A shape measured by the edge of a shadow

An asteroid two hundred kilometres across, three astronomical units away, subtends about a tenth of an arcsecond. From the ground that is a smear; from space it is a few pixels. Nothing about its shape is visible.

Wait for it to pass in front of a star, and the situation is transformed. The star vanishes, and some seconds later it comes back. The interval, multiplied by the speed at which the body’s shadow sweeps across the ground, is a length — a chord across the silhouette, measured to a kilometre or two by anybody with a telescope, a video camera and a clock.

The geometry #

The body casts a shadow of the star, and because the star is effectively at infinity the shadow is the same size as the body and travels at the body’s velocity relative to the Earth projected onto the sky plane. For a main-belt asteroid that is typically fifteen to twenty-five kilometres a second; for a trans-Neptunian object, more like twenty-five.

An observer inside the shadow sees the star disappear when the leading limb crosses the line of sight and reappear when the trailing limb does. The chord’s length is the interval times the shadow speed, and its offset — which chord across the body it is — is set by where the observer stands relative to the shadow’s centreline. The precision comes from the fact that a time is being measured. At twenty kilometres a second, a tenth of a second is two kilometres, and a video camera with a GPS-disciplined clock does considerably better than a tenth of a second. That is a fractional precision of about one per cent on a two-hundred-kilometre body — comparable with what a spacecraft flyby delivers, from equipment that fits in a car.

Chords, and how many are needed #

One chord gives one length and says nothing about shape. Two give a size and an orientation if the offsets are known. Getting a profile takes a spread of chords across the body, and the arithmetic is unforgiving: the chords are parallel, so their spacing is set by how far apart the observers are perpendicular to the shadow’s path, and that has to be organised in advance.

The result is one of the few genuinely coordinated amateur–professional activities in astronomy. A predicted shadow path a few hundred kilometres wide is announced, observers distribute themselves across it, and the chords are combined afterwards. The distribution matters more than the number: ten observers clustered near the centreline produce ten nearly identical chords and almost no shape information, while five spread evenly across the path constrain the limb everywhere. The information in a set of chords is in their spread, not their count, and the campaigns are organised accordingly.

Three things are then extracted.

A size. The equivalent-area diameter of the fitted limb, which is the quantity a thermal model or an albedo needs — and which is otherwise available only from an angular measurement at the very limit of what a telescope can resolve, multiplied by a distance.

A shape. With enough chords the limb is fitted with a low-order harmonic series rather than an ellipse, and features of a few kilometres are resolved.

A position. The shadow’s path on the ground is determined by the body’s position relative to the star, so a successful occultation is an astrometric measurement of the body at the milliarcsecond level — far better than direct imaging gives.

What a size is worth #

A diameter measured this way is not an end in itself. It is the missing half of two other measurements.

With a mass, it is a density. An asteroid with a satellite has a mass from the satellite’s orbit and a volume from a shape model, and the ratio separates a rubble pile from a solid body. Densities below about 1.5 grams per cubic centimetre for an object made of rock mean the interior is half empty, which is a statement about collisional history rather than about mineralogy.

With a brightness, it is an albedo. An object’s apparent magnitude is the product of its cross-sectional area and its reflectivity, and neither is separable from the other by photometry alone. An occultation supplies the area, so the albedo falls out — and albedo is the closest thing to a compositional classification available for a body too faint for spectroscopy. The albedos of the outer solar system’s small bodies, which range from three per cent to nearly ninety, were largely established this way and by thermal radiometry, and where the two disagree the occultation is believed.

The most striking application was a target selection. New Horizons’ second target, Arrokoth, was observed in three occultation campaigns in 2017 by teams distributed across Argentina and South Africa, chasing a shadow whose predicted path was uncertain by more than its own width. The chords showed a body about thirty kilometres long and strongly bilobed — a shape confirmed in every detail when the spacecraft arrived eighteen months later. A silhouette obtained from a dozen portable telescopes was the mission’s only image of its target until the encounter itself.

What sets the floor #

The edge of a shadow is not perfectly sharp, and the reason is diffraction rather than any property of the body.

Light passing the limb diffracts, and the pattern has a characteristic scale $\sqrt{\lambda D/2}$ where $D$ is the distance to the occulter. At the distance of the main belt that is about half a kilometre; at Pluto’s distance it is a little over a kilometre. A disappearance is therefore not instantaneous but takes the time for that scale to pass — some tens of milliseconds — and the light curve shows fringes on either side of the event.

This is a floor, not a nuisance. Nothing about a chord can be measured better than the Fresnel scale, however good the clock. It is also useful: the fringe pattern encodes the angular diameter of the star, because a star large enough to smear the fringes is a star whose size can be recovered from how much they are smeared.

An atmosphere, from the shape of the ingress #

A body with no atmosphere cuts the star off in the time the Fresnel scale takes to pass. A body with one does not: refraction bends the starlight away from the observer gradually as the ray’s closest approach descends into denser air.

For an isothermal atmosphere the transmitted flux obeys $\varphi = 1/(1+u)$ with $u$ growing exponentially inwards, so the fall from 90 per cent to 10 per cent takes exactly $H\ln 81 = 4.39H$ — and measuring that interval measures the scale height with no model of the body in it at all.

Pluto’s atmosphere was found this way in 1988, from an occultation observed by an airborne observatory, and the several dozen events observed since have tracked its surface pressure rising by a factor of three as Pluto receded from perihelion — a seasonal measurement of a body nobody had visited, made entirely from the shapes of light curves.

There is a further prize for an observer near the shadow’s centreline. Rays passing the far limb are bent back towards the axis, and directly on the centreline they converge: the star briefly brightens in the middle of the event. That central flash probes the deepest layers the method can reach, and its shape carries the oblateness of the atmosphere and any haze in it.

Rings, which were found by accident #

In March 1977 an aircraft-borne team observing an occultation by Uranus recorded a series of brief dips before the planet itself covered the star, and an answering set afterwards. The dips were symmetric about the planet, and nothing but a ring system produces that.

The same signature turned up in 2013 around Chariklo, a body 250 kilometres across in an orbit between Saturn and Uranus, in a run recorded to measure a diameter. Two narrow rings, seven and three kilometres wide, at 391 and 405 kilometres from the centre. Nobody had suggested that such an object could have rings; the observation preceded the theory entirely.

Haumea’s rings were found the same way in 2017, and Quaoar’s in 2023 — the last of these at a radius well outside the body’s Roche limit, which is a genuine puzzle rather than a detail. Inside the Roche limit a satellite cannot hold together and outside it material should accrete into one, so a ring at twice that distance is either very young or is being held apart by something.

What the four discoveries have in common is that none of them was looked for. Every one was found in a light curve recorded to measure a diameter, by someone who noticed a dip where nothing should have been — which is a reasonable argument for recording the whole light curve rather than only the interval around the predicted event, and for looking at it.

The occulter that is not the target #

The Moon occults something almost continuously, and lunar occultations are a distinct sub-technique with a different set of virtues and a different limiting nuisance.

The virtue is availability. The Moon covers about half a per cent of the sky and moves through its own diameter in an hour, so it passes in front of a usable star somewhere every few minutes, and no campaign or coordination is needed — a single observer with a fast photometer records events on any clear night.

What such an event measures is the star, not the Moon. The Moon’s limb sweeps across the star at about half a kilometre per second at the observer’s distance, and the diffraction fringes described above pass in tens of milliseconds. A point source produces a clean fringe pattern; a resolved source smears it, and the degree of smearing gives the star’s angular diameter. Hundreds of stellar diameters were measured this way before interferometry could reach them, and the technique still holds its own for the brightest late-type giants.

It also finds companions. A binary whose separation is a few milliarcseconds is unresolvable by any direct means and produces two separate fringe patterns in a lunar occultation, offset in time by the projected separation divided by the limb speed. A large fraction of the known close binaries among bright stars were found this way.

The nuisance is the Moon’s own limb, which is not a smooth circle. Mountains and crater rims mean that the height of the limb varies by kilometres along its length, so the exact time of an event depends on which piece of terrain happened to arrive first. For an ordinary occultation that is a systematic of a few tens of milliseconds; for a grazing event, where the star passes along the limb rather than across it, the star winks in and out repeatedly as peaks and valleys cross it, and the sequence of winks is a profile of the lunar terrain at that longitude.

The irregularity that spoils one measurement is the whole of another, which is the standing pattern of this subject and is the reason grazing occultations were organised as expeditions for decades.

What the technique cannot do #

It needs a star in the right place. The event happens when it happens, and for a given body events bright enough to be useful come along a few times a year at best.

It gives a silhouette, not a shape. A chord profile is a two-dimensional outline at one instant. Building a three-dimensional shape requires several events at different viewing geometries, combined with a rotational light curve — and the combination is what has produced the shape models for a few hundred asteroids. A silhouette also cannot see a concavity that happens to point away from the observer, so a profile is strictly an upper bound on the cross-section at every angle.

And the timing is the whole measurement. An observer whose clock is wrong by a second has produced a chord displaced by twenty kilometres, and there is nothing in the data to reveal it. That is why the field standardised on GPS time insertion into the video stream, and why a chord from an unverified clock is discarded rather than downweighted. Six kinds of second are in circulation and the one wanted is straightforwardly UTC, but a camera that stamps the time at the end of an exposure rather than the middle has introduced a systematic of half a frame, which at twenty kilometres a second is a kilometre — comparable with the Fresnel floor and quite invisible in the data.

A shadow path has a width and a duration and both are small. A body two hundred kilometres across casts a shadow two hundred kilometres wide moving at twenty kilometres a second, so the event lasts ten seconds at most for any one observer and the whole path sweeps a given latitude in minutes. There is no second chance and no way to integrate longer.

Counting what cannot be seen #

There is an inversion of the technique that turns it from a way of measuring a known object into a way of detecting unknown ones, and it is the only method that reaches the smallest bodies in the outer solar system.

A Kuiper belt object a kilometre across at forty astronomical units is far too faint to image — twenty magnitudes below anything a survey detects. Its shadow, however, is a kilometre wide and sweeps past the Earth at about twenty-five kilometres a second, so it occults a background star for four hundredths of a second. Nothing about the occulter’s brightness enters; only its size.

Monitoring a large number of stars at high cadence therefore samples the population of small bodies directly. The events are brief, they are unpredictable, and any one of them is a marginal detection — so the technique is statistical from the start: what is measured is a rate of events per star-hour, which converts into a surface density of objects above a size threshold.

The difficulty is that at these scales the shadow is comparable with the Fresnel zone, so the event is not a simple dip but a diffraction pattern, and the pattern’s shape carries the object’s size. That is a gift and a hazard: it means a sub-kilometre object is detectable at all, and it means the light curve of a real event looks like the light curve of a cosmic ray hit or a bird, and distinguishing them requires observing the same star with two telescopes simultaneously.

A handful of detections have been reported, from space-based photometers monitoring stars for other purposes and from dedicated ground arrays, and they imply a break in the size distribution around a kilometre — fewer small bodies than an unbroken power law predicts, which is a statement about collisional evolution in a population nobody can image.

The method’s insensitivity to brightness is what makes it unique, and it is the same property that makes an occultation profile a length rather than a ratio: what is being measured is a geometry, and a geometry does not care how much light the object reflects.

Where this ladder goes next #

This rung has established the observable and what is extracted from it: a chord, a profile, an atmosphere from a gradient, a ring from a symmetry.

The rung above is the shape model: combining occultation profiles from several epochs with rotational light curves to invert for a three-dimensional convex shape and a spin state, which is the standard route to an asteroid’s volume and hence to its density when a satellite gives a mass.

Beside it lies the use of occultations as astrometry — the milliarcsecond positions they yield are now good enough to improve the orbits of the bodies observed, so each event pays for the prediction of the next.

And below it, the habit: an interval of time is a length once something is known to be moving. Every result in this essay comes from that single conversion, applied to an object that cannot be resolved, by observers who cannot see it.