Three observations and no orbit at all

Counting equations against unknowns is the first thing anybody does with a measurement problem, and for orbit determination the count is encouraging. An orbit in space has six elements. An astrometric observation gives two angles. Three observations give six numbers, the count comes out even, and the problem looks settled before any work is done.

It is not settled, and the reason is worth the whole essay: a problem with as many equations as unknowns can still have several solutions, because the equations are not linear. Gauss’s reduction of three directions to an orbit terminates in a polynomial of the eighth degree. Most of the time it has one root that means anything. Sometimes it has three.

What one observation contains, and what it does not #

An astrometric plate, or the pixels that replaced it, records a direction. It records that direction extremely well — a modern position is good to a few milliarcseconds — and it records absolutely nothing about how far away the object is.

That is the entire difficulty. The five-parameter astrometric solution separates parallax from proper motion by their time signatures and needs years to do it; a newly discovered asteroid has been seen for three nights. Over three nights there is no parallax signal worth having, so the distance must come from somewhere else, and the only other thing available is the assumption that the object is moving under an inverse-square force.

The two relations #

Write $\mathbf R$ for the vector from the Sun to the observer, known to metres from an ephemeris, and $\hat{\mathbf L}$ for the unit vector along the line of sight, which is what was measured. The object’s heliocentric position is

$$\mathbf r = \mathbf R + \rho\,\hat{\mathbf L},$$

with $\rho$ the geocentric distance and the one unknown in the expression. Taking the squared length of both sides gives the first relation:

$$r^2 = \rho^2 + 2\rho\,(\mathbf R\cdot\hat{\mathbf L}) + R^2.$$

This is pure geometry. It says nothing about gravity, it holds for a spacecraft and for a bird, and it is a monotone rising curve of $r$ against $\rho$ once the branch with $\rho>0$ is taken.

The second relation is where the dynamics enter, and it is Gauss’s contribution. Consider the three position vectors $\mathbf r_1$, $\mathbf r_2$, $\mathbf r_3$ at the three epochs. All three lie in the orbital plane, so the middle one is a linear combination of the other two:

$$\mathbf r_2 = c_1\mathbf r_1 + c_3\mathbf r_3.$$

The coefficients are ratios of triangle areas, and Gauss’s insight was that those ratios can be approximated by ratios of the sector areas swept out — quantities that Kepler’s second law fixes. To leading order in the short intervals,

$$c_1 \approx \frac{\tau_3}{\tau}\left[1 + \frac{\mu}{6r_2^3}(\tau^2-\tau_3^2)\right],\qquad c_3 \approx -\frac{\tau_1}{\tau}\left[1 + \frac{\mu}{6r_2^3}(\tau^2-\tau_1^2)\right],$$

with $\tau_1$, $\tau_3$ and $\tau$ the intervals between observations. Substituting into the coplanarity condition and eliminating the two outer distances leaves

$$\rho_2 = A + \frac{\mu B}{r_2^3},$$

where $A$ and $B$ are built from the nine numbers the observations supply — three sight directions, three observer positions, three times — and from nothing else. It is a falling curve of $\rho_2$ against $r_2$, and it contains no orbit: no eccentricity, no semi-major axis, nothing that would presume an answer.

Eight #

Eliminating $\rho_2$ between the two relations is arithmetic. Substituting the dynamics into the geometry and multiplying through by $r_2^6$ gives

$$r_2^8 + a\,r_2^6 + b\,r_2^3 + c = 0,$$

with

$$a = -(A^2 + 2AE + R_2^2),\qquad b = -2\mu B(A+E),\qquad c = -\mu^2B^2,$$

and $E = \mathbf R_2\cdot\hat{\mathbf L}_2$.

An eighth-degree polynomial with real coefficients has eight roots. Descartes’ rule of signs bounds the positive real ones at three, since the coefficient sequence $(+,\,a,\,b,\,c)$ with $a<0$, $b<0$ and $c<0$ changes sign exactly once — which gives one positive root, and that is the ordinary case. It is when the intermediate coefficients change sign that the count rises.

The rejection that removes most candidates is not statistical and is worth stating plainly. Each root gives a heliocentric distance, from which the dynamics relation gives a geocentric distance $\rho_2$. If that comes out negative, the candidate places the object behind the observer, along the line of sight extended the wrong way. An object behind the telescope was not the thing that was photographed. No fitting is involved; the root is discarded by sign.

For Ceres over a forty-day arc the octic has three real positive roots and exactly one of them survives the sign test, at 2.5893 AU against a true 2.5890. The residual — one part in ten thousand — is not measurement error, because the observations were exact. It is the truncation in the sector-to-triangle ratios, and it is the reason Gauss’s method is called an initial orbit determination.

When the count is not enough #

Now take an object at nearly the observer’s own distance from the Sun — an Aten-class asteroid on a 1.1 AU orbit, observed three times over ten days.

The geometric reason is not deep. When the object is far away, the geometry curve is steep and the dynamics curve is nearly flat, and two such curves meet once. When the object is at about the observer’s own heliocentric distance, both curves are gentle over the same range, and gentle curves can cross repeatedly.

Charlier, in 1910, gave the condition in closed form: the solution is genuinely ambiguous when the observer lies inside a particular surface defined by the object’s own position, and the criterion is geometric rather than statistical. It does not depend on how good the observations are. Improving the astrometry by a factor of a thousand leaves three roots, three orbits and no way to choose.

That is the sentence worth carrying away from this rung. A degenerate case in a fit is usually a failure of information: more data, or better data, will lift it. This one is a property of the geometry, and the only cure is a fourth observation at a time that changes the geometry.

Which is worse, a short arc or a bad one #

There is a second failure mode that gets confused with this one and is entirely different.

The practical consequence is a discipline about what a newly announced orbit means. A three-night arc on a main-belt object gives a semi-major axis good to a per cent or two, which is enough to say what the object is. A three-night arc on something passing close to the Earth may give an orbit that is wrong by a factor of four in distance, and the announcement of a close approach depends on which root the software converged to.

The Minor Planet Center’s response to that is not a better algorithm but a different presentation. A short-arc object is not published as an orbit with an uncertainty; it is published as a line of variations — a curve through the space of possible orbits, parameterised by the one badly determined direction — with the observer told to search along it. Where the octic has several admissible roots, that line has several disconnected pieces.

What Gauss actually did in 1801 #

The history is usually told as a story about least squares, and the least squares came later. What Gauss did for Ceres was the reduction above.

Piazzi observed Ceres on 24 nights between January and February 1801, over an arc of about three degrees, and then lost it in the Sun’s glare. Recovering it a year later required predicting where on the sky it would reappear, from an arc that was short, low in the sky, and — as Gauss remarked — taken near the object’s stationary point, which is the worst place for it. Gauss’s prediction put Ceres within half a degree of where von Zach and Olbers found it in December 1801, and the achievement was arithmetic rather than instrumental. Every step above was done by hand, including the eighth-degree root, and the iteration to refine it was carried through several times.

The single most quoted sentence about the episode is Gauss’s own remark that he had used a method which had cost him a great deal of labour and which he would not describe. The description came six years later in Theoria Motus, and every initial-orbit routine written since is a variant of it.

What is done instead of an orbit #

When the arc is too short to determine a solution, the modern practice is not to compute a worse orbit. It is to compute the set of orbits the observations permit and carry all of them.

The two quantities a short optical arc leaves unconstrained are the object’s distance and its rate of change — the range and the range rate. Everything else is measured well. So the object is represented by a region in that two-dimensional space, bounded by conditions that are physical rather than statistical: the object must be bound to the Sun, it must not be inside the Earth, and if it were very close and very slow it would have been detected differently.

That region is sampled — hundreds or thousands of points, each of which is a complete orbit consistent with the data — and every one of them is propagated forward and backward.

What the set is used for is the interesting part. Linkage: a new detection elsewhere in the sky is the same object if it is consistent with some member of the set, which turns a matching problem into a geometric one. Recovery: the set propagated to a future date gives the region of sky to search. And impact assessment: if none of the sampled orbits reaches the Earth, the object is safe on the evidence available, and if a few do, their fraction is the probability quoted.

That last use is why the machinery exists at all. An object discovered days before a close approach has an arc far too short for a determinate orbit, and a decision has to be made anyway — so the honest procedure is to enumerate what is possible rather than to fit what is most likely and report its formal error, which for a badly conditioned fit is meaningless.

Losing something already found #

The other consequence of a short arc is administrative, and it accounts for most of the objects that have been discovered twice.

An orbit determined from a few nights’ observations degrades as it is propagated. The uncertainty does not grow isotropically: it stretches along the orbit, because what is worst determined is the period, and an error in the period accumulates as a displacement in position. After a few months the region to search is a long thin curve across the sky — the line of variations — rather than a patch.

Searching along a curve is a different observing programme from pointing at a coordinate, and it costs time on a telescope that could be discovering new objects instead. That is a real trade, and surveys have generally chosen discovery, with the result that a substantial fraction of the objects catalogued in any year are recoveries of objects seen once before and lost.

The remedy has been arithmetic rather than telescopes. Archival searches — looking for an object in images taken before it was discovered, now that its orbit is known well enough to say where it would have been — extend arcs backwards by years, and the improvement is enormous because the strength of an orbit determination grows with the span rather than with the number of observations.

A single detection on a plate from a decade earlier can be worth more than a month of new observations, and finding it costs nothing but computation on data already taken.

The observation that fixes everything at once #

There is one measurement that removes the whole difficulty, and its scarcity explains why the machinery above exists.

Optical astrometry gives two angles and no distance, which is the source of every problem in this essay. Radar gives the distance and the range rate directly, to metres and to millimetres per second, from a single observation.

A single radar detection appended to a short optical arc changes the determination out of recognition. The two quantities it supplies are precisely the two the optical data leave free, so the admissible region collapses to a point and the orbit becomes as well determined as one from years of imaging.

The catch is reach. Radar’s returned signal falls as the fourth power of the distance — the transmitted beam spreads on the way out and the echo spreads on the way back — so a facility able to detect an object at one distance requires sixteen times the sensitivity to detect it at twice that. In practice only objects passing within a few tens of lunar distances are reachable, and only if a large radio dish is available and pointed at the right moment.

So the technique is not a survey tool. It is applied to objects already found, chosen because their orbits matter — an approaching object whose impact probability is not yet zero, or a mission target whose position must be known before a spacecraft is committed to it.

One radar observation is worth years of astrometry, and it can only be made during the few days an object is close enough, which is why the schedule of these campaigns is set by the objects rather than by anybody’s convenience.

The same asymmetry runs through the whole subject: the cheap measurement is plentiful and weakly constraining, the expensive one is scarce and decisive, and the art is in knowing which objects are worth the second kind.

It also explains why the discovery surveys and the follow-up facilities are funded and scheduled separately: they are answering different questions with different instruments, and the objects that pass between them are chosen by an assessment that neither of them makes.

One more reading shows what the determination’s uncertainty looks like once an orbit does exist.

Where the model stops #

Three approximations sit inside the derivation and each has a visible consequence.

The sector-to-triangle ratios are truncated, which is what makes the method an approximation rather than a solution, and the error grows as the cube of the interval divided by the period. Forty days on Ceres is a tenth of a per cent; forty days on an Aten is one and a half per cent, which is why the near-Earth case above uses a ten-day arc. Iteration recovers most of it, and the modern practice is to iterate with exact Lagrange coefficients rather than the series.

Light travel time is ignored above and must not be in practice. An object at 2.6 AU is seen where it was twenty minutes earlier, and twenty minutes of orbital motion is a quantity far larger than the astrometric error. The correction is applied by iterating on $\rho$, which is itself the unknown being solved for.

And the derivation assumes the two-body problem exactly, which for a short arc is fine and for the recovery of a lost object is not. The prediction Gauss made for Ceres a year ahead needed the planetary perturbations, and computing those is the other half of the same subject. Two of the essay’s three claims about short arcs can be checked by moving a parameter, and both of them are claims about conditioning rather than about any particular object.

Where this ladder goes next #

This rung establishes that the reduction is a polynomial and that a determined problem can have several answers.

One rung above follows the refinement: differential correction, in which an initial orbit is improved by least squares against every observation, and the covariance that comes out of it — an ellipsoid in six dimensions, usually enormously elongated along one direction, which is where the line of variations comes from.

Another follows the ambiguity into practice. An object whose orbit admits several roots is not a curiosity but a routine occurrence in near-Earth surveys, and deciding which of several candidate orbits to spend follow-up telescope time on is a question with an answer: rank them by how quickly the next observation would separate them, not by how well they fit the ones already in hand.

And a third goes sideways, to the problem that takes two positions and a time rather than three directions. That one has a unique answer for a given number of revolutions and a whole family of answers when the revolutions are counted — the same lesson from the opposite direction.