An angle too small to measure, read from a colour

A star’s angular diameter can be measured by finding the baseline at which its interference fringes vanish, and that measurement is one of the few in astronomy that rests on nothing but geometry and the wavelength of light. It is also limited to a few hundred stars. The longest optical baselines resolve discs down to about a fifth of a milliarcsecond, and most stars are smaller than that on the sky.

An angle can be had for any star whose light can be measured, if the question is turned around. The light received from a star is the brightness of each unit of its surface times the solid angle of its disc. If the brightness per unit surface can be found some other way, the solid angle follows from the measured flux, and so does the angular diameter. The brightness per unit surface is set by the temperature of the atmosphere, and a colour is a thermometer. So a relation between colour and surface brightness, calibrated once on stars whose angles have been measured, turns a colour and a magnitude into an angle.

The relation is empirical, and the arguments below are about what it can and cannot carry: which colour it should use, how far beyond its calibrators it can be extrapolated, what the dust between here and the star does to it, and what finally limits the distances it is used for.

A quantity that belongs to the surface #

The bookkeeping is simplest in magnitudes. A star’s V magnitude, corrected for extinction, is $V_0 = -2.5\log S_V - 5\log\theta + \text{const}$, with $S_V$ the surface brightness in the V band and $\theta$ the angular diameter. The combination

$$F_V = 4.2207 - 0.1\,V_0 - 0.5\log\theta ,$$

with $\theta$ in milliarcseconds, cancels the angle exactly and leaves a number that depends only on the surface. The constant is chosen so that $F_V = \log T_{\mathrm{eff}} + 0.1\,\mathrm{BC}_V$, where $\mathrm{BC}_V$ is the bolometric correction; for the Sun it is 3.754. The parameter was introduced by Wesselink in 1969, and Barnes and Evans showed in 1976 that it is a tight, nearly linear function of colour.

Given a relation $F_V = a + b\,(V-K)_0$, the angle follows:

$$\log\theta = 2\,(4.2207 - a) - 0.2\,V_0 - 2b\,(V-K)_0 .$$

The version drawn here, fitted by Kervella and collaborators to interferometric diameters of nearby dwarfs and subgiants, is $\log\theta = 0.5175 + 0.2753\,(V-K)_0 - 0.2\,V_0$. Two coefficients, $+0.2753$ and $-0.2$, will carry most of the rest of this essay.

Why V and K #

The colour is taken between the V band, at 0.55 microns, and the K band, at 2.2, rather than between two neighbouring optical bands, and the reason is visible in how each band’s surface brightness responds to temperature.

For stars between about 3,500 and 7,000 K, the K band lies on the long-wavelength side of the Planck peak, where surface brightness is nearly proportional to temperature. The V band lies on the short-wavelength side, where it rises exponentially. The ratio of the two is a sensitive thermometer, and it is sensitive for a physical reason rather than a fitted one.

The same property keeps the relation clean. Surface gravity and metal abundance change a star’s spectrum mostly through absorption lines and molecular bands, which crowd the optical and the blue. They move an optical colour like $B - V$ and the V-band surface brightness in correlated ways that differ between giants and dwarfs, so a $B - V$ relation needs separate calibrations for each. The infrared is comparatively clean, and a $V - K$ relation holds across luminosity classes with a scatter of one or two per cent in angle. The blackbody curve in the first figure departs from the stars’ line by up to 0.035 in $F_V$ — about sixteen per cent in angle — because real atmospheres have limb darkening and molecular absorption; the relation is calibrated on measured stars for exactly that reason.

The leverage weakens for hot stars. Above about 10,000 K the V band too moves toward the long-wavelength side of the Planck peak, the two curves in the figure approach each other, and the whole range of $V - K$ for B-type stars is less than a magnitude. The same error in colour then means a larger error in temperature and in angle. Hot stars are also rare among the stars an interferometer can resolve, and the relation for them is calibrated on few stars, to several per cent.

How far the calibration reaches #

The relation is calibrated where the curves are above the shaded band and applied where they are deep inside it. A giant of the kind used for the most precise extragalactic distances can be resolved out to about 500 parsecs. At the distance of the Large Magellanic Cloud, a hundred times further, its disc is a hundred times smaller than anything resolved, and at the Andromeda galaxy it would be fifteen times smaller again.

This is an extrapolation in distance, not in the physics. The relation connects a colour to a surface brightness, and a star of the same temperature, gravity and composition has the same surface whether it is 100 parsecs away or 50,000. The assumption is that the distant stars are the same kind of star as the calibrators — the same luminosity class, a similar metal content, an atmosphere in the same state — and that assumption, rather than any property of distance, is what the extrapolation rests on.

A distance from a length and an angle #

The relation becomes a distance when a length is measured for the same star. An eclipsing binary supplies one. The light curve gives each star’s radius as a fraction of the orbit’s size, from the durations of the eclipses and the times of their contacts, and the radial velocities of both stars give the orbit’s size in kilometres, once the inclination is known from the eclipses. The product is each star’s radius in kilometres, to about one per cent for well-observed detached systems. The relation gives each star’s angular diameter from its V magnitude and colour, separated from its companion’s by the ratio of surface brightnesses that the relative eclipse depths measure. The distance is the radius divided by half the angle.

This is the same structure as the distance from a visual and spectroscopic binary, with the angle on the sky replaced by the angle of each disc. It is also the structure of the Baade–Wesselink method for pulsating stars, with one important difference. A pulsating star’s radius comes from integrating a velocity that has to be corrected by a projection factor nobody can measure directly. An eclipsing binary’s radius comes from orbital dynamics and geometry, with no such factor, and its atmosphere is in equilibrium rather than pulsating.

Two stars in one point of light #

A binary at 50 kiloparsecs is a single point of light, and the relation needs each star’s own magnitude and colour. The eclipses provide them. When one star passes in front of the other, the depth of the eclipse is set by how much light the hidden part of the eclipsed star was emitting per unit area, so the ratio of the two eclipse depths measures the ratio of the stars’ surface brightnesses directly. Combined with the ratio of their radii from the light curve, that fixes how the system’s total light divides between them, in each band where eclipses are observed.

The division is a second use of the same physics, and it provides a check. The surface-brightness ratio from the eclipse depths in V must agree with the ratio the relation assigns from the two stars’ separate colours; a pair that disagrees has something wrong with its photometry, its reddening or its light-curve model, and can be caught before it enters the average. The same constraint is why the method prefers pairs of similar stars: when both components are giants of similar temperature, their colours are nearly equal, the light ratio is close to unity and well determined, and an error in the split moves both angles in compensating directions.

The infrared light ratio is usually the weakest link. Eclipses are observed intensively in the optical, where the photometry is precise and the sky is dark, and sparsely in the K band. The K light of each component is then partly inferred from its optical colour and a model spectrum, which feeds a model back into a method whose attraction is that it needs none. The most careful analyses measure the eclipses in K as well, at considerable cost in telescope time.

What dust does to the angle #

Every distance to a star beyond the immediate neighbourhood has to contend with interstellar dust. Dust makes everything look further away: it dims the star, and a standard candle whose dimming is under-corrected by $\delta A_V$ magnitudes gives a distance too large by a factor $10^{0.2\,\delta A_V}$ — 4.7 per cent for a tenth of a magnitude.

The surface-brightness route is affected twice, in opposite directions. Dust dims the V magnitude, and at fixed colour a fainter star has a smaller disc: $\log\theta$ falls by 0.2 for every magnitude of unaccounted extinction. Dust also reddens the star. The K band is dimmed only about a ninth as much as V, so $V - K$ grows by 0.886 magnitudes per magnitude of $A_V$, and a redder star is assigned a cooler, dimmer surface and so a larger disc: $\log\theta$ rises by $0.2753 \times 0.886 = 0.244$. The two nearly cancel. The net is $+0.044$ in $\log\theta$ per magnitude — one per cent for a tenth of a magnitude, a fifth of the standard candle’s error and in the opposite sense.

The cancellation is not designed. It happens because the reddening vector in the plane of $F_V$ against $V - K$ runs nearly parallel to the relation itself, so a reddened star slides along the relation instead of off it. It is one of the main reasons the method is used for distances through galaxies full of dust. A standard candle measured in the K band does better still, since there the dust is nearly transparent. But that candle needs its luminosity calibrated somewhere else, on stars with trigonometric parallaxes or other distances; the angle needs only the relation and the stars it was fitted to.

Where the error in one binary comes from #

The same two coefficients decide how photometric errors propagate. An error in V enters through the magnitude, with coefficient $-0.2$, and through the colour, with $+0.2753$, and the two mostly cancel: a hundredth of a magnitude in V costs 0.17 per cent in distance. An error in K enters only through the colour, with the full $0.2753$, and costs more than seven times as much. For a giant star in another galaxy, measured at 2.2 microns through the Earth’s bright infrared sky, 0.02 magnitudes is a good K measurement, and it is 1.3 per cent in distance. The largest error in the most precise extragalactic distance technique is often the infrared photometry, not anything to do with the binary.

The relation’s intrinsic scatter adds about 0.8 per cent for the late-type giants that were chosen for this reason, the radius from the orbit about a per cent, and an extinction uncertainty of 0.05 magnitudes half a per cent. In quadrature, 1.9 per cent for one binary. That is already competitive with most standard candles for a single object, and it is a geometric measurement.

Averaging twenty binaries #

The decisive application has been the Large Magellanic Cloud, whose distance anchors much of the extragalactic scale, including the period–luminosity relation of the Cepheids that carry distances out to galaxies with supernovae. The measurement took a long time to become possible. The binaries that work best contain two red giants, because their $V - K$ colours are in the relation’s best-calibrated range and giants are bright enough to measure spectroscopically at 50 kiloparsecs; but giants in orbit around each other have periods of hundreds of days, and finding and following twenty of them took more than a decade of observing.

An earlier analysis of eight such binaries, published in 2013, had reached 2.2 per cent. Going from eight to twenty reduced the independent part of the error by a factor of 1.6, as the square root predicts, and the published error fell by nearly that factor. Any further gain had to come from the shared part.

The figure shows why twenty was enough and forty would not have been much better. Each binary’s own errors — its photometry, its orbit, the dust along its particular sightline — are independent of the others’, and they average down as $1/\sqrt{N}$: 2.3 per cent for one becomes half a per cent for twenty. The relation’s calibration is shared by every binary. An error in its zero point shifts every angle by the same fraction, and averaging does nothing to it.

The published analysis of twenty red-giant binaries, in 2019, found a distance of 49.59 kiloparsecs, with a statistical error of 0.09 and a systematic one of 0.54: 0.2 per cent from the binaries and 1.1 per cent from everything they shared, of which the relation’s calibration on 41 interferometrically measured red-clump giants was the largest part. The binaries had been averaged down to the calibration, and a better distance to the Large Magellanic Cloud now requires better interferometric diameters of nearby giants, with their limb darkening measured rather than modelled, not more binaries.

What the relation assumes #

The relation is a statement about stellar atmospheres, and three things can make a distant star disobey a relation fitted to nearby ones.

Metal content is the first. The Large Magellanic Cloud’s giants are about half as metal-rich as the Sun’s neighbours, and the Small Cloud’s about a fifth. Fewer metals mean fewer absorption lines in the optical, a slightly brighter V-band surface at a given temperature and a slightly different $V - K$. Model atmospheres suggest the effect on the relation is small for giants in the $V - K$ range used, and the agreement between the eclipsing-binary distances and the Cepheid distances to the two Clouds supports that, but it has not been measured directly, because there are no resolvable metal-poor giants near enough.

The second is the state of the atmosphere. The binaries used are detached, their stars are not tidally distorted beyond what the light-curve models include, and their atmospheres are quiet. Pulsating stars, stars with spots, and stars with circumstellar dust shells can all move off the relation, and each has to be excluded or modelled.

The third is limb darkening, which the relation absorbs into its calibration. Interferometric diameters are converted to limb-darkened ones with model atmospheres, and the model’s limb-darkening coefficient changes the calibrated angle by a few per cent. The same model has to describe the calibrators and the targets; a systematic difference between the model’s limb darkening and the truth enters the zero point directly and is one of the terms averaging cannot touch.

Still open: hot stars and more distant galaxies #

The method’s reach is limited by the stars it can be applied to. Late-type giant binaries are bright enough for the Magellanic Clouds and barely beyond; at the distance of the Andromeda galaxy only hot, massive binaries are bright enough for high-resolution spectroscopy, and for hot stars the relation is poorly calibrated, the colour’s leverage is weaker, and the few distances measured this way to Andromeda and the Triangulum galaxy have used model atmospheres for the surface brightness instead, with uncertainties of several per cent. Whether interferometry can calibrate the hot-star relation to the precision the giants have reached — which requires measuring diameters of B stars, small on the sky and intrinsically rare nearby — decides whether the method extends past the Local Group’s nearest members.

A second question concerns the relation itself. A calibration of about one per cent from interferometry of giants is a systematic floor that every eclipsing-binary distance in the Magellanic Clouds shares. A completely independent geometric distance to the same galaxy — from a gravitational-wave source, which needs no calibration, or from masers orbiting a black hole, or from parallaxes of individual stars in the Clouds with a future astrometric mission — would test that floor directly. None yet reaches one per cent at 50 kiloparsecs.