii subtends 1.5 milliarcseconds, and it swells and shrinks by about 170 microarcseconds. The largest optical interferometers resolve discs of that size and measure changes of a few per cent in them, which is why this is now done directly for the nearest Cepheids rather than through a colour relation.

The two curves have the same shape because they record the same motion. Their scales differ by exactly the distance.

A straight line with a distance for a slope #

The analysis does not need to know the mean radius. Plot the measured angular diameter against the change in radius computed from the velocity at the same phase, and the points fall on a straight line.

The slope of the line is $2/d$, and the intercept is $2R_0/d$, so a fit gives both the distance and the mean radius. With forty measurements of eight-microarcsecond precision, the model star returns 280 parsecs and 45.3 solar radii — within three per cent of the values it was given, the difference being the noise in forty points spread over one cycle. Nothing in the fit refers to any other star: the velocity is a Doppler shift measured against a laboratory wavelength, the angle is an interferometric measurement against the baseline between two telescopes, and the distance is their ratio.

The mean radius is a bonus with its own uses. Combined with the star’s temperature it gives its luminosity directly, and the period–radius relation of Cepheids is a test of pulsation theory that does not depend on distances at all.

The number the velocity has to be multiplied by #

The one quantity in the method that is not measured is $p$, the projection factor. It exists because the spectrograph does not see the surface’s speed. The whole visible hemisphere is moving radially outward, but only the centre of the disc is moving straight towards the observer; points nearer the limb move at an angle, and contribute only the component of their velocity along the line of sight. What the spectrum records is a weighted average of those components, smaller than the speed of the surface, and $p$ is the factor that restores it.

And the distance is proportional to $p$.

If the radius change is computed with a projection factor that is wrong by some fraction, it is wrong by the same fraction, the slope of the fit is wrong by the same fraction, and the distance is wrong by the same fraction. The published values of $p$ for Cepheids lie between about 1.2 and 1.5 depending on the star, the spectral lines used and the method used to derive it — a spread of twenty-five per cent, which is ten per cent either side of the middle. The method that needs no calibration needs exactly one, and it enters every distance at full strength. No improvement in the velocities or the angles reduces it.

What geometry says, and what the atmosphere does #

The first part of $p$ is geometry, and it can be computed exactly.

Each point of the visible disc moves radially at the surface’s speed, so its line-of-sight component is that speed times the cosine $\mu$ of the angle between the radius and the line of sight. The spectrum averages those components, weighted by how bright each point is and by the area it presents. For a disc whose brightness falls towards the limb as $1 - u(1-\mu)$, the average of $\mu$ weighted that way is

$$\langle\mu\rangle = \frac{\int_0^1 I(\mu)\,\mu^2\,d\mu}{\int_0^1 I(\mu)\,\mu\,d\mu},$$

and the geometric projection factor is its reciprocal: exactly 3/2 for a uniformly bright disc, exactly 4/3 for one darkened completely at the limb, and about 1.41 for the darkening a Cepheid’s atmosphere actually has in visible light. A darker limb weights the centre of the disc more heavily, where the motion is along the line of sight, so less correction is needed.

The values inferred for real Cepheids are lower still, around 1.2 to 1.35, and the difference is the atmosphere. The spectral lines that measure the velocity form in layers above the photosphere, and those layers do not move in step with it: a pulsating atmosphere has velocity gradients, with the line-forming gas moving at a slightly different speed from the layer whose size the interferometer sees. Different lines form at different heights and give different velocities, and each needs its own correction, computed from hydrodynamic models of the pulsating atmosphere. The correction is several per cent, and it is where the uncertainty in $p$ lives: in a model of a moving atmosphere, not in geometry.

There is a further subtlety in how the velocity is measured at all. A line’s position can be taken as the centroid of the whole profile, as the minimum of a fitted Gaussian, or from a cross-correlation with a template, and these give different velocities for a line broadened by rotation and turbulence. The projection factor is defined relative to one particular way of measuring, and a factor derived for one cannot be carried to another.

An angle without an interferometer #

Only a few dozen Cepheids are near enough and large enough on the sky for an interferometer to resolve, and for most of the method’s history none were. The angle came instead from the star’s own light. A star’s angular diameter, its apparent brightness and its surface brightness are tied by definition: the flux received is the surface brightness times the solid angle the disc subtends. The surface brightness depends on the temperature, and the temperature can be read from a colour. So a relation between colour and surface brightness, calibrated once on stars whose angular diameters have been measured directly, turns every colour and magnitude into an angle.

The relation is tightest when the colour spans the visible and the near-infrared — the brightness at 0.55 microns against that at 2.2 — because that colour tracks temperature with little sensitivity to surface gravity or metal content, and it has been calibrated to about two per cent on interferometric diameters of giants and Cepheids. Applied through a pulsation cycle, it gives the angular diameter at every phase from photometry alone, and it is what allows the method to reach Cepheids in other galaxies. It also inherits a calibration: the relation’s zero point and slope come from other stars, and a Cepheid’s atmosphere, pulsating and not quite in equilibrium, is not guaranteed to obey a relation fitted to stars that are not pulsating. The interferometric version removes that assumption for the stars it can reach and leaves the projection factor as the only one.

A velocity that belongs to which layer #

The atmosphere’s motion shows in another way that makes the difficulty with $p$ concrete. Different spectral lines of the same Cepheid, measured over the same cycle, do not give the same mean velocity. Their average over a cycle — which ought to be simply the star’s motion through space — differs from line to line by up to a few kilometres a second, depending on how deep in the atmosphere each line forms. The lines are not measuring the same gas. Some form in layers that are still falling while deeper ones have begun to rise, and the atmosphere compresses and stretches through each cycle as a shock passes through it.

That is the same effect that lowers the projection factor below geometry, seen in the mean rather than in the amplitude. It means the “velocity of the surface” in the integral is a quantity defined by a choice of lines, and the projection factor that goes with it is defined for the same choice. A Baade–Wesselink distance is therefore exactly as good as the model that connects the velocity of a particular set of lines to the motion of the layer the interferometer sees — which is a model of a pulsating atmosphere, tested against the very spectra it is used to interpret.

A small error in timing #

The two measurements must also be made of the same moments in the cycle, and in practice they are usually made years apart.

The velocity curve comes from spectroscopy and the angular diameters from interferometry, often in different seasons or decades, and they are paired by folding each on the star’s period. If the period is slightly wrong, or has changed — and Cepheid periods do change, measurably over decades, as the stars evolve through the instability strip — the two curves are paired at slightly different phases. The fitted line then becomes a loop, the correlation between the two quantities weakens, and the fitted slope flattens. A flatter slope is a larger distance. The bias is quadratic in the shift and always in the same direction: nine-tenths of a per cent for two hundredths of a cycle, four per cent for four. It is smaller than the projection factor’s uncertainty and, unlike it, entirely avoidable by measuring both in the same season.

The method turned round #

For most of a century the Baade–Wesselink method was used to measure distances with an assumed projection factor. The arrival of precise parallaxes has reversed the logic.

A Cepheid within a few hundred parsecs now has a parallax from space astrometry good to a few per cent. A Baade–Wesselink analysis of the same star, with the distance fixed at the parallax value, returns not a distance but a projection factor: the one value of $p$ for which the velocity-integrated radius changes and the interferometric angle changes agree at that distance. Applied to δ Cephei with an earlier space-based parallax, it gave $p$ = 1.27 with an uncertainty of about five per cent; applied to a few dozen Cepheids with modern parallaxes, it gives values that scatter by about ten per cent from star to star, more than their individual errors.

That scatter is the finding. The projection factor is not a single constant, and whether it depends systematically on the period has been argued in both directions: analyses based on atmospheric models find a weak decrease with period, analyses that calibrated it by requiring the Large Magellanic Cloud’s Cepheids to agree on one distance found a much steeper one. The two disagree by up to twenty per cent at the longest periods, which are the Cepheids used to reach other galaxies. The distance that needed no calibration became, once other distances were good enough, a measurement of the one number it had assumed, and that number turned out to vary.

Why it matters beyond one star #

The Baade–Wesselink method is no longer the main way the period–luminosity relation is calibrated; parallaxes are. But it remains the independent check, and the only one that works at any distance where a Cepheid’s velocity can be measured. The infrared version, which infers the angular diameter from a surface-brightness relation between colour and brightness rather than measuring it, can be applied to Cepheids in the Magellanic Clouds, where no interferometer can resolve them, and it has been used to test whether the period–luminosity relation depends on the stars’ metal content — a correction that enters the local measurement of the Hubble constant and its disagreement with the one from the microwave background. A ten per cent uncertainty in $p$ is a large lever on a disagreement of eight per cent.

The method’s structure is shared with another geometric distance: an eclipsing binary gives a distance from its stars’ radii and their angular sizes, with the radii from the orbit and the eclipses and the angles from the same surface-brightness relations. That route has no projection factor, because the radii come from timing rather than from a velocity averaged over a disc, and it is why the most precise distance to the Large Magellanic Cloud now comes from eclipsing binaries rather than from pulsating stars.

What the model star leaves out #

The model is a single star with a four-harmonic velocity curve of δ Cephei’s amplitude and a perfectly known period, observed without the velocity-gradient effects the text describes, integrated with a constant projection factor that is correct by construction. Real analyses fit the velocity curve and the angular diameters together, allow the projection factor to vary with phase as the atmosphere’s structure changes, and must account for the star’s companions — many Cepheids have them, and a companion’s light alters both the colour and the interferometric visibility. The geometric projection factor assumes a spherical, uniformly expanding surface with a linear limb-darkening law; real Cepheids have extended atmospheres and some have circumstellar envelopes that contribute to the interferometric signal. And the phase-shift bias is computed for a noiseless curve; with real noise it is harder to separate from the other sources of scatter.

Still open: what the projection factor depends on #

The projection factor sits in every Baade–Wesselink distance at full strength, geometry fixes it only to within its first twenty per cent, and the measurements that now calibrate it with parallaxes show a star-to-star scatter larger than their errors. Whether that scatter is a dependence on period, on metal content, on the particular spectral lines and velocity definitions used, or on something about each star’s atmosphere that no single parameter describes is not settled. Model atmospheres that follow the pulsation hydrodynamically, spectra that measure lines forming at many heights at once, and interferometric parallaxes for Cepheids at a range of periods are each closing part of it. Until they agree, a distance measured from a star’s own swelling is geometric in every step but one.