Each event pays for the prediction of the next

An occultation prediction is one of the shortest calculations in observational astronomy. Take the star’s position, take the body’s position, draw the line between them, and find where it meets the Earth. Everything else — the time, the duration, the shadow’s speed and width — falls out of the same geometry.

The calculation is short and the inputs are the difficulty. An error in either position is an error in the direction of the line, and by the time that line has travelled forty astronomical units a milliarcsecond has become twenty-nine kilometres.

For most of the technique’s history that arithmetic was fatal. Positions good to fifty milliarcseconds put the shadow fifteen hundred kilometres from where it was predicted, and a body a hundred kilometres across casts a shadow a hundred kilometres wide. Campaigns were lotteries.

The arithmetic, and the number worth memorising #

The relation is $\Delta = \theta D$ and everything follows from one conversion.

A milliarcsecond is $4.848\times10^{-9}$ radians. An astronomical unit is $1.496\times10^{8}$ kilometres. Their product is 0.7255 kilometres, and every requirement in this subject is that number times a distance in astronomical units divided by a shadow width in kilometres.

Three consequences drop out.

Big bodies are easy. Uranus is fifty-one thousand kilometres across at nineteen astronomical units, so the shadow is wide enough that an error of a few arcseconds still lands somewhere on it. The 1977 ring discovery needed no precision at all by these standards, and that is why it happened when it did.

Small bodies are hard in proportion to their size. A hundred-kilometre body is five hundred times harder than Uranus at the same distance.

And distance hurts linearly while size does not help. A Centaur at fifteen astronomical units and a Kuiper belt object at forty, both two hundred kilometres across, differ in their requirement by a factor of two and a half — which is far less than the difference between a two-hundred-kilometre body and a fifty-kilometre one at the same distance. Size is the dominant variable, which is why the technique reached the large trans-Neptunian objects a decade before it reached the small ones.

What a recorded event measures #

The feedback that makes the technique compound is that an event, once recorded, is itself an astrometric measurement of unusual quality.

What the observers have is a set of chords: each station’s disappearance and reappearance times, converted to positions on the sky plane through the shadow’s known velocity. Fitting a limb profile to those chords gives not only the body’s shape but the position of its centre relative to the star — and the star’s position is known to microarcseconds from an all-sky astrometric catalogue.

So the event returns the body’s astrometric position at one instant, with an uncertainty set by the timing precision times the shadow speed, divided by the distance. At twenty kilometres a second and a timing good to a tenth of a second, that is two kilometres, which at forty astronomical units is 0.07 milliarcseconds.

Compare that with imaging astrometry, which measures a faint moving point against a star field that is itself moving and does well to reach thirty milliarcseconds for a trans-Neptunian object. The occultation is better by more than two orders of magnitude.

One event is worth more than the entire imaging history of the object, and that is not an exaggeration about a well-observed body: several trans-Neptunian objects’ orbits are now dominated by a handful of occultation positions among hundreds of imaging observations.

The loop, and why it converges quickly #

Put the two halves together and the technique has a positive feedback in it.

An ephemeris predicts an event. The event is observed, at whatever rate the prediction’s quality allows. The observation returns a position two orders of magnitude better than anything else in the orbit fit. That position is added, the orbit is refit, and the ephemeris improves — not by a little, because one measurement a hundred times better than its neighbours dominates a least-squares fit.

The next prediction is then better, so more stations record the next event, so the position is better still.

The convergence is fast because the improvement is multiplicative rather than statistical. A campaign does not average down as the square root of the number of events; each successful event replaces the dominant error term.

What limits it is the orbital period. A position measured at one epoch constrains the orbit best near that epoch and degrades away from it, at a rate set by how well the orbital elements are known. For a body with a three-hundred-year period, two positions a year apart constrain a small arc of a long orbit, and extrapolating forward a year is reliable while extrapolating a decade is not.

So the loop has to be maintained. An object with a recent occultation is predictable; one whose last event was fifteen years ago has drifted back into uncertainty, and the drift is dominated by the along-track direction — the body is where the ephemeris says, in the direction across its orbit, and not quite when.

What changed, and it was the stars #

The improvement that transformed the technique was not in the ephemerides. It was in the catalogue.

Before all-sky astrometric surveys the reference frame was defined by ground-based catalogues whose systematic errors ran to tens of milliarcseconds and varied across the sky in ways that were mapped only crudely. A prediction carried that error twice — once for the star and once for the body, whose own position had been measured against the same frame.

An astrometric space mission measuring a billion stars to tens of microarcseconds removed both terms at once. The star’s position became essentially exact for this purpose, and the body’s imaging astrometry, re-reduced against the new frame, improved by the same factor its old reference stars had been wrong by.

The second effect was larger than the first. Re-reducing decades of archival plates and CCD images against a new reference frame improved the orbits of thousands of small bodies without a single new observation, and the improvement was in the systematic rather than in the random part.

That is a shape worth noticing. A measurement made against a frame inherits the frame’s errors, and those errors do not average down with repetition — so a population of old measurements can be improved retroactively by fixing the frame, and the improvement can exceed anything new observations would have achieved.

What is left, and it is the body rather than the star #

With the star’s position exact, the remaining error is the body’s, and it has two parts that behave differently.

The cross-track error — perpendicular to the body’s motion on the sky — is what moves the shadow north or south, and it is what the campaign has to overcome. It is constrained by imaging astrometry over a long arc and improves slowly.

The along-track error moves the event earlier or later. It is usually larger, because it accumulates from an error in the mean motion that grows linearly with time since the last observation, and it is usually less damaging: an event predicted a minute early is still observed by a station that is watching.

So the practical statement is that the timing is uncertain by more than the position, and the campaigns are planned accordingly — stations observe for long enough before and after the prediction to absorb the along-track uncertainty, and are spread across the cross-track direction to absorb that one.

A prediction is quoted as a path with an uncertainty band across it and a window in time, and the two uncertainties come from different terms in the orbit fit and are improved by different observations.

What an observer has to record, and what is easy to get wrong #

The astrometric value of an event rests entirely on the timing, and the timing has three parts that fail differently.

The absolute time has to be tied to a standard to better than the precision claimed, which for a tenth of a second is trivial with satellite navigation and was not trivial before it. Historical events recorded against a shortwave time signal and a stopwatch carry uncertainties of a second or more, and their astrometric value is correspondingly reduced.

The shutter or frame timing has to be characterised. A camera reports the time it was commanded to expose, not the time the exposure began, and the offset between them can be tens of milliseconds and can vary with the exposure length. That offset is a systematic shared by every frame, so it does not average down, and it translates directly into a shift of the whole chord — which is a shift of the astrometric position. Calibrating it against an independent time source is standard practice and is the step most often skipped.

The station’s position has to be known to a fraction of the precision being claimed, which at two kilometres is easy and at two hundred metres is not automatic for a portable telescope.

Of the three, the second is the one that produces a wrong answer rather than a noisy one, and it is the reason published occultation astrometry quotes a systematic term separately from the fit residuals.

What the astrometry is used for #

The positions are worth having for their own sake and for three other things.

Spacecraft targeting. A flyby of a distant small body has to be aimed years in advance, and the aiming is limited by the body’s ephemeris. The encounter with a Kuiper belt object after the Pluto flyby was made possible by a campaign of occultations that pinned the target’s position — and the same campaign discovered that the object was a contact binary, which changed the encounter planning.

Ring and satellite detection, which is the use the earlier essays here are about. A well-predicted event puts stations in the right place to catch material at several body radii, which is where rings are, and the two discoveries of rings around small bodies both came from campaigns organised to measure a size.

And the frame itself. A body whose orbit is known to this precision is a test particle in the solar system’s gravity field, and enough of them over enough time constrain the masses of the perturbers, which is what a residual in an ephemeris is read for. That is the same use a planetary ephemeris puts its own residuals to, at a smaller scale and with a different population.

Where the technique stops #

A position is not an orbit. One event gives one instant. The orbit is fitted to everything, and a single superb measurement among many poor ones improves the fit most in the directions that measurement constrains — which for an occultation is the two sky-plane coordinates and not the distance. The radial direction is constrained only through the dynamics.

The star has to be checked for duplicity. An astrometric catalogue’s position for an unresolved binary is the photocentre, which is not where either component is, and a binary with a period comparable to the catalogue’s baseline has a position that is a fit to a model. An occultation of such a star produces two shadows, offset by the separation, and a station recording one of them returns a position offset by the same amount. The most precise astrometric technique there is can be defeated by a star that is two stars, and screening the target list is part of the preparation.

And the limb is not a circle. The astrometric position is the centre of a fitted profile, and for an irregular body the centre of the silhouette is not the centre of mass — it moves as the body rotates. For a small elongated object that offset can be tens of kilometres, which is larger than the timing error, so the dominant uncertainty in an occultation position is a shape effect rather than a measurement one.

The other body in the line, and it moves too #

The prediction contains two positions and this essay has treated the star’s as exact. That is now nearly true and it was not always, and the way it failed is instructive.

A catalogue position is quoted at a reference epoch with a proper motion, and using it at another date means propagating it forward. For most stars the propagation is small and well determined. For three classes of star it is not.

A high-proper-motion star moves by an appreciable fraction of the required precision — one of five numbers a single astrometric fit returns — between the catalogue epoch and the event, so an error in the proper motion — not in the position — dominates. At a hundred milliarcseconds a year, a five per cent error in the proper motion is five milliarcseconds after a decade, which is the whole budget.

A nearby star has a parallax comparable with the requirement, so its position at the moment of the event depends on where the Earth is. That is a known correction and it is only correct if the parallax is.

And an astrometric binary has a photocentre that orbits. Its catalogue position is a fit to a single-star model over the mission’s baseline, and the residual of that fit is a systematic that appears nowhere in the quoted uncertainty.

The third of those is the reason occultation target lists are screened for astrometric excess noise — a catalogue flag indicating the single-star fit was poor — and stars carrying it are avoided or given an inflated uncertainty.

The catalogue’s precision is not the same as the position’s accuracy on the night, and the difference is entirely in what has happened to the star since it was measured.

The general shape #

The structure worth extracting is a feedback loop between a prediction and a measurement, and it is rarer than it sounds.

Most measurements do not improve the conditions for the next one. An occultation does, because the observation’s precision so far exceeds the prediction’s that the prediction is limited by the last observation rather than by anything else. A technique whose output is better than its own input converges, and the convergence is geometric rather than statistical.

The same structure appears in pulsar timing, where every recorded pulse arrival sharpens the model that predicts the next one and the model’s precision is what makes the next arrival identifiable at all. It appears in spacecraft navigation, where tracking during an approach improves the ephemeris of the body being approached in time to aim at it.

What such loops share is a jump in precision at the moment of first success. Before the loop closes, progress is slow and incremental; after it closes, the limiting error is replaced rather than reduced. The history of occultation astrometry has exactly that shape — decades of marginal campaigns, then an astrometric catalogue, then a technique that became routine within a few years.

The lesson for anything similar is that the binding constraint is getting the first measurement at all, and that effort spent reaching the threshold is worth more than effort spent improving what is already above it.

Still open: the along-track drift nobody can remove #

The one error that resists the loop is the along-track one, and the reason is dynamical rather than observational.

A body’s position along its orbit depends on its mean motion, which depends on its semi-major axis, which is constrained by the whole observed arc. For a body with a period of hundreds of years, an observed arc of thirty years is a small fraction of one orbit, and the semi-major axis is correspondingly poorly determined — so extrapolating the position forward accumulates error linearly in time.

Occultation astrometry does not fix that directly. It measures the sky position superbly at one instant and says nothing about the distance, which is what the semi-major axis needs.

What would fix it is a second kind of measurement — a distance, or a position from a very different vantage point. Parallax from a spacecraft in the outer solar system would do it; so would a radar range, for the few bodies close enough. For the trans-Neptunian population neither is available, so every prediction more than a few years past the last observation carries a timing uncertainty that grows, and the campaigns are scheduled around it rather than against it.