A diameter smaller than the fringes can see

The surface-brightness method turns a star’s colour into its angular size. A star’s surface brightness depends only on its temperature, a colour measures the temperature, and so a colour and an apparent magnitude together give the angle the star subtends. For an eclipsing binary, whose components’ physical radii come from the eclipses and the velocities, angle and radius together give the distance. For cool giants the method reaches a per cent, and for the Magellanic Clouds it has given the most precise distances known to another galaxy.

For hot stars it does not. The essay on the cancellation that leaves with the hot stars showed why: once both of the colour’s bands sit on the Rayleigh–Jeans tail of a hot star’s spectrum, the colour hardly changes with temperature, the relation between colour and surface brightness turns steep, and a colour error of a few hundredths of a magnitude costs several per cent in angle. Beyond the Magellanic Clouds the bright eclipsing binaries are hot, so the method’s reach to Andromeda and beyond depends on doing better. One route is an empirical calibration: measure the angular diameters of a few dozen nearby, slowly rotating hot stars to a per cent, and fit the relation directly instead of taking it from model atmospheres.

That essay ended by noting that the calibration needs baselines long enough to resolve fractions of a milliarcsecond. This essay works out how long, at which wavelengths, and with what precision, and why the measurement has been waiting for an instrument that was abandoned fifty years ago.

Why the hot binaries matter #

The surface-brightness method is attractive beyond the Local Group’s nearest members because it is nearly geometric. An angle too small to measure can be read from a colour, and with the radii an eclipsing binary supplies, the distance follows from the angle without any assumption about how luminous the stars are. Every other route to Andromeda’s distance passes through a calibrated candle — a Cepheid, the tip of the red giant branch — whose luminosity was established on nearer stars and transferred, and a brightness is a distance only if something is known about how faithfully it transfers.

Andromeda’s distance has been measured four ways, and the four agree at about five per cent. The eclipsing-binary route there used hot stars and model atmospheres, and its error is set by the hot stars’ surface-brightness relation, not by the binaries’ orbits, which are measured far better. A one-per-cent calibration of that relation would make it the most precise of the four, and the only one with no candle in it — a check on all the others at a precision none of them reaches.

What the fringes see of a small disc #

An interferometer measures the contrast of the fringes formed by combining light from two telescopes a baseline apart. A point source gives perfect contrast; a resolved disc gives less, because light from different parts of the disc makes fringes slightly displaced from each other, and they wash out. For a uniform disc the visibility is a Bessel function of the disc’s angular size times the baseline divided by the wavelength, falling from one to zero at the first null.

A hot main-sequence star of four solar radii at a hundred parsecs subtends about 0.37 milliarcseconds; at three hundred, 0.12. In the near-infrared H band, where most modern stellar diameters have been measured, the longest baselines now operating, about 330 metres, see such a disc reduce the fringe contrast by a few per cent to ten per cent. The diameter is then encoded not in where the fringes vanish, which the measurement never reaches, but in a small departure of the contrast from one.

That is a different kind of measurement from the one Michelson made on Betelgeuse, where the fringes disappeared at a baseline that could be read off a scale. A small departure from perfect contrast has to be measured absolutely, against the instrument’s own loss of contrast from the atmosphere and the optics, and that calibration is the limit.

Shorter wavelengths resolve more #

The resolution is set by the ratio of wavelength to baseline, so the same baseline resolves three times finer in visible light than in the H band.

In visible light the 0.3-milliarcsecond disc is well resolved on 330 metres, its fringe contrast falling to a third, and its diameter can be fitted from the shape of the curve rather than from a small departure at its top. Visible-light interferometry is harder than infrared: the atmosphere’s turbulence decorrelates the wavefront faster and over smaller scales at shorter wavelengths, so the telescopes’ apertures must be smaller or corrected, the fringes must be tracked faster, and fewer photons arrive per coherent interval. That is why most interferometric diameters have been measured in the infrared even though visible light would resolve smaller stars. For hot stars the trade tips the other way: they are bright in the blue and faint, relative to cool stars, in the infrared, and their small sizes demand the resolution only short wavelengths give.

The precision a per cent needs #

How precisely must the fringe contrast be measured to give a diameter good to one per cent?

Below the resolution limit, the fringe contrast departs from one by about the square of the disc’s size in units of the resolution, so the fractional error in the diameter is half the visibility error divided by that small departure. A diameter good to a per cent needs the visibility good to a fiftieth of its departure from one. For discs that are well resolved, at the best baseline, the requirement levels off at about a per cent; for smaller discs it falls as the square of the diameter.

A single calibrated visibility is typically good to one to three per cent: the instrument’s response is measured on reference stars of known size observed just before and after, and the atmosphere changes between them. Averaging ten or more measurements reduces the random part to a few tenths of a per cent; the systematic part does not average. At an effective precision of 0.3 per cent, a one-per-cent diameter is reachable down to about 0.13 milliarcseconds in visible light and 0.38 in the H band. A B star at a hundred parsecs is measurable in the near-infrared only at the limit; at three hundred, only in the visible.

Baselines and the stars within reach #

The smallest diameter measurable to a per cent falls in proportion to the wavelength and in inverse proportion to the baseline.

The stars that would make an empirical calibration are main-sequence B stars that rotate slowly and carry no circumstellar gas. B stars are rare — the nearest few dozen suitable ones lie between one and three hundred parsecs — and of those, most rotate rapidly and many are Be stars with gaseous discs. A useful sample of thirty or forty needs to reach to two or three hundred parsecs, where the stars subtend 0.12 to 0.2 milliarcseconds. In visible light that needs baselines of three to six hundred metres; in the infrared, kilometres.

The longest optical baselines in operation are a few hundred metres. A kilometre-scale optical array has been proposed several times and not built, because holding the light paths of separate telescopes equal to a fraction of a wavelength over a kilometre, through turbulent air, is an engineering problem that grows with the baseline. The calibration the surface-brightness method needs for hot stars sits just beyond what the operating arrays can do at the wavelengths they mostly use.

The calibrator problem #

Every visibility is measured relative to a reference star whose own visibility is assumed known, and its diameter enters the result.

The instrument’s loss of contrast is measured on a calibrator star observed close in time and direction, and the target’s visibility is the ratio of the two raw contrasts, multiplied by the calibrator’s visibility computed from its assumed diameter. If that diameter is wrong by five per cent, the calibrator’s visibility is wrong by an amount proportional to how much the calibrator is itself resolved, and the error passes into the target. A calibrator half the target’s size passes on a fifth of its own uncertainty; one the target’s size passes on all of it.

So calibrators must be smaller than the targets, and when the targets are a few tenths of a milliarcsecond, the calibrators must be smaller still — fainter stars or more distant ones, whose own diameters come from the surface-brightness relations the measurement is meant to establish. For hot-star targets the calibrators are often hot stars too, their diameters taken from model atmospheres, and the calibration risks inheriting the very model dependence it was meant to remove. The way out is to use calibrators so small that their diameters barely matter, which needs bright, distant, compact stars near each target on the sky, and those are scarce.

Stars that spin #

Most B stars rotate fast, and a rapidly rotating star is not a disc.

A star rotating near its critical rate is flattened by centrifugal force, its equator bulging out by a third beyond its polar radius at the extreme, and darker at the equator than at the poles because the surface gravity is lower there. Seen at an angle, its outline is an ellipse whose axis ratio depends on the inclination; seen equator-on, it is markedly oval. A diameter measured on one baseline orientation is a diameter along one axis, and a surface brightness computed from it applies to a part of the star whose temperature is not the star’s average.

The same gravity darkening that complicates the diameter is measured by other means: a planet crossing a rapidly rotating star dims it differently going in than coming out, because it crosses bright poles and a dark equator in an order set by the star’s tilt. Interferometric images of the nearest rotators, which resolve the bright poles directly, and transit light curves of the distant ones measure the same law from two directions, and a calibration that modelled each rotator could borrow its darkening from them.

That is why the calibration wants the slow rotators. They are a minority among B stars, which narrows the sample further and pushes it further away. The alternative is to model each star’s shape and gravity darkening, which interferometry can do for the nearest, brightest rotators by measuring on many baseline orientations — and which turns a one-number calibration into a model fit with its own assumptions.

The same diameters fix a temperature scale #

A measured angular diameter does more than calibrate a colour relation. Combined with the star’s total flux, it gives the effective temperature directly, since the flux is the surface brightness times the solid angle and the surface brightness of a blackbody-like photosphere is $\sigma T^4$. That is how the temperature scale of cool stars was anchored — and how a hundred kelvin of disagreement between the interferometric scale and the one read against a star’s infrared tail was found.

For hot stars the temperature scale rests largely on model atmospheres and on the Narrabri diameters of fifty years ago. The same measurements that would calibrate the surface-brightness relation would re-anchor the hot stars’ temperatures, and with them the luminosities, masses and ages derived for every massive star and every young cluster whose distance or age depends on its hottest members.

The intensity interferometer returns #

The first systematic diameters of hot stars were measured not by combining light coherently but by correlating the intensity fluctuations seen by two separate detectors. Robert Hanbury Brown and Richard Twiss showed in the 1950s that the arrival of photons at two telescopes is correlated when the source is unresolved and uncorrelated when it is resolved, and that the correlation can be measured with fast electronics, without bringing the light together at all. The intensity interferometer built at Narrabri in Australia, with two large light-collecting mirrors on a circular track up to 188 metres apart, measured the diameters of thirty-two bright stars at blue wavelengths by 1974, most of them hot.

Those measurements are still among the foundations of the temperature scale for hot stars, and they were never extended, because the method needs enormous collecting areas for faint stars and was overtaken by coherent interferometry. It has now come back. The arrays of large mirrors built to detect the Cherenkov flashes of cosmic-ray showers in the atmosphere are exactly the instrument the method needs — big, crude mirrors with fast photon detectors — and in the last few years they have measured the diameters of hot stars in the blue by intensity interferometry, to a few per cent. The arrays being built for the next generation of gamma-ray astronomy will have dozens of telescopes on baselines of kilometres.

At a kilometre in the blue, the smallest diameter measurable falls to a few hundredths of a milliarcsecond. The intensity interferometer is insensitive to the atmosphere’s turbulence, because it never combines the light’s phases, and it works best for hot, bright stars in the blue — precisely the stars and wavelengths the surface-brightness calibration needs. The calibration that was out of reach for the optical arrays may be finished by instruments built for gamma rays.

What the figures leave out #

The discs are uniform. Real stars are darker at the limb, and the diameter that comes out of a fringe measurement depends on a model atmosphere through the limb-darkening correction. For hot stars in the blue the correction is a few per cent; for partly resolved discs, the first lobe of the visibility hardly distinguishes limb darkening from size, and the correction has to be assumed.

The circumstellar gas is absent. Be stars and stars with winds add light from gas outside the photosphere, which makes the source look larger, more in the infrared than in the visible. A calibration sample has to exclude them or measure them at several wavelengths.

Reddening is ignored. Interstellar dust between the star and the observer dims and reddens it, and a colour corrected for the wrong amount of dust gives the wrong surface brightness. For the nearby calibrators the dust is slight; for the hot binaries in other galaxies it is the dominant uncertainty, and the calibration’s value depends on transferring a relation measured with little dust to stars seen through a great deal.

Companions are absent. Many B stars are binaries, and a faint companion within the interferometer’s field changes the visibility in a way that can mimic a different diameter. Their exclusion needs spectroscopic or interferometric screening.

The averaging is ideal. The effective visibility precision assumes errors that average down as the square root of the number of measurements. Systematic calibration errors do not, and they set a floor that more nights do not lower.

Still open: whether the hot stars’ diameters can be finished #

The surface-brightness method’s reach beyond the Magellanic Clouds depends on hot stars, and its precision there depends on a calibration nobody has finished: a few dozen nearby, slowly rotating B stars with diameters good to a per cent. Those stars subtend a tenth to a third of a milliarcsecond, which the longest coherent arrays resolve only in visible light and at the limit of their precision, with calibrators whose own diameters come from the relation being calibrated.

The obstacle is not one difficulty but several that compound: the stars are small, so the fringes barely change; the calibrators are smaller still and their sizes are the thing being calibrated; the stars spin, so their outlines depend on how they are tilted; and the suitable ones are few, so every exclusion — a companion, a disc, a fast rotation — shrinks a sample that was small to begin with. Each obstacle on its own costs a factor of two in precision; together they have kept the hot-star calibration at several per cent for half a century.

Two instruments could finish it: coherent arrays extended to longer baselines or pushed to shorter wavelengths, and intensity interferometers on the large arrays of Cherenkov telescopes, which work in the blue on kilometre baselines and are indifferent to the atmosphere. If either delivers a per cent for forty stars, the hot binaries of Andromeda and Triangulum could be measured as precisely as the cool ones of the Magellanic Clouds, and the surface-brightness method would become a geometric distance to the nearest spiral galaxies independent of every calibrated candle beneath it.