A population counted by shadows that never repeat

Every occultation described here so far has been predicted. A body’s orbit is known, a star’s position is known, and the shadow’s track across the Earth is computed in advance so that telescopes can be put in it.

There is a whole population for which none of that is possible. Bodies a few kilometres across in the outer solar system are far too faint to have been catalogued — a one-kilometre object at forty astronomical units is around magnitude 35 on a scale that runs backwards — so their orbits are unknown, their positions are unknown, and no event involving one can ever be predicted.

They still cast shadows, and a shadow does not care how faint its caster is. The approach is therefore to stop predicting and start waiting: monitor many stars at high cadence, record everything, and treat an occultation as something that happens rather than something that is arranged.

Why a small body casts a shadow bigger than itself #

The edge of a shadow is a wave, and that is usually a limitation. Here it is what makes the measurement possible at all.

Light passing an obstacle diffracts, and the scale over which the geometric shadow is blurred into a fringe pattern is the Fresnel zone, $\sqrt{\lambda D/2}$ — the geometric mean of the wavelength and the distance. At Kuiper belt distances in visible light that is a little over a kilometre.

A body much larger than that casts a shadow of its own size with fringed edges. A body much smaller than that does not cast a shadow of its own size; it casts a diffraction pattern about a Fresnel zone across, shallower than a full obscuration but no narrower.

The consequences run in both directions.

Against the survey: the events are shallow. A half-kilometre body dims the star by a fraction rather than blocking it, so the detection threshold is a photometric one rather than a geometric one, and a marginal event is hard to distinguish from noise.

For the survey: the events are not shorter. The duration is the shadow’s width divided by the relative speed — an interval of time is a length once something is known to be moving — and the width has a floor, so an event by a sub-kilometre body lasts about the same fraction of a second as an event by a kilometre one. A survey does not have to sample faster to reach smaller bodies, which it would if the shadow shrank with the caster.

What a rate measures #

The measurement is not a detection. A single serendipitous event is by construction unrepeatable and uncheckable: the body will not be found again, its orbit will not be determined, and nothing about it can be confirmed by any other means.

What is measured is a rate — events per star per hour — and what a rate carries is an integral over the population:

$$\text{rate} \;\propto\; \int n(a)\,2\!\left(a + F\right)v\,da,$$

with $n(a)$ the number density per unit size, $F$ the Fresnel scale, and $v$ the relative speed. The observable is that integral, evaluated above whatever size the survey can detect.

So a survey with a lower detection threshold measures a larger rate, and the ratio between two surveys with different thresholds is the slope of the size distribution over the interval between them. That is the quantity the whole exercise exists to get.

The size distribution is what a collisional history predicts, and it is the one prediction of collision theory that can be tested at kilometre scales in the outer solar system. Larger bodies are catalogued by imaging and their distribution is measured directly; below a few tens of kilometres nothing else reaches.

The prediction a cascade makes #

The reason a size distribution is worth this much effort is that it is the fossil of a collisional history, and the theory makes a sharp prediction.

A population in collisional equilibrium — where every size is being destroyed by collisions with smaller bodies as fast as it is being resupplied by the fragmentation of larger ones — settles into a power law. For bodies whose strength is independent of size the exponent is 3.5, a result of Dohnanyi’s from 1969 that follows from the self-similarity of the cascade rather than from any detail of the collisions.

Real populations depart from it in ways that carry information. Bodies large enough to be held together by their own gravity rather than by material strength are harder to disrupt, so the distribution steepens above that size. Bodies small enough to be removed by radiation forces are depleted, so it turns over at the bottom. And a population that has not had time to reach equilibrium retains the slope it was born with.

For the Kuiper belt the question is whether the sub-kilometre population is in equilibrium at all. The collision rate there is low — the bodies are far apart and moving slowly relative to one another — so the cascade may not have processed the small end over the age of the solar system. A measured slope of 3.5 would say it has; anything else would say it has not, and would be a measurement of the belt’s original size distribution.

That is a statement about what the outer solar system was made from, reached by counting shadows.

What has been found, and what was withdrawn #

The history of this technique is a record of marginal detections, and the pattern is worth setting out because it shows what a rate measurement has to survive.

Early searches using ground-based photometry of single stars reported candidate events at rates implying a very large population of small bodies. Most did not survive scrutiny: the events were single-station, at low significance, and the statistical treatment of a search that examines millions of samples for a rare dip is unforgiving.

The problem is a familiar one in a form that is especially severe here. A survey recording a hundred stars at twenty hertz for a hundred hours takes $7\times10^{8}$ samples. A threshold that admits one false positive per $10^{8}$ samples still admits seven, and the expected number of real events may be smaller than that.

Three defences are used and all three are necessary.

Multi-station coincidence. A real occultation is recorded simultaneously at telescopes separated by less than the shadow’s width and not at telescopes outside it. A detector artefact, a cosmic ray, a bird, or a scintillation event is recorded at one station only. This is the strongest test and it is why modern searches use paired telescopes.

The shape. A real event by a body near the Fresnel scale has a specific diffraction profile — a dip flanked by brightenings — set by the wavelength, the distance and the body’s size. Matching that profile is a much stronger requirement than exceeding a threshold, and it is what distinguishes an occultation from an atmospheric scintillation, which has its own characteristic timescale and no fringes.

The ecliptic latitude dependence. Kuiper belt objects are concentrated toward the ecliptic, so a real population produces a rate that falls with latitude. A survey that observes both on and off the ecliptic has an internal control, and any instrumental cause produces the same rate in both.

The searches that apply all three report rates consistent with a population, at the smallest sizes ever probed in the outer solar system, and the numbers remain uncertain by factors of several.

The rate, computed from the other direction #

There is a consistency check available and it is worth doing, because it shows how few events the technique expects.

Take a survey monitoring stars near the ecliptic. The number density of kilometre-scale bodies implied by extrapolating the catalogued population downward gives a certain optical depth along a line of sight to a background star — the fraction of the sky covered by their shadows. That optical depth is tiny: of order $10^{-9}$ or less.

The rate is that optical depth times the rate at which shadows sweep past, which is the shadow width divided by the relative speed of the shadow across the star’s apparent position. For a Kuiper belt object the Earth’s own orbital motion dominates, at about thirty kilometres a second, so a 1.3-kilometre shadow crosses in a twentieth of a second.

Multiplying through gives of order one event per star per thousand hours. A survey monitoring a hundred stars for a hundred hours expects around ten.

Ten expected events against a hundred million samples is the arithmetic that makes the false-positive discipline above mandatory rather than careful.

Why the duration is the diagnostic it cannot quite be #

A single event carries one number beyond its depth, and it is the duration. In principle that number locates the body.

The shadow crosses the observer at the relative velocity of the shadow across the line of sight, which for an outer solar system body is dominated by the Earth’s own orbital motion — thirty kilometres a second, modulated by the body’s motion and by the angle between the line of sight and the Earth’s velocity. So the duration is the shadow width divided by a speed that depends on the observing geometry and on the distance.

Both the width and the speed depend on distance, and they do so in opposite directions. The Fresnel scale grows as the square root of the distance; the apparent relative speed falls as the body’s own contribution becomes negligible and the geometry changes. The net dependence is weak — roughly the square root — so a factor of four in distance is a factor of two in duration.

A duration measured to twenty per cent therefore locates a body to a factor of about one and a half in distance, which is enough to separate the Kuiper belt from the asteroid belt and nowhere near enough to separate the classical belt from the scattered disc.

The events are also short enough that the duration is measured across a handful of samples. At twenty hertz a fiftieth-of-a-second event is one sample; at two hundred hertz it is ten. Sampling rate buys distance discrimination, and that is the main argument for the fastest cameras rather than the most sensitive ones.

Where it works better #

The obstacles are photometric, so the escapes are photometric, and two of them are in use.

Above the atmosphere. Scintillation is the dominant source of false positives from the ground, and it is an atmospheric effect. A space telescope monitoring a star field at high cadence has no scintillation, and the two events reported from stellar-photometry missions at kilometre scales are the least ambiguous in the subject — obtained, in both cases, from instruments built for something else.

In the occultation of a bright star by a known body. A survey of a different kind watches the star field around a predicted event and catches shadows of the satellites or fragments near the target, which is how several small bodies have been found — and how rings around a Centaur were.

And one escape is proposed and not yet realised: at longer wavelengths the Fresnel scale is larger, so an infrared or radio survey has a bigger cross-section per body and a lower size threshold in the sense that matters for the rate. It also has worse photometric precision, and the trade has not yet come out in favour.

What the catalogued population already says #

The small end is what this technique reaches; the large end is measured by imaging, and the two have to join up.

Deep imaging surveys of the ecliptic have catalogued several thousand trans-Neptunian objects, each one an orbit fitted from a short arc of angles and the luminosity function they trace is well measured down to about magnitude 27, corresponding to bodies of a few tens of kilometres. Over that range the size distribution is steep — a slope well above the collisional-equilibrium value — with a distinct break at around a hundred kilometres, above which it is steeper still.

The break is the interesting feature and it has a natural reading. Bodies above it are the ones that were never destroyed; bodies below it are fragments. If that is right, the break marks the largest size at which collisions have processed the population over the age of the solar system, and it is a measurement of the belt’s collisional history rather than of its formation — the same reading an asteroid family’s age gets from a scatter plot.

An alternative reading makes the break primordial: planetesimals formed at about that size directly, by a streaming instability, and the small bodies below it are indeed fragments while the ones above it are original.

The two readings make different predictions about the slope below a kilometre, which is exactly where the imaging stops and the shadows begin. The occultation surveys are aimed at the one interval that distinguishes two accounts of where planetesimals came from, and that is why a technique yielding two confirmed events in two decades is still worth building instruments for.

Where the picture stops #

A rate is a product and the factors are not separable. The measured rate depends on the number density, the size distribution, the distance and the relative speed. A survey that detects events cannot tell whether they came from bodies at forty astronomical units or from a nearer, sparser population — the durations differ, but only by the square root of the distance, and a marginal event has a poorly measured duration.

The Fresnel scale assumes a point source and monochromatic light. A real observation is over a band, and a broad band washes out the fringes because each wavelength has its own pattern. That reduces the contrast and therefore the detectability, and the effective Fresnel scale for a broadband observation is not the monochromatic one.

And the shadow speed is the Earth’s, not the body’s. Which means the technique’s sensitivity depends on the time of year and the direction of observation, since the component of the Earth’s velocity across the line of sight varies through the year and vanishes twice.

The habit, and where else it applies #

The structure here is one worth naming because it recurs wherever individual objects are beyond reach.

When no single detection can be confirmed, the confirmable quantity is a rate, and the design problem becomes controlling the false-positive rate rather than characterising an event. Everything about the survey — the multi-station requirement, the profile matching, the latitude control — is aimed at the denominator rather than at the numerator.

The same shape governs the microlensing surveys, where an event is also unrepeatable and the science is in the statistics of many; it governs gravitational-wave searches before the loudest events arrived; and it governs every search for a rare transient against a large background.

What distinguishes the good instances from the bad is whether the survey has an internal control — some direction, time or configuration in which the real signal must be absent and the instrumental one must not. A search with no null direction is a search that cannot report a rate, only an upper limit, and the difference between those two outcomes is a design decision made long before any data are taken.

Still open: how many there are #

The measured rates are consistent with a population and are not yet a measurement of one.

The published constraints at the sub-kilometre end differ by an order of magnitude between surveys, and the two space-based detections are two events. The slope over the interval from a hundred metres to a few kilometres — the quantity that would say whether the belt’s small bodies are collisionally processed — is unconstrained.

What would settle it is a survey with enough simultaneous stations and enough hours to accumulate tens of unambiguous events at two different thresholds. That is an instrument rather than an observation, and several have been proposed: arrays of small telescopes monitoring thousands of stars, with the coincidence requirement built into the design rather than assembled afterwards.

The obstacle is not the telescopes, which are cheap. It is that the data rate from a thousand stars at twenty hertz is large and the events are rare enough that nothing can be thrown away until it has been examined, so the analysis is the instrument.