An error that is nearly all in one direction

The rung below this one found a determined problem with several answers: three perfect angles-only observations of a near-Earth asteroid, reduced by Gauss’s method, terminating in an equation of the eighth degree that can admit three admissible orbits at once.

That is the pathological case. The ordinary case is worse in a quieter way. The solution is unique, the fit is good, the residuals are small — and the uncertainty around that unique answer has a shape so lopsided that treating it as a blob throws away almost everything it says.

After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging. — Fig. 1 The two semi-axes of a fitted orbit’s position uncertainty, against elapsed revolutions. The **radial** extent does not grow at all: a body on a slightly larger orbit is slightly further out, and stays so. The **along-track** extent grows linearly, and after a single revolution it is already 3π ≈ 9.4 times the radial one. There is nothing subtle in the mechanism, and everything in the consequence.

Where the asymmetry comes from

Take an orbit fit that has produced a semi-major axis good to some small $\delta a$ and everything else in proportion. The mean motion follows from the semi-major axis and from nothing else:

n = \sqrt{\frac{\mu}{a^3}}, \qquad \frac{\delta n}{n} = -\frac{3}{2}\frac{\delta a}{a}.

An error in the size of the orbit is therefore an error in the rate at which the body goes round it. That is the whole of it. After time $t$ the two candidate orbits have accumulated a mean-anomaly difference $\delta M = \delta n\,t$ , and the physical separation along the track is $a\,\delta M$ , which after $N$ revolutions is

a\,\delta M = 3\pi N\,\delta a.

Meanwhile the radial separation is still $\delta a$ . The ratio is $3\pi N$ and it is exact. Nothing in that derivation is specific to asteroids. The same thing happens to a satellite, to a comet, to a spacecraft between planets. It is why a satellite catalogue’s predictions degrade in time of pass long before they degrade in altitude of pass, and why an amateur who points a telescope at a predicted position and finds nothing should try again ten minutes later before concluding anything.

What the fit actually delivers

An orbit determination returns six numbers and a $6 \times 6$ covariance matrix. The matrix is where the information is, and the six numbers are the least interesting part of the output.

Its eigenvectors are directions in the space of orbits, and its eigenvalues are how well each is constrained. For a short observed arc the picture is stark: one eigenvalue is enormous and the rest are small. The direction with the enormous eigenvalue is very nearly “make the orbit slightly bigger and put the body slightly further back along it” — the combination the observations cannot separate, because a short arc of sky motion is compatible with a whole family of orbits differing in exactly that way.

Why 41 days was hard: the recovered orbit's size against the length of the arc. The fractional change in the recovered semi-major axis of Ceres for one arcsecond of error in the observed directions, against the length of the observed arc, on five sightings spaced as the drawn arc is. Each point is a determination: the sightings are regenerated at that arc length, the orbit is recovered from the directions alone, and the response of the four elements to an arcsecond in each direction is read off the least-squares fit — at 41 days the value is 0.60%, and re-solving the whole determination with one angle moved by a full arcsecond gives 0.60%, so the linearisation is the thing it claims to be. The fall is 3.0 decades over 7× in arc, between 12 and 80 days: an exponent of -3.69, steeper than the inverse square the short-arc argument gives, because a short arc loses the curvature and the perspective shift together. It is still falling at that rate at 80 days, the right-hand end of the sweep. The curve stops at 12.1 days on the left because that is where a single arcsecond — about the best a 1801 transit circle could do — moves the recovered a by 41% of itself; at the 14 shorter arcs tried, down to 5.0 days, an arcsecond moves it by half the orbit's size or more and there is nothing left to be precise about. The point of the panel is not that 41 days is too few but that it is barely enough — 0.60% of 2.7658 AU is 0.017 AU, which is the difference between a lost object and a found one. — Fig. 2 The same fact seen as a conditioning problem. Gauss’s reduction of three observations depends on differences between nearly parallel sight lines, and the shorter the arc the more nearly parallel they are: the condition number of the solution rises steeply as the arc shortens, so a fixed observational error is amplified into a larger and larger family of admissible orbits. **The arc length is not a detail of the observing programme; it is the single quantity that decides whether there is an orbit at all.**

The practical form of that family is a set of clone orbits — a few hundred or a few thousand solutions, each drawn from the covariance and each fitting the observations about as well as the nominal one. They are then integrated forward individually, which is the only honest way to propagate an uncertainty through a nonlinear system.

What comes back is not a cloud. It is a thread.

The line of variations

The dominant eigenvector, propagated forward, traces out a curve in space that the clones lie along. It is called the line of variations, and the name is exact: the clones vary along one line and hardly at all across it.

That changes the arithmetic of every question one might want to ask.

$An impact probability of 0.64 per cent, computed along a line. The target plane at a close approach — the plane through the Earth perpendicular to the incoming relative velocity — with 601 statistically equivalent solutions of the same orbit fit plotted on it, in Earth radii, at equal scales. They do not fill an ellipse; they fill a line, because by the time of the encounter a semi-major-axis error has become an along-track error and the covariance has collapsed onto one direction. The filled circle is not the Earth but the focused capture cross-section, √(1 + v_esc²/v∞²) = 1.99 Earth radii at v∞ = 6.5 km/s; the dashed circle inside it is the Earth itself. The nominal solution misses by 9.4 radii — 3.1σ — and the impact probability is the weighted fraction of the line inside the circle, 0.64 per cent, a one-dimensional integral rather than a volume. That is why these numbers move in jumps as observations arrive: a new observation shortens the line, and the answer changes discontinuously depending on whether the capture circle is still on it. The marked segment is a keyhole — an interval 1.0 per cent as long as the drawn line, carrying 0.2861 per cent of the probability — which misses now and is deflected onto a resonant orbit that comes back.$

Fig. 3 The target plane at a close approach, with six hundred statistically equivalent solutions on it. They fill a line, not an ellipse. The filled circle is not the Earth but the focused capture cross-section, √(1 + v²_esc/v²_∞) Earth radii — a body aimed at 1.99 radii is pulled in, and the focusing is the same effect that makes a planet a larger target than its own disc. The impact probability is the weighted fraction of the line inside that circle: a one-dimensional integral, not a volume.

Two consequences follow, and both are counterintuitive enough to have caused public confusion more than once.

Impact probabilities move in jumps. A new observation shortens the line. If the capture circle is still on the shortened line, the probability rises, because the same total probability is now spread over less line. If the circle falls off the end, the probability drops to zero at a stroke. That is why a well-observed asteroid’s impact probability characteristically climbs for a few days and then vanishes — and why the climb is not evidence of anything getting worse.

The observation worth taking is not the one that fits best. A measurement that reduces the uncertainty across the track buys almost nothing, because there was almost none there. A measurement that shortens the line — a recovery at a widely different epoch, a precovery on an old plate, a radar range that fixes the distance directly — is worth orders of magnitude more. Ranking follow-up targets by how quickly the next observation would separate the candidates, rather than by how uncertain they currently are, is the standing recommendation and it falls straight out of this geometry.

3 orbits from three perfect observations of an Aten-class asteroid. Gauss's reduction of three directions to a distance, drawn as the two relations whose intersection it is. Three observations of an Aten-class asteroid on days 0, 5, 10 of an arc, generated from its elements and used only as sight directions — no range, no radial velocity. The rising curve is geometry: the heliocentric distance a candidate at geocentric distance ρ₂ would have, r₂² = ρ₂² + 2ρ₂(R₂·L̂₂) + R₂², which contains no dynamics at all. The falling curve is dynamics: ρ₂ = A + µB/r₂³, with A and B built from the three sight vectors, the three observer positions and the three times, and containing no orbit. Eliminating ρ₂ between them gives r₂⁸ + a r₂⁶ + b r₂³ + c = 0 — an eighth-degree equation, from a problem with exactly as many equations as unknowns. Here the curves cross 3 times at positive ρ₂: 1.004 AU, 1.075 AU, 4.572 AU, of which 1.076 is the truth. All 3 are consistent with the observations, which are exact, and no amount of care with these three nights distinguishes them. A fourth observation does, immediately — and that is why an initial orbit is quoted from three and confirmed from four. — Fig. 4 The rung below, in this light. When the arc is short enough, the ambiguity is not a long thin ellipse but genuinely separate roots: three admissible orbits from three perfect observations, each of which would propagate to its own line of variations. The continuous case above and the discrete case here are the same phenomenon at two strengths, and the remedy for both is the same — a fourth observation, taken where it separates them.

The observation that is worth a hundred others

The asymmetry of the covariance has a mirror image in the asymmetry of the data, and it is the practical half of the subject.

Optical astrometry measures two angles and no distance. Over a short arc those angles constrain the direction well and the size of the orbit hardly at all, which is exactly the combination that leaves the line of variations long. Adding more optical observations from the same apparition shortens the line slowly, because they carry the same kind of information as the ones already in hand.

Three things carry a different kind.

A radar range measures the distance directly, at a precision of tens of metres, and the distance is the quantity the angles are worst at. A single radar detection of a newly discovered near-Earth asteroid routinely shrinks the along-track uncertainty by a factor of a hundred or more, and a radar Doppler measurement — the line-of-sight velocity, good to a millimetre a second — does the same for the rate.

A precovery is an image of the object taken before it was known to exist, found afterwards in an archive. Its precision is no better than any other optical position, and often worse; what it has is epoch. An observation twenty years old extends the arc by twenty years, and since the along-track error grows linearly with elapsed time, the leverage runs the other way too: a twenty-year baseline divides the mean-motion uncertainty by roughly the ratio of the baselines.

A second apparition does the same job without the archive. An object recovered at its next close approach has an arc spanning whole revolutions rather than a few weeks, and the difference in the covariance is not incremental — it is the difference between an orbit that is a family and an orbit that is a number.

Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc. — Fig. 5 What the angles do and do not say. Five sight lines from a moving Earth to a moving body, with no distance among them: the geometry that makes an orbit determinable at all is the change in the observer’s own position during the arc. That change is what supplies the missing scale, which is why the arc length matters so much more than the number of observations in it, and why a hundred images taken in one night constrain an orbit scarcely better than three.

The general statement behind all three is that the information a measurement carries is not its precision but its projection onto the direction that is poorly known. A very precise measurement of a well-determined quantity adds nothing; a mediocre measurement of the ill-determined one can be decisive. That is a property of least squares rather than of astronomy, and it is why observing programmes for near-Earth objects are scheduled by covariance rather than by magnitude.

What the observations are made of

The covariance is an output of a least-squares fit, and a fit is only as good as its statement of what the errors are. That statement has changed completely in the last decade, and the change is worth describing because it moved every orbit in the catalogue.

An optical astrometric observation is a position measured relative to catalogued stars in the same frame. Its error therefore has two parts: how well the object’s image was centred, and how well the reference stars’ positions were known. Until recently the second dominated. Catalogues in general use had systematic errors of a few tenths of an arcsecond, varying across the sky in patterns peculiar to each catalogue, so an observatory’s residuals carried a signature of which catalogue it had used rather than of how good its telescope was.

Those systematics do not average down. A thousand observations reduced against the same catalogue in the same region of sky share the same offset, and a least-squares fit that assumes independent errors treats the agreement among them as evidence and shrinks the covariance accordingly. The result is a formal uncertainty smaller than the true one — the failure mode that matters most here, because the whole apparatus of clones and lines of variations is a statement about the covariance being right.

The repair was twofold. Old observations are debiased, by applying a correction derived from comparing each historical catalogue against a modern one region by region. And new observations are reduced against a reference frame good to a fraction of a milliarcsecond, which removes the term entirely.

An orbit’s uncertainty is not measured, it is asserted — computed from a model of the observation errors — and improving that model changed thousands of published covariances without a single new observation being taken.

The same point applies to the weights the fit assigns. An observatory that reports a hundred positions from one night is not supplying a hundred independent constraints, and treating it as though it were is another route to a covariance that is too small.

When the arc is too short for a line

Everything above assumes there is an orbit to have a covariance about. Below a certain arc length there is not, and the object joins a population whose size is rarely appreciated: the ones that were seen, measured, and never found again.

A few nights of astrometry on a moving object give a direction and a rate of change of direction — four numbers — and an orbit needs six. The two that are missing are the distance and the rate of change of distance, and they are missing in the strong sense that the data are consistent with a continuous two-dimensional range of them.

So the object’s possible orbits do not form a line. They form a region, and the region is bounded only by physical arguments: the object must be bound to the Sun, it must not be inside the Earth, and it must be large enough to have been detectable at the distance being posited. That bounded region is what the follow-up is chasing, and it can be large enough that the object’s predicted position a month later spans tens of degrees.

The consequence is a linkage problem rather than a prediction problem. An object seen on two nights in one month and two nights in the next is two short arcs, and deciding whether they are the same object requires searching for an orbit consistent with both — against a catalogue of hundreds of thousands of other objects any of which might be the match. That search is the reason surveys are designed to revisit the same field on a cadence rather than to cover the most sky.

The line of variations is what a short arc becomes once it is long enough to have one, and the transition between the two-dimensional region and the one-dimensional line is where most of an object’s uncertainty is actually removed.

Keyholes

Inside the line of variations there are intervals that miss the Earth now and return to strike later.

The mechanism is a resonance. A body that passes at a particular distance is deflected onto an orbit whose period is a simple ratio of the Earth’s, which brings it back to the same point in space at the same time of year some whole number of years later. The interval of the target plane that produces such an orbit is narrow — kilometres, sometimes hundreds of metres — and it is called a keyhole.

The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed. — Fig. 6 Why a resonant return is a specific place rather than a general risk. A flyby cannot change the encounter speed, only its direction, so the post-encounter orbit lies on a contour of the quantity that survives the encounter. Landing on a *resonant* period — one of the discrete values that brings the body back — is therefore a matter of hitting a particular point on that contour, which corresponds to a particular narrow interval of the aim. The keyhole is the preimage of a point.

A keyhole is a small target, which is why the probability attached to one is usually tiny. It is also the reason a deflection mission would be planned decades in advance and would need to move an asteroid by only a few hundred metres: the object is not being pushed away from the Earth, it is being pushed off a keyhole.

The best-known case is Apophis, which was found in 2004 and briefly carried an impact probability of about one in thirty-seven for 2029 — the highest ever assigned to a sizeable object. A precovery image from March 2004, located after the fact, extended the arc backwards and removed the 2029 possibility entirely within days. What remained for the next fifteen years was a set of keyholes for 2036 and 2068, each a few hundred metres wide in the 2029 target plane, and radar observations in 2021 closed all of them.

None of that sequence is a story about the object changing. It is a story about a line getting shorter.

Why the growth eventually stops being linear

The $3\pi N$ law is a linearisation, and it holds while the clones stay close enough together that the same linear map applies to all of them. Two things end it.

The first is a close approach. Passing a planet at a distance that differs by a few thousand kilometres from clone to clone produces post-encounter orbits that differ by far more than the pre-encounter ones did — the encounter multiplies the spread, sometimes by two or three orders of magnitude. After that, the line of variations may fold or break into disconnected pieces.

The second is chaos in the ordinary sense. For a body whose orbit crosses several planets, the Lyapunov time can be a few decades, and past a few Lyapunov times the notion of “the orbit” has stopped meaning anything.

4 metres where the data is, and 4.9 kilometres at 700 BC. The difference between two solutions of the same ephemeris problem, against epoch, on a logarithmic scale. Inside the fit interval — 1913 to 2022, shaded — the observations hold them together at the 4 metres the ranging measures. Outside it nothing does, and two terms take over at different rates: a residual difference in the mean motion grows linearly with elapsed time, and a mismodelled acceleration — several hundred asteroid masses are the usual culprit — grows quadratically. Each is fixed here to contribute half the data precision at the edge of the interval, which is all a fit guarantees. By 3000 the two disagree by 756 metres and at 700 BC by 4.9 kilometres, 1222 times the precision they were fitted to, and 98 per cent of that is the quadratic term. An ephemeris used outside its interval is an extrapolation, and a newer one is not automatically a better one there — it is a different extrapolation, fitted to data that says nothing about the epoch being asked about. Which is why a Babylonian eclipse record is not predicted from a modern ephemeris but used to constrain one. — Fig. 7 The same growth in a different guise. Two solutions of a planetary ephemeris agree to metres where the observations hold them and diverge outside — linearly from a mean-motion difference, quadratically from a mismodelled acceleration. An ephemeris is a fit like any other, and the reason it degrades away from its interval is the reason a single asteroid’s prediction degrades away from its arc.

What is being reported

The number quoted for an asteroid — “one chance in sixteen thousand in 2046” — is therefore the output of a specific and rather elaborate calculation: a fit, a covariance, a few thousand clones, a numerical integration of each of them through every planetary perturbation, and a weighted count of how many land inside a circle whose radius is set by gravitational focusing.

It is not a statement about the asteroid. It is a statement about what the current observations fail to exclude, and it changes when they change. The Palermo and Torino scales exist to convey that, and they are routinely reported as though they were properties of the object.

There is one further wrinkle, and it is the reason the numbers are not purely a matter of geometry. Small asteroids are pushed by sunlight. Thermal re-emission from a rotating body is not symmetric — the afternoon side is warmer than the morning side — and the resulting recoil produces a tiny along-track acceleration that changes the semi-major axis by metres per year. Over decades that is a displacement of thousands of kilometres, comparable with the width of the target the calculation is aiming at. So the fit has to carry that acceleration as yet another parameter, determined from the astrometry itself; and since it depends on the body’s size, shape, spin and thermal inertia, none of which is usually known, it is often the largest single term in the error budget for a prediction fifty years out. The same class of small non-gravitational force is why a spacecraft’s trajectory has to be tracked rather than computed, and here it is being measured on a body nobody can visit. One more covariance calculation shows what a worse astrometric error does to the same ellipse.

Where this ladder goes next

This rung has established the shape of the uncertainty and what follows from it: a line rather than a ball, a probability that is an integral along it, and an observing strategy that follows from the geometry rather than from intuition.

The rung above is the deflection problem proper, where the question stops being what the covariance is and becomes what a given impulse does to it — and where the answer turns out to depend on the keyhole structure far more than on the size of the impulse.

Beside it lies the survey-design question. Given a telescope and a fixed amount of time, the optimal observing programme is not the one that observes each object best; it is the one that shortens the most lines of variation per hour, and that is a different allocation entirely.

And below it, as the habit: an uncertainty has a shape, and the shape is usually more informative than the size. A single number quoted for the accuracy of a position has thrown away the one thing a covariance was computed to say.

An error that is nearly all in one direction

Where the asymmetry comes from

What the fit actually delivers

The line of variations

The observation that is worth a hundred others

What the observations are made of

When the arc is too short for a line

Keyholes

Why the growth eventually stops being linear

What is being reported

Where this ladder goes next

What links here

The objects this essay names

Where the asymmetry comes from #

What the fit actually delivers #

The line of variations #

The observation that is worth a hundred others #

What the observations are made of #

When the arc is too short for a line #

Keyholes #

Why the growth eventually stops being linear #

What is being reported #

Where this ladder goes next #

Where this goes next

What links here

The objects this essay names

Where the asymmetry comes from

What the fit actually delivers

The line of variations

The observation that is worth a hundred others

What the observations are made of

When the arc is too short for a line

Keyholes

Why the growth eventually stops being linear

What is being reported

Where this ladder goes next