A distance between orbits that depends on how many there are

A meteor shower is a reading of an orbit: the date gives the node of the stream, the speed its inclination, and the point in the sky the direction it comes from. That account took the existence of the stream for granted. The Perseids and the Geminids can be picked out by eye, in a sky full of meteors, because they are numerous and because they come from one place. But most streams are nothing like that. Camera networks now triangulate many tens of thousands of orbits a year, and the large majority of those orbits belong to nothing known. Some of the rest belong to streams too thin to notice on any one night. Finding those is a statistical question, and it has to begin with a definition: what does it mean for two orbits to be close?

Six numbers fix an orbit, and for a body that is crossing the Earth’s path one of them is a time that does not matter here and another is partly set by the requirement of crossing. The rest are not all of the same kind, since one is a length, one a shape, and three are angles, one of which can be meaningless if the orbit is nearly in the ecliptic or nearly circular. No obvious ruler measures across them. The solution proposed in 1963 was to write down a formula anyway, with a sensible weight for each element, and to calibrate it against a chance.

The formula has lasted over sixty years with modifications. Its threshold has lasted less well, and the reasons are instructive beyond meteors. A threshold set on a distance is a statement about how many neighbours would be found by accident, and that depends on how many objects are being compared. The criterion was calibrated correctly for the catalogue it was given. What has changed is the size of the catalogue, by a factor of a thousand.

A ruler across unlike numbers #

Richard Southworth and Gerald Hawkins were working with 360 meteor orbits from the Harvard small cameras. They needed a way to decide which belonged together, and they built up a sum of squares:

$$D^2 = (\Delta e)^2 + (\Delta q)^2 + \left(2\sin\tfrac{I}{2}\right)^2 + \left(\frac{e_1+e_2}{2}\cdot 2\sin\tfrac{\Pi}{2}\right)^2.$$

The first two terms compare eccentricity and perihelion distance directly. The third is the angle $I$ between the two orbital planes, and the fourth is the angle $\Pi$ between the two perihelion directions measured from where those planes cross, scaled by the eccentricity because the direction of perihelion means little for a round orbit. The use of a sine of a half-angle means that the term is the straight-line chord between two points on a unit sphere, a nice choice with a real geometric meaning.

This is not a distance in any fundamental sense. The weights are chosen and could have been chosen otherwise, and alternative forms by Drummond and by Jopek differ in how much they stress perihelion distance and eccentricity. All of them answer a question nobody can derive from dynamics, which is how much difference in eccentricity should count as equal to how much difference in inclination. They were all calibrated in the same way, by comparison with randomised data. Southworth and Hawkins took the actual orbits and shuffled the nodes, which removes real associations while keeping the population’s other properties, and they chose a threshold of 0.2, at which chance pairs were acceptably rare.

How many strangers an orbit can match #

How rapidly chance neighbours grow with the threshold is set by how many independent dimensions the sporadic orbits fill. One can measure this directly. Draw a schematic population of Earth-crossing orbits, measure the distance between every pair, and ask what fraction of pairs lies within a distance $D$.

This is a very general way to measure the dimension of a cloud of points. The population is effectively four-dimensional, or a little less, because the requirement of intersecting the Earth’s orbit ties $q$, $e$ and the argument of perihelion together. It follows that the expected number of chance partners for a given orbit is proportional to the number of orbits in the catalogue times the fourth power of the threshold:

$$\langle n \rangle \propto N D^4.$$

Put in the numbers, and the history of the method can be read in one figure.

At 360 orbits, the threshold did what it was meant to do: a typical orbit has less than one chance partner, so finding one or more is mildly interesting and finding ten is clear evidence. With a hundred thousand orbits the same threshold returns over a hundred chance partners for every orbit in the catalogue. It would then call every meteor a member of a stream. This is no defect of the formula. A number chosen as a false-alarm limit has been applied outside the conditions that gave it that meaning, and it is precisely the error warned of by the statistics of counting.

Modern surveys know this, and they use one of two remedies. The more common uses a lower threshold, such as 0.1 or 0.05, chosen by hand for the job at hand, or else a method that uses radiant positions and time of year, in place of orbital elements. Another is to keep the chance rate constant, and that requires the threshold to fall as the catalogue grows.

A threshold that shrinks until it cuts the stream #

Hold the expected number of chance partners constant, and the threshold must scale as the inverse fourth root of the catalogue’s size, a rule that has been suggested explicitly as $0.2(360/N)^{1/4}$. It falls slowly, which sounds like good news.

The trouble is that a stream has a width of its own. Grains leave a comet at a range of speeds, are pushed by sunlight according to their size, and are spread by planetary perturbations over thousands of years. So members of a real stream sit some way from each other in the element space, by an amount that grows with the stream’s age. Measurement errors add to the width. So a threshold shrinking past that width starts to discard genuine members, and old diffuse streams get cut first. There is a best catalogue size for finding any given stream with a single threshold, and beyond it a fixed false-alarm rate is bought by losing the very thing being sought.

This defines a trap in the method rather than an impassable limit. Different thresholds suit different streams, and a stream’s own members can be used to fit its width after it has been found. Yet there is no choice of threshold that treats all streams fairly, which is why two analyses of the same catalogue can report different lists.

What a larger catalogue does not buy #

It might seem that more orbits should make a stream stand out more clearly against the background. In one important sense they do not.

The members of a stream and the chance neighbours both grow in proportion to $N$, so their ratio stays where it was. A stream making up two-tenths of a per cent of the orbits is contaminated at the same level by chance partners, at a given threshold, however large the catalogue becomes. What improves is the statistical significance of the excess, because a larger count has a proportionally smaller fluctuation. So a large catalogue can detect faint streams it could not otherwise have seen. It cannot give a clean list of who belongs to them. Any individual meteor assigned to a faint stream has a substantial chance of being a stranger, and the assignment is a probability that no amount of further counting will sharpen.

That difference matters for what is done with membership next. A shower’s mean orbit is robust against a modest contamination. But the study of parent bodies relies on tracing a few members, and the search for the highest-precision orbits, as in integration back to a comet’s ejection, is sensitive to which orbits are chosen.

The chain that links everything #

There is a further way for the method to fail, and it comes from an algorithm instead of a threshold. The simplest grouping rule, known as single linkage, joins two orbits if they are within the threshold, and then follows the links. A group is everything reachable by any chain.

The behaviour is a percolation transition, the same mathematics that describes water finding its way through porous rock. Below a critical threshold there are many small groups, and above it almost everything joins one giant group. The original threshold of 0.2 was safe for 360 orbits, and it lies at the transition’s edge for only a few times that number. A catalogue with no streams at all then yields a single enormous stream. Once it has formed, no significance test applied to the group can recover anything useful, because the group is a real structure of the linking rule, and a test comparing its size with chance will find it far too large.

This has a close counterpart in the asteroid families, whose membership list depends on a linking speed in the same way and for the same reason. In that case a family’s size is a choice, because each family grows as the cutoff rises, until the families begin to merge. The problem has more bite with meteors, since a stream does not have a dynamical boundary, and because the asteroid belt’s elements are stable over millions of years, whereas a stream’s are changed by the measurement of each meteor as much as by the stream itself. Modern work replaces single linkage with density-based methods, which ask whether an orbit’s neighbourhood is more crowded than the background, and these reduce the chaining though they do not remove it.

A measurement error is already a distance #

A distance of 0.05 is also the size of an error that can be produced by a modest mistake in measurement. In the case of a meteor, the orbit is computed from a direction and a speed. The direction is good to a small fraction of a degree. The speed, measured from the frames of two cameras as the meteor decelerates in the air, needs a correction for that deceleration and for the Earth’s gravity, and it can be off by a per cent or so.

The sensitivity grows sharply as an orbit approaches escape, since the semi-major axis there depends on the difference between two nearly equal numbers. For a high-speed stream, a comet’s orbit in all but name, measurement error can broaden the stream in this distance by an amount comparable to the real spread. When a catalogue spans decades and cameras of different kinds, a systematic difference in the speed correction can separate two sets of observations of a single stream, or merge two streams with similar orbits. The threshold is measuring the instrument as well as the sky.

So several different effects are folded together in the width of any stream. The width has a dynamical part, set by ejection speeds and age, and a measured part, and the error in an orbit lies mostly in one direction, so that a circular threshold in element space treats a long thin error ellipse as if it were round. A more rigorous approach would weight each pair by its known uncertainties, as is done for the linking of asteroid observations, and it would then give a probability that two orbits are the same.

What the orbit leaves out #

Part of the remedy has been to stop relying on orbital elements alone, and the change is more interesting than it first appears. A meteor is recorded with three numbers that come almost directly from the cameras: the date, which is the longitude of the Sun, the direction of the radiant, and the speed. These are the coordinates in which a shower is actually seen, and working in them has been the choice of the largest recent surveys. Each is measured with an error of its own, and the errors are nearly independent, which is untrue of elements computed from them. A cluster in those observed coordinates has none of the near-parabolic amplification described above, since the speed is compared directly and not through the difference of two nearly equal numbers.

Observed coordinates also bring in something the elements omit by construction. The set of orbits belonging to a real stream shows a radiant that drifts slowly from night to night as the Earth moves through the stream, at a rate set by the stream’s geometry. A chance clump in elements has no reason to show that drift, and so the drift becomes a second test. It has some of the same character as a family checked by a colour it was not chosen by. A grouping defined by one set of numbers is confirmed only by another set that played no part in defining it.

The elements then come back at the end, in a more useful role. Once a stream has been established from what the cameras recorded, its mean orbit is the thing to compare with a list of comets and asteroids, and the comparison is a much smaller statistical problem: one orbit against some thousands, where the expected number of chance matches can be computed directly. That is how the Geminids came to be tied to the asteroid Phaethon in 1983. The order of the steps matters, because each step asks only as much of the distance as its false-alarm rate allows.

A rule about a catalogue #

The general lesson goes beyond meteor astronomy. Most of the classifications in this subject have a threshold in them somewhere, and a threshold that serves well for one sample can mean something different for the next, without anyone having changed the rule. A shower list has grown enormously in the past fifteen years, reaching hundreds of proposed entries, of which only about a hundred have been established with confidence. Some of that growth reflects real discoveries of faint streams, enabled by very large numbers. Some of it may be a consequence of the search itself, because a larger catalogue offers more chances for a chance clump to look significant. Without knowing the false-positive rate, those two can’t be separated, and that rate depends on the very size of the catalogue.

The surprising part is how close the connection is between a definition and a census. Questions of the form “how many showers are there?” sound as if they have an answer waiting to be found. In fact the answer is a function of $N$ and $D$, with a sharp feature at a percolation threshold. A count of streams has meaning only relative to the catalogue and the method. The same issue arises with occurrence rates of planets found near a detection limit and with a population biased by what could be seen, and it has the same remedy, namely that the rate of false positives has to be computed for the sample at hand and stated alongside the result.

What the picture cannot show #

The population in these figures is schematic: uniform in perihelion distance and eccentricity over chosen ranges, with an inclination distribution made up for convenience. The real sporadic background is highly structured. It falls into a few broad radiant sources tied to Jupiter-family and Halley-type comets, and that structure makes chance partners far more common in some parts of the element space than in others. So the numbers of partners in the figures are only an average guide, and an orbit near a dense source will have many more.

The stream in the purity figure is a four-dimensional Gaussian of one width, whereas a real stream is elongated, sometimes split, and spread unequally among the elements, so a single spread cannot describe it. The speed figure takes one element’s error and ignores deceleration, and errors in direction are also larger for near-ecliptic orbits. Nothing in these figures distinguishes between kinds of orbit-similarity function, as they would show the same behaviour with altered constants, and no conclusion about any particular shower is implied.

Still open: what a stream with no parent is made of #

The catalogues now contain many streams with no known parent. Some of these may be the remains of a dead comet, a family of objects from one breakup, or a parent that has simply not been found, and others will turn out to be fluctuations in the background. Sorting them requires an answer to the question of false positives that this argument has raised, and a stronger test than proximity in the elements. One such test is independent: a real stream should return each year on the same dates with the same radiant, and a chance clump should not, since nothing would tie its members to a recurring date. Another is backward integration to a common origin. With decades of camera data now accumulated, some candidates can already be tested in this way, and how many survive is a direct measurement of how much of the current list is the sky and how much is the size of the catalogue.