Astronomy & Celestial Mechanics
the sky, computed
Concept

Braking index — where it appears

The exponent in the law by which a pulsar's rotation slows, predicted to be three for pure magnetic dipole radiation. The handful of measured values fall between two and three, which means every characteristic age is systematically wrong.

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

Also named here as characteristic age — the same set of essays touches all of them, so they are one junction rather than several.

Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

A clock that runs down and says what it is

A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it.

stars · Pulsars Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

A neutron star born turning too slowly

Collapse an iron core a few thousand kilometres across down to twelve, and conservation of angular momentum multiplies its rotation rate by about eighteen thousand. A model with no transport in it delivers a newborn pulsar at the break-up limit; the ones that are observed turn twenty times slower, which is a measurement of the core before it fell.

stars · Supernovae A formula everyone uses, and the number no pulsar has. Above: the braking index measured for the 4 pulsars whose spin-down has been followed long enough to give a second derivative, against the value a magnetic dipole rotating in vacuum requires. That value is exactly 3, and it is what every catalogued field strength and every characteristic age assumes. Not one measurement reaches it: they run from 1.4 to 2.839, and all of them fall short in the same direction, which is the signature of a systematic rather than of noise. Below: what that costs. The age a spin-down history gives is the period divided by (n − 1) times its derivative, so the ratio to the quoted characteristic age is 2/(n − 1) — 5.00 for Vela. The numbers are not thereby useless: an exponent recovered from the data is exactly the kind of correction a measurement can absorb. What has gone is the claim that the field strength printed beside a pulsar is a measurement of a field. It is a measurement of a spin-down rate, read through a model the same pulsar refutes.

The exponent no pulsar has

Every field strength in the pulsar catalogue comes from one formula, which assumes the star is a magnetic dipole rotating in a vacuum and therefore that its spin-down obeys an exponent of exactly three. Where that exponent has been measured it is 2.51, 2.84, 1.4 — never three, and always short.

stars · Pulsars

Named alongside it

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

Characteristic ageMoment of inertiaNeutron starGlitchMagnetarSpin-downAngular momentumAngular momentum transportAsteroseismologyBirth periodCollapsarCore-collapse

All concepts