Generator

The binary-solution generator

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.

6 essays call binary-solution. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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

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.

Stars

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

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

A distance of 52.0 parsecs with nothing underneath it. Two ways to a distance for the same pair. The orbital parallax needs no iteration and no assumption: a double-lined spectroscopic orbit gives the relative orbit's linear size as (K₁+K₂)P√(1−e²)/2π sin i = 0.2268 AU, an astrometric orbit gives its angular size as 4.36 milliarcseconds, and the ratio is 52.0 parsecs — a length divided by an angle, with no rung of the distance ladder below it and no property of the stars assumed. The curves show the dynamical parallax, the version available when only one spectrum can be measured: guess the mass sum, take the linear size from the harmonic law, divide by the angular size, convert the apparent magnitude to an absolute one and read a new mass sum off a mass–luminosity relation. Three starting guesses spanning a factor of 10 in mass converge to the same distance in 8 passes and agree to 0.001 per cent. It converges because the distance depends on the assumed mass only as its cube root — the measured exponent here is 0.3333 — so a factor of two in the mass is 26 per cent in the distance, and one pass removes most of that. What it converges to is not the orbital parallax: the iteration settles at 54.2 pc against 52.0, 4.2 per cent away, because the fixed point is set by the mass–luminosity relation and the apparent magnitude rather than by anything measured about this orbit. The same insensitivity that makes it converge is why it is never better than the relation it leans on.

Stars

Two orbits of one pair, and a distance falls out

Measure the same binary spectroscopically and astrometrically and the orbit comes back twice — once as a length in kilometres and once as an angle on the sky. The ratio is a distance that owes nothing to parallax, nothing to a standard candle, and nothing to any assumption about the stars.

Stars

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.

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.

Stars

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.

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