Astronomy & Celestial Mechanics
the sky, computed
Concept

Perturbation theory — where it appears

An expansion of a solution in a small parameter. Its corrections do not come with a sign, so an estimate improved in one respect is not automatically an improved estimate, and the place to check it is where the neglected quantity is largest.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A blur undone with the field that made it. The acoustic feature in the correlation function, as it was laid down and as bulk flows leave it, with three reconstructions. The broad band is left out entirely — an analysis divides it away and fits the position of what remains — so what is drawn is the feature itself. Every galaxy has moved since recombination, by several megaparsecs, and the two ends of a pair separated by 150 megaparsecs move almost independently: the pairwise dispersion of those displacements, Σ = 8.5 Mpc here, convolves the feature and broadens it from 35 to 43 megaparsecs. Reconstruction estimates the displacement from the observed density and moves every galaxy back along it. The estimate has to be smoothed, because the relation between density and displacement is only linear on large scales, and whatever the smoothing removes survives as a residual — Σ falls to 1.1 at a 10-megaparsec kernel and would fall further at a finer one, until the noise the kernel is there to suppress took over. The feature is sharpened by the same field that blurred it, and what limits the sharpening is not the idea but how well the field can be measured.

A blur undone with the field that made it

The acoustic feature is smeared by the same bulk flows whose displacements can be estimated from the density they produced. Moving every galaxy back along that estimate sharpens the peak and removes most of its second-order shift — and what limits the procedure is not the idea but how well a field can be measured from the objects it moved.

cosmology · Baryon acoustic oscillations 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 The first-order perigee rate is right for a slower Sun, and half wrong for ours. The rate at which an orbit's perigee turns, in turns per orbit, against the ratio m of the Sun's motion to the orbit's, from separate integrations of a satellite with the Moon's eccentricity under a solar tide on a circular path. The first-order rate (3/4)m² gives 0.0042 at the Moon's m of 0.0748, a perigee period of 18 years. The integrations give 0.0085, 8.9 years; the Moon's perigee takes 8.85. At m = 0.01 the integration gives 8.3e-5 against a first-order 7.5e-5. So the disagreement is not a matter of neglected forces: it depends on how large m is.

A perigee that moved twice as fast as Newton

The Sun's tide on the Moon should turn the line of its perigee once in eighteen years. It takes less than nine. For sixty years that was the outstanding failure of the law of gravitation, and in 1747 Clairaut told the Paris Academy that the inverse-square law would have to be amended. Two years later he withdrew the remark. The law was right and the calculation was unfinished: the series for the perigee has a small parameter that is not small, and its second term is seventy per cent of its first.

orbits · Perturbations

Named alongside it

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

Disturbing functionApsidal precessionBaryon acoustic oscillationsConvergenceCorrelation functionDalemberts ruleDisplacement fieldEvectionGreat inequalityIndirect termInverse square lawLaplace coefficient

All concepts