Astronomy & Celestial Mechanics
the sky, computed
Concept

Hierarchical clustering — where it appears

A method that builds groups by repeatedly linking any object within a chosen distance of one already in the group. Applied to asteroid orbits it finds collisional families, and the cutoff distance decides how many members, and how many strangers, each family has.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

orbits · Asteroid families 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.

orbits · Asteroid families

Named alongside it

The objects these essays reach for when they reach for this one.

Asteroid familyInterloperProper elementsYarkovsky effectAlbedoCollisional familyEjection velocityMain beltPercolationSingle linkageSize frequency distribution

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