A family checked by a colour it was not chosen by

An asteroid family is a hypothesis about a set of orbits. A collision broke a parent body apart, the fragments left with velocities of a few tens of metres a second, and their orbits — once the fast oscillations have been averaged away into proper elements — still lie close together, hundreds of millions of years later. Find a tight cluster of proper elements and the family is found.

The trouble is that the asteroid belt is not empty between families. A clustering algorithm that gathers every asteroid within some velocity distance of another member does not know which nearby orbits are fragments and which are unrelated asteroids that happen to orbit there. How many members a family has depends on the cutoff chosen, and the same choice decides how many strangers are admitted. Those strangers are interlopers, and they do not announce themselves: they have proper elements, which is all the clustering looks at, and those proper elements are among the family’s.

This essay is about the one property the clustering ignores and the fragments share. A collision breaks one body, and one body has one composition. The fragments of a carbonaceous parent are carbonaceous; the fragments of a stony one are stony. Composition shows in reflected colour, and the large sky surveys that found most of the belt’s asteroids measured their colours in the same exposures. Colour is a membership test that was never used to choose the members, which is what makes it a test rather than a restatement.

How many strangers a cutoff admits #

The family drawn throughout this essay is synthetic: 260 fragments of a carbonaceous parent, spread by a collision and then pulled apart in semi-major axis by the slow thermal drift that disperses every family, sitting in a uniform background of unrelated asteroids — two-thirds stony, one-third carbonaceous. The clustering is single linkage in the standard velocity metric: start from the fragment nearest the family’s centre, absorb anything within the cutoff of anything already absorbed, and stop when nothing more can be reached.

The shape of the true curve is the familiar one. At a small cutoff the algorithm finds a dense core of fragments and almost nothing else. As the cutoff grows it admits the family’s outskirts and, among them, the occasional background object. Past some cutoff it reaches a background object that is close to another background object, and the list grows by bridging rather than by absorbing the family. A published family is chosen at a cutoff where the membership is stable, below the bridge — and even there, in this synthetic case, roughly one object in five is not a fragment.

That fraction depends on how dense the background is relative to the family, which varies enormously across the belt. A family in a sparsely populated region at high inclination has few neighbours to absorb; one in the crowded inner belt has many. The figure is drawn for a crowded case, and the published estimates of interloper contamination in real families range from a few per cent to a few tens of per cent.

Where the strangers sit #

The interlopers are not distinguishable by position — that is what makes them interlopers.

The picture shows why positions alone cannot clean a family. An interloper is admitted because it is near a member, and once admitted it is near members by construction. Nothing about its location marks it out.

It also shows where the interlopers concentrate: towards the ends of the band in semi-major axis, where the fragments have been spread thinnest by the drift. The family’s density falls away there while the background’s does not, so the ratio of strangers to fragments rises.

That gradient matters for everything a family is used for. The V-shaped plot that dates a family is read from its edges: the outermost fragments of each size have drifted furthest, and the slope of the envelope they trace is the elapsed time. An interloper beyond the envelope widens the V and makes the family look older; an interloper among the edges blurs the envelope and makes its slope uncertain. The most informative members of a family for its age are the ones most likely not to belong to it.

The test the clustering never applied #

Fragments of one parent body share its composition, and composition shows in colour. The Sloan Digital Sky Survey, imaging most of the sky in five filters for galaxies and quasars, recorded every bright asteroid that crossed its fields — more than a hundred thousand of them — and their colours fall into a few groups that correspond to the main compositional classes. The most useful single number combines two of the survey’s colours into one index that runs from about −0.1 for the carbonaceous, dark, primitive asteroids to about +0.1 for the stony ones, with little overlap between them.

The colour cut does what it is designed to do. Two-thirds of the background is stony, two-thirds of the interlopers are stony, and they sit in a separate peak where no fragment of a carbonaceous parent can be. Removing them is not a judgement; the colour says the object cannot be a fragment of this parent.

The cost is small. A colour measured with an uncertainty of a few hundredths scatters some true fragments across the cut, two in 260 here. That is a loss of members, not a gain of interlopers, and it is a known rate that can be corrected for.

The test’s limit is equally clear. A carbonaceous background asteroid linked into a carbonaceous family has the family’s colour, and nothing in the histogram distinguishes it from a fragment. Twenty of the sixty-one interlopers are of that kind. They stay.

Families were colour-checked from the start #

The idea is older than the surveys. Kiyotsugu Hirayama identified the first families in 1918 from orbital elements alone — Koronis, Eos and Themis among them — and argued that each was the debris of a single body. When reflectance spectra of asteroids began to be taken in the 1970s and 1980s, the families were among the first things checked, and the check came out the way the hypothesis required: Themis members are dark and carbonaceous, Koronis members are stony, and Eos members belong to a rarer intermediate class that is uncommon in the background around them. A grouping found by orbits alone turned out to be a grouping by composition as well, and that agreement is a large part of why families are believed to be collisional at all.

What the surveys added is scale. A few spectra confirm that a family’s core is homogeneous; a hundred thousand colours let every member of every family be checked, including the small faint ones at the edges that dominate the counts. The colour test changed from a confirmation of the family hypothesis into a cleaning step applied to each family’s list.

The scale also brought the colours’ own systematics into view. A colour measured on one night is a snapshot of a rotating body at one phase angle. Surfaces exposed to space darken and redden over time — stony surfaces more than carbonaceous ones — so a young stony family is measurably bluer than the old stony background it sits in. That makes colour a slightly better test than the two-class model drawn here for young stony families, and introduces a dependence on age that the model ignores.

A parent that was not one colour #

The test rests on an assumption that is usually true and occasionally interesting when it fails: that the parent body was compositionally uniform, so that every fragment carries the same colour.

A body large enough to have melted and differentiated is not uniform. It has a metal core, a mantle of olivine-rich rock and a basaltic crust, and a catastrophic collision that broke it apart would scatter fragments of all three, each with its own colour and spectrum. A colour cut set at the crust’s colour would reject the mantle and core fragments as interlopers.

This is not hypothetical. The fragments of Vesta are the basaltic crust excavated by a cratering impact, and they are colour-uniform because only the crust was excavated. Differentiated parents that were broken completely should have left families containing olivine-dominated mantle fragments, and such fragments are strikingly rare in the belt — a long-standing puzzle sometimes called the missing mantle problem. One proposed resolution is that most differentiated bodies were destroyed so early that their debris has been ground down or removed since. Another is that their mantle fragments have been sorted out of their families by exactly the kind of colour cut drawn here, because nobody expected them.

A family cleaned by colour is therefore cleaned under a hypothesis about its parent. For an undifferentiated parent the hypothesis is safe; for a differentiated one the colour cut discards the members that would have said so.

Estimating what cannot be removed #

The hidden interlopers can be counted even though they cannot be identified, because the background’s composition is known from the regions around the family.

If a third of the background asteroids near the family are carbonaceous and two-thirds stony, then the linked interlopers should be in the same proportion, since the clustering chooses them by orbit and not by colour. Every two stony interlopers detected and removed imply about one carbonaceous interloper undetected. Scaling the detected count by the background’s own mix gives an estimate of the hidden count, and the dashed curve in the sweep is exactly that estimate.

It tracks the true interloper fraction closely at every cutoff, which is the result the method needs. It does not track it exactly, because the hidden interlopers are a few dozen random draws from the background and their number fluctuates about its expectation. For a single family the estimate is good to about the square root of the number of interlopers; for the statistics of many families it is unbiased.

What the estimate cannot do is remove anything. A family’s list after the colour test is shorter and cleaner, and still contaminated at a known rate by objects that cannot be named. For any measurement made from the list as a whole — a size distribution, an age from the V, a count of members — the known rate is enough to correct for. For any measurement made of an individual member — a spectrum taken because the object is thought to be a fragment, a rotation period thought to record the collision — it is not, and the colour-cleaned list still holds the strangers.

A family that looks like its neighbourhood #

The test’s power depends on how different the family is from its surroundings, and it can be close to useless.

This asymmetry runs through the real belt. The inner belt is dominated by stony asteroids and the outer belt by carbonaceous ones, so a carbonaceous family in the inner belt and a stony family in the outer belt are the easy cases, and their opposites are hard. The cleanest examples are families whose parent was unusual. The family of Vesta, the large differentiated asteroid whose basaltic crust was excavated by an impact, has a distinctive colour and spectrum that almost no background object shares, and colour separates its fragments from the background with almost no ambiguity. A family whose colour is the commonest colour in its region gets almost nothing from the test.

Colour is one of several such tests #

The same logic applies to any property the fragments share and the clustering did not use.

Albedo. The fraction of light an asteroid reflects is a strong function of composition — a few per cent for carbonaceous bodies, twenty or thirty for stony ones — and infrared surveys that measured asteroids’ thermal emission gave albedos for over a hundred thousand of them. An albedo is a better separator than a visible colour for dark families in bright neighbourhoods, and it comes with a diameter as a by-product. The families identified by clustering and then checked against albedo show the same pattern as the colour checks: a clean core and a contaminated periphery.

Spectra. A full reflectance spectrum distinguishes many more classes than two, but is available for only a few thousand asteroids, mostly the larger ones. It is the test used to confirm the colour test, rather than an alternative to it.

Rotation and shape. These are not inherited from the parent in any simple way and are not membership tests at all.

What all of these share is the logic of a control. The clustering is a selection on orbits. Any property that is correlated with membership but not used in the selection can be used to measure how good the selection was — and the measurement says something different from the selection, because the interlopers were admitted by the orbits and are caught, when they are caught, by something else.

What the cleaning changes #

The size distribution of a family is one of the two or three things a family is studied for, and it is where the interlopers do the most damage, because they bring the background’s sizes with them.

A young family’s fragments follow a steep size distribution: many small pieces for every large one, as a collision leaves them. The background has a shallower distribution, eroded by billions of years of collisions that grind small bodies faster than they are replenished. Interlopers drawn from the background therefore over-represent the large sizes relative to the fragments, and a contaminated family’s size distribution is flatter than the true one — biased towards the background’s and towards the appearance of a family that has been eroding for longer. The size distributions of the bodies that collisions grind down are read for exactly that erosion, so the bias runs in the direction of the quantity being measured.

And the age from the V-shaped plot, as above, is biased by interlopers beyond the true envelope. There is a second, quieter route by which composition enters the V: the drift that spreads the family depends on each fragment’s size, density and thermal properties, and the drift rate itself says what a surface is made of. A size is usually computed from brightness through an assumed albedo, and an interloper of the other composition is assigned the family’s albedo and therefore the wrong size — a stony interloper in a carbonaceous family, given the family’s low albedo, is assigned a diameter two or three times larger than its own, and lands in the V as a large fragment far from the centre — where the envelope says no large fragment has had time to drift. Both biases are reduced by the colour cleaning and neither is removed. The careful family studies now quote membership lists that are cleaned by colour or albedo, with an estimate of the remaining contamination, and fit the V and the size distribution with an explicit background term rather than assuming every listed object is a fragment.

Where the model stops #

The background is uniform. The synthetic background has the same density everywhere in the slab. The real belt has structure on every scale — other families, resonances that empty strips of semi-major axis, gradients in density — and the interloper fraction depends on what the family happens to sit next to.

Two compositions, one colour. The model has two classes with well-separated colours. The real belt has more classes, some of which overlap in a two-filter colour, and space weathering reddens stony surfaces over time so that a young stony family is bluer than an old stony background. That last effect makes colour slightly better than drawn for young stony families.

Colours measured once. Each object’s colour is drawn with one measurement uncertainty. A real survey colour is a snapshot of a rotating body that may vary across its surface, measured at a phase angle that affects it, and the photometric errors grow for fainter objects — which are the small fragments, at the family’s outskirts, where the test is most needed.

The clustering is single linkage from one seed. Published family lists use more elaborate procedures — multiple seeds, cutoffs that vary with object size, steps that merge or split candidates — but every one of them is a selection on orbits, and every one admits interlopers by the mechanism drawn here.

The surprising connection #

The colour check on an asteroid family is the same move as the null tests of a weak-lensing survey: find a quantity the selection did not use, predict what it should look like if the selection worked, and measure it. In lensing, the null test is a component of the shear that gravity cannot produce; in families, it is a colour that a fragment cannot have. In both cases the comparison exposes only one kind of error, and the error it cannot catch — shear that is gravitational in form but not in origin, an interloper of the family’s own composition — has to be estimated from the one it can.

What makes the comparison worth stating is that both fields learned the same thing about it. The tests are not used to certify a sample as clean. They are used to measure how dirty it is, and then the dirt is carried through the analysis as a term with an uncertainty rather than removed. A family list with a colour test is not a list of fragments; it is a list with a known fraction of strangers.

Still open: how often the belt breaks #

The cleaned families are the inputs to a different question, one the previous essays on this subject have pointed towards: how often large collisions happen in the belt now. The handful of young families whose ages are measured to high precision by their converging orbital angles are small and recent; the large old families are dated by the V, which the interlopers bias. A census of families by age and parent size, with interlopers removed or modelled, gives a rate of catastrophic collisions over the belt’s history — and whether that rate has been steady, or was higher early on, or shows a recent spike from a single event, depends on ages that are only as good as the membership lists they are fitted to.