Astronomy & Celestial Mechanics
the sky, computed
Concept

Proper elements — where it appears

The nearly constant amplitudes underlying a body's oscillating osculating elements, obtained by removing the terms forced by the other planets. Members of an asteroid family share them millions of years after the family formed, when their instantaneous elements have nothing in common.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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.

orbits · Secular theory The divisor is 0.129″/yr and the theory's own error is 0.24. Six frequencies of the secular solar system on one logarithmic axis, in arcseconds per year. The top two are the pair whose near-equality is the whole story: the perihelia of Mercury and Jupiter separate at 1.333″ a year, the nodes of Mercury and Venus at 1.462, and the difference of those two differences is 0.129 — a resonant argument that turns once every 10.0 million years. A term with that argument in the disturbing function acts in one direction for five million years at a stretch, which is what pumps Mercury's eccentricity, and it is the reason the inner solar system's Lyapunov time is what it is. The bottom three bars are why this figure exists. The divisor is smaller than the corrections the theory that computes it leaves out. Relativity contributes 0.4298″ a year to g₁ alone — the same 43 arcseconds a century that broke Newtonian gravity — which is 3.3 times the divisor; the fourth-order terms in the eccentricity that Laplace–Lagrange truncates come to about 0.24″, which is 1.8 times it; and the second-order solution computed on this page gets 0.35″, missing the published value by more than the value itself. A theory cannot bound what it cannot resolve. Laplace's proof that the eccentricities stay bounded is a proof about a system whose frequencies are constants, and the frequency that decides the question is not one.

The bound that holds only in the linear theory

Laplace proved the planetary eccentricities bounded, and the proof is a proof about a linearised system with constant frequencies. One combination of those frequencies is nearly zero — and it is smaller than the terms the linearisation threw away, which is why the stability of the solar system is a probability rather than a theorem.

orbits · Secular theory 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.

orbits · Asteroid families 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

Named alongside it

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

Asteroid familyAngular momentum deficitChaotic diffusionEjection velocityInterloperLaplace–Lagrange theoryYarkovsky effectCollisional ageCollisional familyCosmic-ray exposure ageDisturbing functionEccentricity

All concepts