Astronomy & Celestial Mechanics
the sky, computed
Concept

Apsidal motion — where it appears

The slow rotation of an orbit's line of apsides, so that the point of closest approach moves around the orbit. It is driven by the companions' tidal distortion and by general relativity, and its rate measures how centrally condensed the stars are.

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

The light curve of AI Phoenicis, computed from its elements. Total light against orbital phase, computed by overlapping two discs of radius 1.805 and 2.9303 solar radii at an inclination of 88.5°, each weighted by its own surface brightness. The two eclipses hide the same area of sky and have different depths — 48.0 and 19.1 per cent — because what is lost is the light of whichever star is behind, and the ratio of the depths is therefore the ratio of the two surface brightnesses. Two things are left out and both matter to a real solution: this is the bolometric light rather than the light in a filter, and the discs are uniform, where a real one is limb-darkened and so has a deeper, rounder eclipse than the flat-bottomed one drawn here.

Two radii, from a light curve alone

The radius of a star is not measured. It is inferred, from a temperature and a luminosity, through a model. There is one exception — a pair of stars that eclipse each other, whose light curve and velocity curves between them give both radii, both masses and the ratio of temperatures with no model of a stellar interior anywhere in the chain.

stars · Binary stars Three causes, three shapes, one residual. An eclipse-timing residual: the observed minus the computed time of each eclipse, against cycle number, in days. Three effects are superposed and each has its own functional form. A slow change in the orbital period — from mass transfer or from magnetic braking — integrates to a parabola. A third body in a wide orbit moves the whole binary towards and away from the observer, so its light-travel time adds a sinusoid at the third body's period. And an eccentric orbit whose apsides are precessing moves the two eclipses in opposite directions, which is a sinusoid that changes sign between primary and secondary minima. The last is why both eclipses have to be timed: a third body moves them together and apsidal motion moves them apart, and a series of primary minima alone cannot tell the two apart at all.

Three causes with three shapes in one curve

The times of eclipse in a binary star are a clock, and the clock runs late and early. Three completely different things make it do so — a third body, a changing period, and a slowly turning orbit — and they are separable only because each imposes a different shape on the residual.

stars · Binary stars What DI Herculis's orbit should do, and what it does. The rate at which the periastron of the eclipsing binary DI Herculis advances, in degrees per century. Top: the prediction for stars whose spins are aligned with the orbit — 2.34 from general relativity, computed from the masses (5.17 and 4.52 solar masses), the 10.550164-day period and the eccentricity of 0.489; and 1.93 from the stars' tidal and rotational distortion, as published stellar models give it, of which the rotational flattening is 1.50 and the tides 0.43. Total 4.27. Bottom: the observed rate, between about 0.6 and 1.4 degrees per century in the published timing analyses — a quarter to a seventh of the prediction. For a quarter of a century after the discrepancy was established it was cited as possible evidence against general relativity, which contributes more than half the predicted total by itself.

A precession slowed by stars lying on their sides

The eclipsing binary DI Herculis turns the long axis of its orbit at a quarter to a seventh of the rate relativity and the stars' own distortion predict, and for a quarter of a century that gap was offered as evidence against general relativity. The answer was in the stars: both spin nearly on their sides, and a flattened star tipped past 54.7 degrees drags the orbit's periastron backwards instead of forwards. Explained, the slow precession becomes a measurement of something no spectrum shows — how far the spins lean towards the Earth.

stars · Binary stars

Named alongside it

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

Eclipsing binaryApsidal constantBenchmark starDegeneracyDetached binaryDistance determinationEclipse timingEphemerisFractional radiusGeneral relativityInclinationLight-travel time

All concepts