Astronomy & Celestial Mechanics
the sky, computed
Concept

Central force — where it appears

A force directed along the line to a fixed point whose magnitude depends only on the distance, under which angular momentum and the plane of motion are conserved. Only the inverse-square and the linear laws close every bound orbit; every other central force produces a rosette that precesses.

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

Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

orbits · Angular momentum A vector that does not change, drawn five times. The Laplace–Runge–Lenz vector A = v × L − GM r̂, constructed at five points of one orbit at e = 0.45 under a force ∝ r^−2, each drawn from the body rather than from the focus so that its length and direction can be compared point by point. Under the inverse square every one of them is the same vector: identical to machine precision in length and in direction. Its length is 0.4500, which is the eccentricity, and it points at pericentre — so the orbit's orientation is a conserved quantity and not a constant of integration, which is why the ellipse does not turn.

The third thing that is conserved

Energy fixes an orbit's size and angular momentum fixes its shape. Neither fixes which way it points — so a curve that closes needs a third conserved quantity, and only two force laws in the universe supply one.

orbits · Conic sections Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

Every orbit one force allows, and the number that picks between them

Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

orbits · Conic sections The two-body problem, and the one-body problem it is. Left: two bodies with mass ratio 0.4 on ellipses of eccentricity 0.5 about their common barycentre, the heavier one on the smaller orbit. Right: the same system as one body of the reduced mass on a single ellipse of the same eccentricity about a fixed centre, at the separation of the two. The right-hand curve is the point-by-point difference of the two left-hand curves, so the substitution is drawn rather than asserted.

Two bodies replaced by one that does not exist

The two-body problem is solved by turning it into a one-body problem about a fixed centre. The substitution is not an approximation — it is exact, and the body it invents has a mass no object in the system has.

gravitation · The two-body problem

Named alongside it

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

Angular momentumEccentricityConic sectionsOrbital elementsAnomalyApsidal precessionBarycentreBinary starsConserved quantityDegeneracyDynamical massEffective potential

All concepts