The two-body problem — the series · Astronomy & Celestial Mechanics

Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

part 1 · gravitation

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, not merely stated.

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.

part 2 · gravitation

A floor under the unseen companion

A star that swings back and forth in velocity with nothing visible beside it is orbiting something. One velocity curve gives one number about the something — the mass function — and that number is not an estimate of its mass but a floor under it, reached only if the visible star weighed nothing and the orbit were seen exactly edge-on. When the floor stands higher than any neutron star can, the companion is a black hole, whatever the inclination turns out to be.

part 3 · gravitation

A black hole weighed by a wobble on the sky

A velocity curve weighs an unseen companion only up to the tilt of its orbit. The same orbit followed on the sky, as a loop a few thousandths of an arcsecond across riding on a star's parallax, shows the tilt as the shape of the ellipse — and with the parallax turning the angle into astronomical units, Kepler's third law gives the companion's mass outright. That is how the nearest black holes were found: silent, unfed, and on orbits no X-ray survey could have reached.

part 4 · gravitation

Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one.

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

part 5 · gravitation