The edge of a shadow is a wave

The rung below this one treated an occultation as geometry: a shadow with a boundary, a set of observers at known places, and a silhouette reconstructed from the chords they timed. Everything in it was ray optics, and everything in it was correct, because an asteroid’s shadow really does have an edge.

The Moon’s does not. A shadow boundary is only sharp at scales larger than the Fresnel scale $\sqrt{\lambda D/2}$, and at 550 nanometres and 384,000 kilometres that is ten metres. So what sweeps across a telescope during a lunar occultation is not an edge at all: it is a diffraction pattern, it has structure on the scale of the aperture, and it goes past in a tenth of a second.

Why a knife edge does not cast a sharp shadow #

The half-plane is the oldest problem in diffraction and one of the few with a closed solution. Light arriving at the observation plane comes from every unobstructed point of the wavefront, and the contributions add with phases proportional to the extra path each one travels. Writing that sum in the standard variable

$$w \;=\; x\sqrt{\frac{2}{\lambda D}}$$

gives an intensity built out of the Fresnel integrals $C(w)$ and $S(w)$,

$$I(w) \;=\; \tfrac{1}{2}\Big[\big(C(w)+\tfrac{1}{2}\big)^2 + \big(S(w)+\tfrac{1}{2}\big)^2\Big],$$

and the whole of the observable behaviour is in that expression.

Three of its values are worth carrying. At the geometric edge, $I = 1/4$. Half the wavefront has been removed, so the amplitude is halved, and intensity is amplitude squared — the factor of two that a ray-optics intuition supplies is the wrong one, applied to the wrong quantity. The first maximum is 1.37, at 1.217 Fresnel scales into the illuminated side: the light removed from the shadow has to go somewhere, and it goes just outside. And deep in the shadow the intensity falls as $1/w^2$ rather than to zero, so a star behind the Moon’s limb does not vanish, it fades with a tail.

None of that depends on anything about the star or the Moon. It is the same pattern behind a razor blade in a laboratory, scaled by one length.

The one length #

The Fresnel scale is what sets every practical number in the technique, and it is short.

At $\lambda = 550$ nm and $D = 384{,}400$ km,

$$\sqrt{\lambda D/2} \;=\; \sqrt{(5.5\times10^{-7})(3.84\times10^{8})/2}\;\mathrm{m}\;=\;10.3\ \mathrm{m}.$$

The Moon’s limb crosses a fixed observer at between about 0.2 and 0.8 kilometres a second depending on the geometry, so one Fresnel scale passes in about seventeen milliseconds. The first few fringes — the part that carries the information — are over in a tenth of a second.

That is the reason the technique is a photometric one rather than an imaging one and the reason it needs a particular kind of instrument: a photometer sampling at a kilohertz, on a telescope large enough to collect useful counts in a millisecond. It is also why it was, for decades, the only way to reach a milliarcsecond.

How a diameter comes out of it #

A star is not a point. Each point on its disc lies in a slightly different direction, so each casts the same diffraction pattern displaced by its own offset — the offset being the angular separation times the distance to the Moon. The observed trace is the knife-edge pattern convolved with the star’s projected brightness distribution.

The projected width of a stellar disc of angular diameter $\theta$ is $\theta D$. For a star of one milliarcsecond at lunar distance that is

$$(1\ \mathrm{mas})(4.85\times10^{-9}\ \mathrm{rad/mas})(3.84\times10^{8}\ \mathrm{m}) \;=\; 1.9\ \mathrm{m},$$

against a fringe scale of 10.3 metres. A one-milliarcsecond star therefore blurs the pattern slightly; a ten-milliarcsecond star blurs it away completely. Between those two the contrast is a monotone function of the diameter, and measuring the contrast measures the diameter.

The sensitivity is set by the ratio of the two lengths, and it is worth naming what that means: the technique’s resolution is roughly $\lambda/D_{\rm Moon}$ expressed as an angle, which is about half a milliarcsecond. That is the angular resolution of a telescope 384,000 kilometres across, obtained with a telescope one metre across, because the aperture doing the work is the distance to the obstruction rather than the size of the collector.

What a millisecond of photometry costs #

The instrumental demand is worth stating in photons, because it is the reason the technique arrived when it did rather than in 1920.

A fringe has to be sampled several times as it passes, so the integration is of order a millisecond. In a millisecond a one-metre telescope collects, from a star of sixth magnitude through a broad filter, something like ten thousand photons — which sounds ample and is not, because what has to be measured is a contrast of a few per cent in a curve sampled a hundred times. Counting statistics alone put the noise at one per cent per sample, and the scintillation of the atmosphere at that timescale is worse.

Scintillation is the awkward one. It is the same turbulence that blurs an image, seen in intensity rather than in position, and its characteristic frequency is tens of hertz — squarely inside the band the fringes occupy. It is beaten by aperture: a larger telescope averages over more independent patches of the wavefront, and the fractional scintillation falls as $D^{-2/3}$. So the technique’s practical floor is not a photon count but a variance imposed by the air, and the way through it is the same one that made the atmosphere’s own speckles into a measurement rather than a nuisance.

Two further conveniences make it work at all. The event is short, so the atmosphere and the instrument barely drift during it, and the before and after levels of the same star supply their own calibration — nothing about the absolute throughput enters. And the star’s own brightness cancels, because the fitted quantity is a shape rather than a level: the whole measurement is of a ratio between adjacent samples of the same light curve.

What was actually measured #

The first stellar diameter from a lunar occultation was obtained in 1939, four decades after the first interferometric one, and the technique became routine in the 1970s once fast photometers and minicomputers existed together.

The observation is a light curve at a kilohertz through a broad filter, lasting a few seconds and containing perhaps a hundred milliseconds of signal. What is fitted to it is a model with the star’s angular diameter, the limb speed, the local slope of the lunar limb, the effective wavelength of the filter, and the star’s brightness. Several of those are nuisances and one of them is a genuine problem.

The limb is not a knife edge on all scales. The Moon has mountains, and a limb profile with a metre-scale slope changes the effective geometry — the shadow is cast by a surface at an angle rather than by a straight boundary. In practice this is handled by observing the same star at several limb positions, or by preferring events at the dark limb near the poles where the topography is better known.

The effective wavelength is not the filter’s centre. The pattern’s scale goes as $\sqrt{\lambda}$, so a broad filter smears the fringes in the same way a finite source does, and the two effects are degenerate unless the bandwidth is known. Narrow filters cost photons; broad ones cost precision.

And the star may not be a uniform disc. What is fitted is a uniform-disc diameter, which is a Fourier-domain summary rather than a physical radius. Limb darkening makes the true photospheric diameter some few per cent larger, and the correction is a model rather than a measurement — the same correction, and the same difficulty, as in interferometry.

Where the technique stops #

The limitation is not precision. It is the Moon.

An occultation happens when the Moon passes over a star, and the Moon stays within about five degrees of the ecliptic. Only stars in that band are ever occulted, which is roughly ten per cent of the sky, and within that band which stars and when is decided entirely by the lunar orbit. A star of interest may be occulted several times in a year or not for a decade.

The position angle is not chosen either. What is measured is the star’s extent perpendicular to the limb at that event, so an elongated star — a rapid rotator, or a close binary — gives different answers at different events, and mapping the shape requires waiting for the geometry to come round.

The record it left #

Some seven hundred stellar angular diameters were measured by lunar occultation between 1970 and 2000, and that catalogue is still doing work.

Most of the entries are red giants and supergiants, because those are the stars large enough to resolve and bright enough to sample at a kilohertz. Their diameters, combined with distances that were mostly parallaxes, gave the first substantial set of stellar radii not inferred from a model — and radii are the quantity a stellar-structure calculation is hardest to check against, because everything else it predicts has already been used as an input.

The catalogue also produced a result nobody was looking for. A significant fraction of the events showed two diffraction patterns superimposed, displaced by a few milliarcseconds and with different amplitudes: unrecognised binaries, resolved by an accident of geometry at separations no imaging survey of the time could reach. Lunar occultations turned out to be one of the more productive ways of finding close companions, and the statistics of how many stars have them — which matters for everything from the mass function to the interpretation of a spectrum — rest partly on that sample.

What ended the technique’s dominance was not a better version of it. Long-baseline optical interferometry reached the same angles from the ground in the 1990s, on stars of the observer’s choosing rather than the Moon’s, and space astrometry then supplied the distances that turn an angle into a radius. The occultation catalogue became a calibration set for the instruments that replaced it, which is a respectable end.

The generalisation #

The quantity being measured here is not a size but a coherence, and that is why the method reaches angles no image can.

A perfectly point-like source produces a diffraction pattern of full contrast, because every part of the wavefront arriving at the observer carries the same phase relationship. An extended source produces overlapping patterns whose fringes are displaced relative to one another, so the contrast falls; and the rate at which it falls with the geometry is a Fourier transform of the source’s brightness distribution. That is the van Cittert–Zernike relation, and it is the same statement that underlies an interferometer’s visibility and the phase closure that survives what corrupts it.

The three techniques differ only in what supplies the baseline: two telescopes, in an interferometer; the atmosphere’s own patches, in speckle interferometry; and a rock 384,000 kilometres away, here. In every case the resolution is set by the largest separation over which the wavefront is compared, and in every case the object being measured is a correlation rather than a picture.

The rings that turned up in a light curve #

The most consequential result this technique produced was not a diameter and was not looked for.

In March 1977 several groups observed the occultation of a ninth-magnitude star by Uranus, intending to measure the planet’s atmosphere from the shape of the ingress and egress. What they recorded instead began some thirty-five minutes before the planet reached the star: a series of brief, sharp dips in the star’s brightness, five of them, lasting a second or two each.

The same pattern repeated after the planet had passed, in mirror image and at the same distances from the planet’s centre. A symmetric set of narrow occulting features on both sides of a planet is a ring system, and Uranus was not known to have one.

The observation is worth setting beside the essay’s earlier discussion, because it is the geometric regime rather than the diffractive one. The rings are narrow — some are only a few kilometres wide — and at Uranus’s distance the Fresnel scale is about a kilometre, so the events are at the boundary between the two: partly a shadow, partly a diffraction pattern, and fitting them requires both.

What the light curves gave was extraordinary for the epoch. Each dip’s duration times the relative speed is a chord across a ring, so the widths came out directly, in kilometres, for a system that no image of the time resolved at all. The radii came from the timing, and the shapes came from comparing chords at different events — which is how the epsilon ring was found to be eccentric and to vary in width from about twenty kilometres to ninety.

A comparable discovery followed for a small body between Saturn and Uranus in 2013, again from a stellar occultation, again as symmetric dips flanking the main event — the first rings found around anything that is not a planet.

A technique built to measure atmospheres and stellar diameters has produced two ring systems, both as unexplained features in a light curve somebody was recording for another reason.

Two consequences of the same length are worth following before the ladder, and only the first of them was anybody’s intention.

The scale that moves with the colour #

The Fresnel scale carries a square root of the wavelength, and following that dependence explains why the same technique behaves so differently in different bands.

At visible wavelengths and lunar distance it is ten metres. In the near infrared at two microns it is twenty, so the fringes are twice as wide and pass twice as slowly — which relaxes the timing requirement, at the cost of resolution, since the smallest source distinguishable from a point scales the same way.

The trade is usually worth taking for a cool star, because a red giant is far brighter in the infrared than in the visible and the photon budget is the binding constraint. Several of the later occultation programmes worked entirely in the near infrared for that reason.

Push the wavelength up further and the regime changes completely. A spacecraft’s radio signal at a few centimetres, passing through a planet’s atmosphere on its way to Earth, has a Fresnel scale of kilometres — so what is measured is not an edge at all but a large-scale refraction, and the analysis becomes an inversion for the atmospheric density profile rather than a fit to a diffraction pattern.

That is radio occultation, and it is the standard way of measuring the atmospheres of the outer planets and of Titan. The same technique pointed at the Earth, using the signals of navigation satellites received by a satellite in low orbit, gives temperature profiles through the terrestrial atmosphere and is now a routine input to weather forecasting.

One length, varying as the square root of the wavelength, decides whether an occultation is a diffraction experiment, a photometric one or an atmospheric sounding — and the three communities that do those things do not generally realise they are doing the same measurement.

Where this ladder goes next #

Later rungs on this anchor: occultations by the outer planets’ moons and by Kuiper belt objects, where the events are minutes long and the Fresnel scale is kilometres; the discovery of rings by exactly this method, at Uranus in 1977 and around a small Centaur in 2013; the atmospheric profiles that stellar occultations give for Pluto and Titan, which are the only direct measurements of those atmospheres between spacecraft visits; central flashes, where refraction focuses light through the far limb and probes the deepest layers; and the astrometric use of the technique, where the event’s timing fixes the occulting body’s position to a few milliarcseconds and improves its ephemeris more than a year of imaging would.