Astronomy & Celestial Mechanics
the sky, computed
Concept

Binary stars — where it appears

A pair of stars bound to one another, whose orbit is the only route to a stellar mass that assumes nothing about stellar structure. An eclipsing double-lined system gives both masses and both radii with no stellar model assumed, and those systems are what every model is calibrated against.

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

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.

gravitation · The two-body problem 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 The surface a star stops at. The equipotential through L₁ — the Roche lobe — at mass fractions 0.50, 0.20, 0.05, in the frame that rotates with the pair. Each is a level set of exactly the same function the zero-velocity curves are level sets of, at exactly the critical value, so nothing here is a new construction: the Roche lobe is the last closed equipotential, and it is closed only because the two lobes touch at a single point. Material that reaches that point is no longer bound to the star it came from, and it leaves through an opening of zero area. The lobes are drawn in the orbital plane; in three dimensions each is a teardrop, and its volume-equivalent radius is what "the size of a Roche lobe" means. As the mass ratio becomes extreme the smaller star's lobe shrinks towards it, which is why a white dwarf accreting from a companion has a lobe smaller than the Sun.

The surface a star stops at

Around each star of a close pair there is a last closed equipotential, and the two touch at a single point. A star that swells to reach it hands its outer layers to its companion through an opening of zero area — and the transfer, once started, makes itself worse.

gravitation · Lagrange points Two bodies at a mass ratio of 2.6 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 2.6 to 1, so the heavier body's path is 2.6 times smaller.

The companion survives, and it is moving

When one star of a close pair explodes, the other is left holding an orbital velocity and no orbit. Whether the pair stays bound turns on a single question — did more than half the total mass leave? — and the stars that answer no are running across the Galaxy at a hundred kilometres a second with nothing visible behind them.

stars · Supernovae

Named alongside it

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

Mass ratioBarycentreEccentricityReduced massAccretionAlgol paradoxAngular momentumAstrometryCentral forceCentre of massDynamical massEquipotential

All concepts