An angle of five hundredths of an arcsecond

The 2.54-metre Hooker telescope on Mount Wilson, the largest instrument in the world in 1920, could just separate two points 57 milliarcseconds apart in yellow light. The largest stellar disc in the sky, R Doradus, is 57 milliarcseconds across, and every other star is smaller — so no telescope then or since has resolved a stellar disc, and a five-millimetre pupil, held to the same diffraction limit that decides whether a planet can be seen beside its star, is out by a factor of five hundred.

Betelgeuse’s diameter was measured on that telescope regardless, and came out at five hundredths of an arcsecond. Nothing about the measurement resolved the star. What it resolved was an interference pattern — a different object, at a different scale, plainly visible at an eyepiece.

The whole measurement is the position of that zero: not a brightness, not a photograph of anything, but a length in metres at which a pattern stops being visible.

The claim, stated as a length #

The claim is that a stellar angular diameter is obtained by measuring a distance between two mirrors, and that this is the result’s actual content, not a picturesque description of it. Two apertures separated by $B$ combine light from one star, and a point source gives full fringe contrast at every separation while a disc gives fringes that vanish at a separation inversely proportional to its angular size. Push the mirrors apart until the fringes are gone, read the tape measure, divide.

Everything else said about such a star — a radius in solar units, a luminosity, a place on a diagram — is a conversion applied afterwards to that angle, and each conversion imports a quantity nobody measured here. The angle is the measurement; the radius is the angle multiplied by somebody else’s distance. Parallax is the only distance measurement that assumes nothing, and it is where the radius’s error will turn out to live.

Two mirrors, and the separation that erases the fringes #

The coherence of the light field at two points separated by $B$ is the Fourier transform of the source’s brightness distribution on the sky, evaluated at the spatial frequency $B/\lambda$. For a uniformly bright disc of angular diameter $\theta$ that transform is

$$V(B) = \left|\frac{2J_1(x)}{x}\right|, \qquad x = \frac{\pi\theta B}{\lambda},$$

with $J_1$ the first-order Bessel function. $V$ is the fringe visibility: the contrast between bright and dark bands, one for a point source and zero when the pattern has washed out. The first zero of $J_1$ is at $j_{1,1} = 3.8317059702$, so the fringes vanish when $x$ reaches that value, which is to say at

$$B_{\text{null}} = \frac{j_{1,1}}{\pi}\,\frac{\lambda}{\theta} = 1.21967\,\frac{\lambda}{\theta}.$$

Read backwards, $\theta = 1.21967\,\lambda/B_{\text{null}}$. At 575 nm a 47-milliarcsecond disc puts its null at 3.08 metres, and that is the entire calculation: a wavelength, a measured length, and a pure number out of a Bessel function. The inverse proportionality is the harsh part — halving the diameter doubles the instrument.

Unresolved by one mirror, and longer than one mirror #

A single aperture of diameter $D$ resolves a disc when $\theta > 1.21967\,\lambda/D$. An interferometer reaches the disc’s null when $B_{\text{null}} = 1.21967\,\lambda/\theta < B$. Rearrange either and the other appears: “too small for the telescope to resolve” and “needs a baseline longer than the telescope’s mirror” are one inequality written twice.

That is why the instrument was a beam laid across the top of a telescope rather than a bigger telescope. The two requirements are the same length in metres — but two small flats on a girder can be set six metres apart for the price of the girder, and a six-metre paraboloid in 1920 could not be made.

The Sun subtends 1,919 arcseconds, so Betelgeuse’s disc is smaller by a factor of 40,800: sorted by apparent size, the stars put their second entry four and a half orders of magnitude below the first.

What was actually measured, in December 1920 #

The instrument was a steel beam twenty feet long bolted across the front of the Hooker telescope, carrying four flat mirrors. The outer two slid along it, catching the starlight and folding it inwards to the inner pair, which sent it down through the aperture to an eyepiece. The telescope was a light bucket and a mount; the measurement was made by the two flats on the beam.

What an observer recorded was whether fringes could be seen, and at what mirror separation they stopped being seen: not a flux, not a magnitude, not an image, but a yes-or-no judgement of pattern contrast, made by eye, repeated as the outer mirrors moved apart. On 13 December 1920 the fringes on Betelgeuse were gone at a separation of 121 inches, or 3.07 metres, and $1.21967\,\lambda/B$ at that separation is 0.047 arcseconds.

Four things had to be assumed to get from the one to the other. An effective wavelength, because a broad visual response looking at a red star has no wavelength but a band whose effective wavelength depends on the star’s own spectrum, and 575 nm enters the answer linearly. A uniform disc, since 1.21967 belongs to a flat disc and nothing else. That the lost contrast was the star’s rather than the atmosphere’s — a distinction available only by turning to a smaller star and finding its fringes still crisp. And that the star is circular and unchanging, the assumption with no defence at all. The published result carried a radius too, and that is where the point shows. Michelson and Pease quoted a diameter of 240 million miles, which is what 0.047 arcseconds means at about 56 parsecs — the distance the parallax then credited to the star implies — and amounts to a radius near 280 times the Sun’s. A century later the angle has shifted by a few per cent and the radius has tripled. Nothing was wrong with the interferometry.

The radius is a second measurement, and a worse one #

Converting the angle needs a distance: $R = \theta d/2$, with $d = 1/\varpi$ for a parallax $\varpi$. Both factors are measured, so their fractional errors add in quadrature — and they are not comparable in size.

Two published parallaxes for one star, differing by a third in the radius they yield, is the honest state of the number rather than a caveat appended to it. The interferometry is good to a few per cent; the astrometry is not.

The 4.9 per cent limb-darkening term is smaller but it is a different kind of error, because it does not shrink with better observing — it is a choice of model. An interferometric diameter is always quoted as either uniform-disc or limb-darkened, and the two differ by more than the uncertainty of a good modern determination, so comparing them without checking which is which manufactures a five per cent discrepancy out of nothing. The null is a fact and the diameter is an interpretation of the null. Every distance in astronomy is calibrated on the rung below it; every diameter is calibrated on a choice of disc.

An angle larger than the star’s own parallax #

Betelgeuse’s disc is 47 milliarcseconds across; its parallax is 5.95 milliarcseconds, or 4.51 on the revision. The star’s size on the sky is therefore about eight times its parallax, and four times the full width of the ellipse its position traces in a year. That inversion has a cause, and the cause is the disc. Astrometry measures the position of a photocentre, and a red supergiant’s photocentre is not a fixed point: the outer layers of such a star are convective, the cells are a large fraction of the disc across, and as they brighten and darken the centroid of the light moves by an amount comparable to the parallax being sought. The revision that produced 4.51 mas combined radio positions with the optical astrometry, the radio photosphere being a steadier fiducial than the boiling optical one.

So the two factors in $R = \theta d/2$ are not independent difficulties. The reason the angle is easy is that the star is enormous, and the reason the distance is hard is that the star is enormous.

The one quantity a distance cannot touch #

One combination of the measured angle and a measured brightness cancels the distance exactly, and it is the most useful thing an angular diameter delivers.

Received flux is $f = L/4\pi d^2$ and luminosity is $L = 4\pi R^2\sigma T_{\text{eff}}^4$, so $f = (R/d)^2 \sigma T_{\text{eff}}^4$. But $R/d$ is $\theta/2$, and therefore

$$T_{\text{eff}} = \left(\frac{4f}{\theta^2\sigma}\right)^{1/4}.$$

The distance is gone. A bolometric flux and an angular diameter give an effective temperature with no parallax, no model atmosphere and no colour calibration — which makes the handful of stars with measured discs the calibrators for every temperature read off a colour instead. The fourth root is generous, too: the 4.9 per cent ambiguity in $\theta$ becomes 2.4 per cent in the temperature. The two ends of that connection are usually taught in different chapters. The reason a low-surface-brightness galaxy cannot be found by moving closer to it is the reason a stellar temperature can be measured without knowing how far away the star is: one statement about $d^{-2}$ against $d^{-1}$, at 200 parsecs and at 200 megaparsecs.

Two routes to a radius, and what each of them costs #

A radius can also be had without any interferometry: take a temperature from the spectrum, a luminosity from the apparent brightness and the distance, and invert the Stefan–Boltzmann law. That is the route almost every stellar radius in the literature took. The instructive point is that the indirect route does not escape the distance, or even escape it more gently. Luminosity goes as $d^2$, so a radius from $\sqrt{L}$ goes as $d^1$ — the same linear dependence as $R = \theta d/2$. The choice is not about the parallax at all but about which model is imported: a disc profile for the direct route, or a temperature scale, a bolometric correction and an extinction estimate for the indirect one.

Which makes the agreement of the two a weaker check than it looks. Divide one by the other and the distance cancels, leaving the identity of the previous section, so the comparison tests the temperature scale rather than the distance. A genuinely distance-free radius needs another method, and there are only two: an eclipsing binary, where a velocity multiplied by a time gives a length in kilometres, and the seismic route through a star’s oscillation frequencies.

What the picture cannot show #

The null is a zero in the model, not an observation. Nobody has ever seen a visibility of zero. What is recorded is fringes becoming indistinguishable from noise, at a threshold of a few per cent contrast, so the measurement brackets the null rather than landing on it — and the quoted uncertainty on the angle is the width of that bracket.

The disc is neither uniform nor circular nor constant. The two curves are a flat disc and one linear limb-darkening law, 4.9 per cent apart at the null. A red supergiant’s photosphere carries convective cells that are a large fraction of the disc across, its apparent size differs between the visual and the infrared because the molecular layers above the photosphere are opaque in some bands and not others, and the diameter is not the same from one epoch to the next.

One visibility is not an image. The visibility is the modulus of a complex quantity, and the phase — where the information about asymmetry lives — is scrambled by the atmosphere and not recorded. A single baseline samples one point of a two-dimensional Fourier transform. A picture needs many baselines and a way of recovering phase, which is why an interferometric image of a stellar surface came sixty years after an interferometric diameter.

The curve is monochromatic and the observation was not. A broadband measurement sums curves whose nulls fall at different baselines, so the real minimum is a shallow smeared trough rather than a zero, and the 575 nm on the axis stands in for a whole visual response folded through a cool star’s spectrum. Change that effective wavelength by five per cent and the diameter moves five per cent, with no observation involved.

What replaced the twenty-foot beam #

The beam on Mount Wilson measured about half a dozen stars and then stopped, and the reason it stopped is instructive: at six metres the fringes were already so unstable that the observation was at the limit of what an eye could judge, and a longer beam made it worse rather than better.

The instability is atmospheric. The two flats sample the wavefront at points metres apart, and the atmosphere corrupts the phase differently at each, so the fringe pattern moves faster than it can be read. Lengthening the baseline samples points that are less correlated, so the fringes move faster still. The technique was not limited by engineering but by the fact that the quantity being measured — a phase difference — is exactly the quantity the atmosphere destroys.

Two escapes were found and they are quite different.

The first abandoned phase altogether. Hanbury Brown and Twiss showed that the intensities received at two separated detectors are correlated, with a correlation that carries the same visibility information, and intensity fluctuations are far less sensitive to atmospheric phase errors than amplitudes are. The intensity interferometer built at Narrabri in the 1960s worked at baselines up to 188 metres — thirty times the Mount Wilson beam — and measured angular diameters for thirty-two hot stars, down to a fraction of a milliarcsecond. The price was sensitivity: the technique needs enormous photon rates, so it reached only bright, hot stars and could not be pushed further.

The second kept the phase and beat the atmosphere with speed. Modern separated-element interferometers record fringes in exposures short compared with the atmospheric coherence time, and combine them in a way that survives the phase noise — the same trick as the closure phase used to make images. Baselines run to 330 metres at CHARA and 130 at the VLTI, and the number of stars with directly measured angular diameters is now in the several hundreds rather than the several.

What did not change is the observable. A modern determination still reports a visibility against a projected baseline and still fits a disc model to it, and it still delivers an angle rather than a radius. The distinction this essay is about survived every improvement in the instrument, because it was never a limitation of the instrument.

Nor did the arithmetic change. A 330-metre baseline reaches the first null of a 0.44-milliarcsecond disc at 575 nm, which is a hundredfold improvement on the twenty-foot beam and exactly the hundredfold ratio of the two baselines — the relation between length and angle has no scale in it, so every gain in this subject is bought at the price it looks like it should cost.

Where the ladder goes next #

This first rung of angular-diameter has spent its whole length on one number. The rungs above it get past a single baseline: the lunar occultation, where the Moon’s limb sweeps across the star and Fresnel diffraction in the light curve gives the same angle from a measurement in time rather than in space; the visibility measured beyond the first null, where flat and darkened discs diverge far more than 4.9 per cent and the limb darkening stops being assumed and starts being observed; and phase closure across three or more apertures, which turns a set of visibilities into a picture of a surface.

Then the rung that pays the collection back: angular diameters predicted from colours, by a relation calibrated on the measured ones, which is what supplies the angular sizes that turn eclipsing-binary radii into distances. Beside it sits the interior read from a comb of oscillation frequencies, which returns a radius with no distance in it at all — and comparing that with a radius built from a null baseline and a parallax is the sharpest available test of both.