Astronomy & Celestial Mechanics
the sky, computed
Concept

Spectroscopic binary — where it appears

A pair of stars recognised as a pair only by the periodic shifting of their spectral lines, since the two are never resolved. A double-lined system gives both masses times the cube of the inclination's sine; a single-lined one gives only the mass function.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

The mass correction against the mass. Each planet's departure from the massless harmonic law, against its own mass in solar units, on logarithmic axes. The exact law puts every point on the diagonal. Jupiter and Saturn are the only planets whose mass correction is larger than the perturbations from everything else, and Saturn's measured departure has the opposite sign.

The third law is wrong by the mass of the planet

Kepler's harmonic law says the square of the period goes as the cube of the size. Newton's version has one more term in it, and the term is the orbiting body's own mass — negligible for a planet, decisive for a binary star, and the reason a period can be converted into a mass at all.

orbits · Harmonic law Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

stars · Binary stars The companion's mass against the visible star's, for f = 6.08 M☉: never below 6.08, and 7.3 M☉ for a 0.7 M☉ star seen edge-on. The mass of the unseen companion against the mass of the visible star, both in solar masses, for a mass function of 6.08 M☉, at inclinations of 90°, 70°, 55°. Every curve starts at or above 6.08 when the visible star weighs nothing and rises as it gets heavier, because a heavier visible star moves less for the same companion. For a visible star of 0.7 solar masses — V404 Cygni's donor is a cool subgiant of about that mass — the companion is 7.3 M☉ at 90°, 8.6 M☉ at 70°, 12.4 M☉ at 55°. The shaded band is the range of neutron-star masses, up to about 2.3 M☉; no curve comes near it. A mass function above that limit is a proof that the companion is too heavy to be a neutron star, without knowing the inclination or the visible star's mass at all.

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.

gravitation · The two-body problem

Named alongside it

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

InclinationMass functionMass ratioRadial-velocityBarycentreBlack holeEclipsing binaryGreat inequalityKepler's third lawLimb darkeningMass transferNeutron star

All concepts