Astronomy & Celestial Mechanics
the sky, computed
Orbits

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.

Assumes Asteroid families, Secular theory and Non-gravitational forces.

16 min read 8 figures Small perturbations, large consequencesScales that defeat intuition

The first rung of this anchor got a date out of a family: the fragments’ semi-major axes spread apart at a rate that depends on their size, so the cloud is a V whose slope is an elapsed time. That argument takes the membership list as given.

There is no observation that produces one. An asteroid does not carry a label saying which parent body it came from, and the belt is not empty between the families. What exists is a list of orbits, and the assignment of some of them to a family is the output of a clustering algorithm with a free parameter in 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.
Fig. 1 Membership of a synthetic family against the cutoff velocity used to define it — 260 genuine fragments in a background of unrelated bodies, clustered by single linkage in the standard proper-element metric. At low cutoff the family is found in pieces. There is then a plateau, near 58 metres a second here, giving 276 members of which 6 per cent are background. Past it the count runs away, because single linkage absorbs an object and then searches from it: one chance interloper bridges the family to the belt. Completeness and contamination rise together, so there is no cutoff at which both are good.

The metric is a velocity, and that is not a convention

The clustering is done in a distance defined between two orbits:

d  =  na54(δaa)2+2 (δe)2+2 (δsin⁡i)2d \;=\; n a \sqrt{\tfrac54\Big(\tfrac{\delta a}{a}\Big)^2 + 2\,(\delta e)^2 + 2\,(\delta \sin i)^2}

with nana the orbital speed. The three coefficients are not chosen. They come from Gauss’s equations for how a velocity impulse applied at an unspecified point on an orbit changes each element, averaged over the orbit — so dd is, to first order, the speed with which one fragment would have to have left the other in order to be where it is.

That is a strong thing to have. It means the cutoff is a physical quantity with an expected value: a catastrophic collision between bodies of a few tens of kilometres produces fragments with ejection speeds of a few tens to a hundred metres a second, comparable to the escape velocity from the parent body. So a cutoff of forty or eighty metres a second is not an arbitrary knob; it is a statement about the collision.

It is also, in a way the metric conceals, wrong for most of the families it is applied to.

The history is worth a paragraph because the method is a century old and its difficulty has been the same throughout. Kiyotsugu Hirayama noticed in 1918 that the asteroids clump in the elements that stay put rather than in the ones that oscillate, and identified three families by eye from a list of a few hundred orbits. Eye and hand remained the method for sixty years. Automatic clustering arrived in the 1980s with the sample in the thousands, and the plateau criterion arrived with it — not as a theory but as an observation that the count of a real clump is flat over a range of cutoffs and the count of a chance grouping is not. That criterion is still what every published list rests on, and it is a heuristic.

Where a family stops being denser than the belt: 82 m s⁻¹. Counts in annuli of transverse velocity distance from the family's centre — eccentricity and inclination only, with semi-major axis excluded because Yarkovsky drift has spread the family in that coordinate and in no other, for the same synthetic slab: the family's own 260 fragments against 417 unrelated bodies. The two histograms have opposite shapes for reasons that have nothing to do with each other. The family's is a Gaussian in each coordinate, so its counts fall away steeply; the background's rises in proportion to the distance, because an annulus of radius d holds an area proportional to d and the background is uniform in it. Their crossing at 82 metres a second is the family's edge in the only sense in which it has one — beyond it, most of what a clustering algorithm absorbs is not related to the parent body at all. That is why the plateau in the previous figure exists and why it is not sharp. A family does not end; it becomes outnumbered. The consequence for anything computed from the members is that the outermost ones, which are the most informative about the ejection velocity and about the age, are also the least likely to belong.
Fig. 2 Counts in annuli of transverse velocity distance from the family’s centre — eccentricity and inclination only. The family’s own fragments fall away steeply; the background’s counts rise in proportion to the distance, because an annulus of radius d holds an area proportional to d. They cross at 82 metres a second, and that crossing is the family’s edge in the only sense in which it has one: beyond it, most of what a clustering algorithm absorbs is unrelated to the parent body. A family does not end. It becomes outnumbered.

Why semi-major axis is not in that figure

The transverse distance in the last figure leaves out the semi-major axis, and the reason is the whole complication.

A fragment’s semi-major axis does not stay where the collision put it. The Yarkovsky effect — a thermal recoil from the asymmetric re-radiation of absorbed sunlight — pushes it inward or outward according to its spin direction, at a rate that goes inversely with its size, for as long as it exists. Over a hundred million years a kilometre-sized fragment drifts by a few hundredths of an au, which in the metric above is several hundred metres a second.

So an old family is a cigar in the metric: a few tens of metres a second across in eccentricity and inclination, and several hundred long in semi-major axis. Its velocity distance from its own centre is mostly a distance the fragments never travelled at, and any statistic that treats the three coordinates alike is measuring the drift and calling it a dispersion.

A family of 261, or of 327, 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 138 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 43 metres a second here, giving 261 members of which 0 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.
Fig. 3 The same family with almost no drift — a young one, or a family of large bodies that Yarkovsky barely moves. The plateau is at 43 metres a second rather than 58, the family is found essentially uncontaminated, and the runaway is postponed. A compact family is easy and an old one is not, which is exactly backwards from what is wanted: the old families are the ones with enough drift to be dated, and they are the ones whose membership is least secure.

The practical response has been to cluster in the two transverse coordinates first and then take a slice in semi-major axis, or to cluster in a space that includes the absolute magnitude so that the V-shape’s own geometry is built into the metric. Both amount to assuming part of the answer, and both are better than not doing it.

There is one further consequence of the cigar shape and it is about the algorithm rather than the physics. Single linkage grows a cluster by absorbing anything within the cutoff of anything already absorbed, so it follows a chain of stepping stones and is perfectly happy with a long thin structure. That is why it finds drift-widened families at all — a method requiring compactness would reject them. The same property is what makes it percolate: a chain is a chain whether the stepping stones are fragments or background. The algorithm’s ability to find the families and its tendency to swallow the belt are the same property.

What a deeper catalogue does

The percolation cutoff is not a property of the family. It is a property of the family and the catalogue.

A family of 290, or of 1088, 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 834 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 63 metres a second here, giving 290 members of which 10 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.
Fig. 4 The same family in a catalogue twice as deep. The plateau has moved to 63 metres a second and the contamination there has risen from 6 per cent to 10; the runaway now reaches 1088 objects rather than 582. Both effects follow from the same fact — the background is denser, so a chance bridge is closer at hand — and neither is an improvement in the data being an improvement in the answer. A family’s published membership grows with every survey, and part of that growth is real and part is not, in a proportion that has to be modelled rather than measured.

There is a related trap in comparing family sizes between surveys. A statement that family A has three times the members of family B is a statement about two clustering runs, and if the two families sit at different distances from the Sun then the survey’s completeness differs between them — a given absolute magnitude is a fainter apparent magnitude further out — so part of the ratio is the survey and not the collision. Correcting for that requires knowing the size distribution being cut into, which is one of the things the family is being used to measure.

Reality is worse than the figure, because the density of the belt is not uniform. A family sitting next to another family has a background that is itself clustered; a family near a resonance has a background with a hole in it. The published lists carry these as caveats and the caveats are not quantitative.

Where a family stops being denser than the belt: 152 m s⁻¹. Counts in annuli of transverse velocity distance from the family's centre — eccentricity and inclination only, with semi-major axis excluded because Yarkovsky drift has spread the family in that coordinate and in no other, for the same synthetic slab: the family's own 260 fragments against 4181 unrelated bodies. The two histograms have opposite shapes for reasons that have nothing to do with each other. The family's is a Gaussian in each coordinate, so its counts fall away steeply; the background's rises in proportion to the distance, because an annulus of radius d holds an area proportional to d and the background is uniform in it. Their crossing at 152 metres a second is the family's edge in the only sense in which it has one — beyond it, most of what a clustering algorithm absorbs is not related to the parent body at all. That is why the plateau in the previous figure exists and why it is not sharp. A family does not end; it becomes outnumbered. The consequence for anything computed from the members is that the outermost ones, which are the most informative about the ejection velocity and about the age, are also the least likely to belong.
Fig. 5 The same construction for a more energetic collision — an ejection dispersion of forty metres a second rather than fifteen. The crossing moves out to 152 metres a second, because the family occupies a larger volume and stays denser than the background further out. So the natural cutoff for a family is a property of the collision that made it, and the practice of applying one cutoff across a whole survey is a compromise made because the collision’s energy is one of the things being inferred.

It is worth saying what would settle the matter and why it is not available. A membership test that did not depend on proximity at all would break the circularity — something intrinsic to the object rather than to its orbit. Two such tests exist in principle. One is composition, which the next rung is about. The other is a backward integration of the orbits to a common epoch, which works beautifully and only for families young enough that the integration stays deterministic. That limit is the expiry date every prediction in a chaotic system has: the belt’s Lyapunov times are a few hundred thousand to a few million years, so a backward integration is reliable for a handful of million and useless beyond — against the hundreds of millions the interesting families have lived.

What proper elements cost

Underneath everything above is a coordinate system that has to be computed rather than observed.

An osculating eccentricity that swings 10 times a family's own width, over a proper one that does not move. One outer-belt asteroid's elements over 800 thousand years. Above, eccentricity; below, sin i. The osculating value — the one an orbit solution reports for today — is the length of a forced vector fixed at 0.0383 by the planets plus a free vector of length 0.06 turning around it once every 188 thousand years, so it breathes between 0.022 and 0.098. The shaded band is the twelve-year term Jupiter adds; it is drawn as an envelope rather than as a curve, because its period is thirty thousand times finer than this plot and any sampling of it would be an alias rather than a signal. The flat line is the proper eccentricity, the length of the free term alone, which is a quasi-integral of the motion and holds to within a fraction of a per cent for tens of millions of years. The swing is 0.077; a collisional family's whole dispersion in eccentricity is about 0.008. That ratio, 10 to 1, is why the asteroid belt drawn in osculating elements is a featureless cloud and the same belt drawn in proper elements has a hundred clumps in it. The families were always there; the coordinates were not.
Fig. 6 Why the catalogued elements are useless for this. An asteroid’s osculating eccentricity — the number an orbit determination reports — oscillates over the secular period as the free vector rotates about the forced one, swinging here across ten times the width of a family. Its proper eccentricity, the amplitude of that rotation with the forced part removed, does not move at all. The whole of family identification happens in the second, which is a quasi-integral computed by averaging away the short-period terms and subtracting the forced ones, and which does not exist as a number until somebody has done that.

The proper elements are also not one thing. There are analytic ones, computed from a truncated perturbation series, and synthetic ones, computed by integrating the orbit for a few million years and Fourier-analysing the result. The two agree well where the theory converges and disagree where it does not, which is exactly the regions the theory has trouble with — high inclination, high eccentricity, and near resonances. The published family lists based on the two do not contain the same objects.

Two things follow. The proper elements have their own errors, from the truncation of the perturbation theory and from the object’s own orbit being imperfectly known, and those errors are of order a metre or two per second — small compared with the cutoff, but not negligible for the tightest families. And more seriously, proper elements fail in and near mean-motion resonances, where there is no quasi-integral to compute. A family straddling a resonance is cut in two by the coordinate system rather than by anything physical, and the part inside the resonance leaks away on a timescale of millions of years, which is a real loss and looks like the same thing. A resonance clears a gap whether or not anybody is trying to count what used to be in it.

Six families and four gaps, in coordinates no telescope reports. The main belt in proper elements: semi-major axis against proper sin i, with 900 background bodies and the members of six families. Nothing here is what an orbit catalogue lists. Every coordinate is a free-oscillation amplitude recovered by averaging away the short-period terms and subtracting the forced ones, and the same objects plotted in the elements they are catalogued with fill this frame as a single smooth cloud. What appears once the averaging is done is structure of two opposite kinds. The clumps are collisions: each is the debris of one parent body, and each drawn family scatters between 6.1 and 12.3 times more tightly than the background at its own distance from the Sun. The gaps are resonances — the vertical lanes at 3:1, 5:2, 7:3, 2:1 with Jupiter — and they are empty because proper elements stop existing there: a body in a mean-motion resonance has no quasi-integral to compute, its eccentricity wanders, and it leaves. The two features are the same argument seen twice: the belt keeps a memory of individual events for hundreds of millions of years everywhere except where the memory is erased, and both the memory and the erasure are invisible until the coordinates are changed.
Fig. 7 The belt in proper elements, with six families and the resonant lanes that empty it. The clumps and the gaps are the same argument seen twice — a memory of individual collisions kept for hundreds of millions of years everywhere except where the memory is erased. Every one of those clumps has a membership list produced by the procedure in the first figure, and a boundary between two of them where they overlap in projection is a boundary drawn by an algorithm.

Why any of it matters outside the belt

It would be possible to read all of this as a specialist’s difficulty with a specialist’s catalogue. It is not, because the families are the input to two arguments that reach much further.

The first is the impact history of the inner solar system. The families are the reservoir from which the near-Earth objects are supplied — fragments drift in semi-major axis until they meet a resonance, the resonance pumps their eccentricity, and they leave the belt. So the size and age distribution of the families is what sets the cratering rate, and a surface dated by counting holes in it is dated against a flux whose supply is estimated this way. A family membership uncertain by a factor of two is a supply rate uncertain by the same factor.

The second is the collisional history of the belt itself. The number of families of each size, against the size of the parent bodies they came from, is a record of how often bodies of each size have been shattered — which is the only observational constraint on the strength of asteroids at kilometre scales that does not come from a laboratory experiment on a rock the size of a fist. Where the mass is and where the light is in a collisional cascade depends on that strength through an exponent, and the exponent is fitted to the families.

What is actually measured

The chain from observation to family is long enough to set out.

An asteroid is discovered as a moving point in a survey image. Enough detections give an orbit, which is a set of osculating elements at an epoch. A numerical integration or an analytic theory then converts those into proper elements, with an accuracy that depends on how long the orbit has been observed and on whether the object is near a resonance. Those proper elements go into the metric above. The clustering runs. A cutoff is chosen by looking for a plateau in a curve of the kind in the first figure. And the resulting list is then used to infer the collision’s age, its energy, and the parent body’s size.

None of the intermediate quantities is checkable against anything else. There is no independent census of how many fragments a collision of a given energy produces, no second method for the age of an old family, and no way to visit one. The internal consistency of the picture is its whole support: the ages come out ordered sensibly against the families’ sizes, the size distributions look like collisional cascades, and the drift rates implied are the ones the thermal recoil predicts from a body’s spin and thermal inertia. That is a good deal of consistency and it is not a measurement of any one thing.

Every quantity in that last sentence depends on the membership list, and the list depends on the cutoff. The age in particular is read off the family’s outer edge in semi-major axis, which is exactly where membership is least certain.

A 1181.4-million-year-old collision, dated from the shape of a scatter plot. The Eos family: 200 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 1300 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 1181.35 million years against the 1300 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.
Fig. 8 The V-shape from which an age is read, drawn for a large old family. The age comes from the slope of the envelope, and the envelope is defined by the outermost members at each size — the objects furthest from the centre in the coordinate the metric handles worst, and therefore the ones most likely to be interlopers. Including a few background objects steepens the V and ages the family; excluding real members flattens it and makes it younger. The published ages of the large families carry uncertainties of tens of per cent for this reason and not because the drift rate is poorly known.

What the numbers actually are

Setting the synthetic figures beside the real catalogue is worth doing, because the orders of magnitude are the thing.

The main belt catalogue holds over a million objects with orbits good enough for proper elements. Recent family analyses identify around a hundred and twenty families, containing between them perhaps a third of all belt asteroids — a third of the belt is collisional debris of identifiable parentage, which is a remarkable statement and one that depends entirely on where the cutoffs were put. The largest families have tens of thousands of members; the smallest that anyone will assert have a few dozen.

The cutoffs actually used run from about twenty metres a second for the tightest families to a hundred and twenty for the most diffuse, chosen family by family rather than globally, on the plateau criterion. Estimated interloper fractions, where they have been estimated at all, run from a few per cent to about thirty. And the number of asserted families has roughly doubled with each major extension of the catalogue, which is the pattern the fourth figure predicts and does not distinguish from genuine discovery.

The generalisation

The structure worth extracting is that single linkage percolates, and that percolation is a threshold rather than a gradual degradation.

Connecting every pair closer than a distance and asking what is connected to what is a classical problem, and its answer is that below a critical density-dependent distance the clusters are local and above it there is one cluster containing almost everything. Between those regimes there is no comfortable middle: the transition is sharp, and it gets sharper as the sample grows. Any method built on single linkage therefore has a parameter it cannot be indifferent to.

There is a second structural lesson, about what it means for a parameter to be chosen by looking at the data. The plateau criterion is defensible — a real clump does produce a flat stretch — but the flatness is not a property of the family alone; it is a property of the family against the local background, and the same family in a different neighbourhood plateaus somewhere else. Choosing a parameter from a feature of the data means the parameter inherits everything that made the feature, and there is no separating the two afterwards. That is the same difficulty an eccentricity that belongs to the whole system rather than to one planet presents in a different guise: a quantity attributed to an object turns out to be a property of the object and its surroundings jointly.

The same shape appears wherever a structure is defined by proximity in a space with a background in it. A galaxy cluster identified by a friends-of-friends algorithm has the same problem with the same cure and the same residual doubt; so does a stellar association identified in position and proper motion. The question to ask of any such catalogue is not how many objects it contains but what happens to that number when the linking length is changed by twenty per cent.

There is a third thing to take from it, which is about how a field lives with a difficulty it cannot remove. Nobody in this subject believes a membership list is exact, and nobody has stopped using them. What has happened instead is that the quantities computed from them have been chosen for robustness: the centre of a family in proper elements is very stable against the cutoff, and so is the family’s existence; the age and the total mass are not. Reading a paper in this field means noticing which of the two kinds of quantity is being quoted, and the good ones say so.

The corollary is about what these lists are for. A family membership list is not a measurement; it is a hypothesis, and it is a good one when the quantity computed from it is insensitive to the cutoff. The age from a V-shape is not, which is why it is the number this essay ends on.

Where the ladder goes next

The next rung takes the interloper problem head on, using the one piece of information the clustering ignores: colour. Fragments of one parent body have the same composition and therefore the same reflectance spectrum, so a broadband colour taken from the same survey that found the object is an independent membership test — and applying it to families identified by clustering removes between five and thirty per cent of the members, depending on the family.

Further rungs on this anchor: the families that are found only in a space including the absolute magnitude, which are old and diffuse and were invisible to the classical method; the halo of a family, which is real and extends far beyond any cutoff; the parent body’s size reconstructed from the fragment size distribution, and the fact that the reconstruction usually accounts for less mass than the family should contain; and the young families, of which a handful are recent enough for the fragments’ orbits to be integrated backwards to a common point in time — the only family ages that are measurements rather than inferences.

Where this goes next

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. Below 0.46 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 0.46 microns; for ice it is 1.15, and for iron 0.15. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola. The drag that sorts a disc by size Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is. Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap. No planet has an eccentricity of its own Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Asteroid familyCollisional familyEjection velocityHierarchical clusteringInterloperMain beltPercolationProper elementsSingle linkageYarkovsky effect