Astronomy & Celestial Mechanics
the sky, computed
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Asteroid families — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A 128.8-million-year-old collision, dated from the shape of a scatter plot. The Erigone family: 165 members drawn at their diameters and their proper semi-major axes, with inverse diameter up the page. The cloud is a V, and the V is a clock. Each member has been drifting in semi-major axis ever since the collision at a rate that goes as one over its diameter, with a sign set by which way it spins — prograde outward, retrograde inward — so after 130 million years the small members have moved far and the large ones have barely moved at all. Plotted against 1/D that envelope is a straight line through the family's centre, and its slope is the drift rate for a one-kilometre body multiplied by the elapsed time. Fitting the two edges of the points actually drawn here returns 128.85 million years against the 130 the members were generated from. The rounding at the bottom is not an artefact: it is the ejection velocity, some 15 metres per second, which every member got at the moment of the collision and which is the same for all sizes. The picture cannot show the interlopers — background asteroids that happen to lie inside the V and have nothing to do with the family — and it cannot show the members that have drifted into a resonance and left the belt entirely, which is the reason the oldest families have the softest edges.

    A collision dated by a scatter plot

    Nothing in the solar system carries a date. A collisional family does — because a force that depends on a body's size has been pushing its fragments apart ever since, so the cloud is a V whose slope is an elapsed time, and one of those dates is confirmed by fossil meteorites in Swedish limestone.

    part 1 · orbits
  2. A family of 276, or of 582, depending on one number. Membership of a synthetic family against the cutoff velocity used to define it, for a family of 260 genuine fragments ejected at 15 metres a second sitting in a background of 417 unrelated bodies. The clustering is single linkage in the standard proper-element metric, started from one object and grown until nothing more is within the cutoff of anything already absorbed. Three curves: the number of objects claimed, scaled to its largest value; the fraction of the real family recovered; and the fraction of the claim that is background. The shape is the whole difficulty. At low cutoff the family is fragmented and only its core is found. There is then a plateau — near 58 metres a second here, giving 276 members of which 6 per cent are background — and that plateau is what every published family list is chosen at. Past it the count runs away, because single linkage absorbs an object and then searches from the object it has just absorbed: one chance interloper bridges the family to the belt and the algorithm returns half the main belt. Completeness and contamination rise together, so there is no cutoff at which both are good and the plateau is a compromise rather than a discovery.

    A family whose size is a choice

    Nothing observable distinguishes a collisional fragment from an asteroid that happens to be nearby. Membership is assigned by clustering at a cutoff velocity, and the same family has 276 members or 582 according to which cutoff is used — with completeness and contamination rising together, so no cutoff is good.

    part 2 · orbits
  3. Perihelia of a young family, run backwards until they meet. The longitude of perihelion of each of 26 members of a synthetic young asteroid family at the Karin family's position, relative to the family's mean at each moment, against time before the present. Today the angles are spread round the whole circle: members only a few thousandths of an astronomical unit apart precess at rates that differ by a fraction of an arcsecond a year, which over millions of years is many full turns. Run backwards — with each member's own drift in semi-major axis included — they converge, and the spread is smallest at 5.75 million years ago, where they were made. The nodes turn the other way, and in the real family, whose rates come from a full theory in which the two frequencies differ, they give a second, independent convergence at the same instant. An age read this way is a measurement of a time in the way a light curve's period is, with an error of a few per cent, rather than an inference from a drift rate.

    An age read from angles that meet

    A family of asteroids a few million years old has had its members' orbits turned round and round by the planets, each at a slightly different rate, until their orientations look random. Turn them back and they come together at one instant — the collision — with an uncertainty of a few tens of thousands of years, but only if each member's own slow drift in sunlight is put back too.

    part 3 · orbits
  4. What colour removes, and what it can only estimate. The share of a synthetic family's linked membership that is not part of the family, against the clustering cutoff, with the parts a colour test can and cannot see. The true interloper fraction rises with the cutoff, from 0 per cent at 20 m/s to 84 per cent at 110. A colour cut removes the interlopers of the other type — 12 per cent of the list at 60 m/s — and leaves those of the family's own type, 6 per cent, which can only be estimated by scaling the detected ones by the background's mix. That estimate (dashed) tracks the truth on average and scatters about it, because a few dozen interlopers are a few dozen random draws.

    A family checked by a colour it was not chosen by

    Asteroid families are found by clustering orbits, and the clustering absorbs background asteroids that happen to orbit nearby. Fragments of one body share its composition, and the survey that found them measured their colours for free — so colour is a second membership test the clustering never used. It removes the interlopers that look different, and it can only estimate the ones that look the same.

    part 4 · orbits

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