Astronomy & Celestial Mechanics
the sky, computed
Concept

Disturbing function — where it appears

The extra gravitational potential one orbiting body feels from another, expanded as a sum of cosines of combinations of their orbital angles. Averaging away the terms containing the fast angles is what leaves the secular problem behind.

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

The osculating semi-major axis of a perturbed orbit. The semi-major axis a test particle would have if the perturber vanished, computed from its position and velocity at every step of an integration over 26 orbits of the perturber. It is not constant: a short-period ripple rides on a slow trend, and only the trend accumulates.

Elements that do not stay constant

Six numbers fix an orbit for all time, and the phrase is only true in a universe containing two bodies. Add a third and the six start moving — some of them wandering and returning, one or two of them drifting in one direction forever, and the difference between those two behaviours is the whole of celestial mechanics after Newton.

orbits · Perturbations 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 two halves of what is left over. The disturbing potential of a perturber on a test particle, against the difference in longitude between them, at a semi-major axis ratio of 0.62. The upper curve is the direct term — the perturber's own attraction, which peaks at conjunction where the separation is smallest and falls to 1/(1+α) half a turn later. The lower one is the indirect term, which exists only because the coordinates are centred on a primary that is itself being accelerated, and which is a pure cosine of the longitude difference. The indirect term is the larger of the two over 3 per cent of the circle, and it averages to exactly zero while the direct term averages to something positive. Everything that happens slowly in a planetary system comes from that asymmetry: the part that survives averaging is not the part that dominates the instantaneous force.

The series that is subtracted

The two-body problem is solved, so nobody solves it twice. Every planetary theory since Newton begins by taking that solution away and asking what is left — and what is left is an infinite series whose terms are stacked in a hierarchy that makes the first half-dozen of them enough.

orbits · Perturbations Jupiter and Saturn trading a degree of longitude over nine centuries. Mean longitudes of Jupiter and Saturn in arcminutes, with their straight trends taken out and a 40-year smoothing applied, from a planar integration of the Sun, Jupiter and Saturn from their elements at the year 2000. Jupiter swings ±20′ and Saturn ±50′, in opposite senses, with a period of about 870 years. Comparing Ptolemy's positions with those of his own day, Kepler could not fit one mean motion to both, and by 1676 Halley reported Jupiter gaining and Saturn losing. Before 1785 that looked as if the Solar System were unstable; the curve shows that it is a very long period.

A term too small to keep that a denominator kept

Every planetary theory is an expansion, and the order of its terms is meant to say which can be dropped. In the 1770s the largest unexplained motion in the Solar System, Jupiter speeding up and Saturn slowing down by a degree over centuries, came from a term that the ordering called negligible. Five of Saturn's orbits nearly match two of Jupiter's, and that one near-match divides a tiny third-order coefficient twice by a tiny frequency. An integration of the three bodies shows the swap directly, and shows what Laplace saw: it reverses every nine centuries.

orbits · Perturbations

Named alongside it

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

Osculating elementsGreat inequalityLaplace coefficientPerturbation theorySecular variationAngular momentum deficitAsteroid familyConvergenceDalemberts ruleEccentricityEigenmodeForced eccentricity

All concepts