Pulsar spin rates

Why does a collapsed star spin hundreds of times a second when its parent turned slowly?

A massive star takes weeks to turn once. What it leaves behind can turn seven hundred times a second — a ball heavier than the Sun, about twenty kilometres across, whipping round faster than a kitchen blender.

The answer everybody reaches for is conservation of angular momentum: squeeze a spinning thing and it must spin faster. That is a true law and it is genuinely at work here. But watch what happens when we ask it to produce the number, because it does not merely fall short — it lands on the wrong side, and then on the wrong side again.

Do the naive sum honestly. Spin rate rises as the square of the shrinkage when the mass keeps its arrangement. Take the Sun — 696,000 kilometres in radius, turning once in about 25 days — and squeeze all of it into a ball 10 kilometres across the radius. The ratio is 69,600; squared, about 4.8 billion. Twenty-five days is 2.16 million seconds. Divide: about 0.45 milliseconds.

Is that right? The fastest pulsar known, PSR J1748-2446ad, turns once every 1.40 milliseconds — 716 times a second — and even at that rate a neutron star is not far from flinging material off its own equator. So the naive calculation does not merely fail to explain the fast pulsars. It over-predicts them threefold, and over-predicts the ordinary ones by a factor of tens of thousands. A law that hands you too big an answer is telling you something was left out.

What was left out is that the star does not collapse. Its core does — roughly 1.4 solar masses of iron, already compressed to a few thousand kilometres, losing its support and falling while the envelope is blown clean away. The question was never "what was the star doing" but "what was the core doing at the instant it let go."

Redo the sum. A core 3,000 kilometres in radius becoming 10 gives a ratio of 300, so the rate multiplies by 90,000. Young pulsars appear to be born turning every few tens of milliseconds — the Crab takes 33 milliseconds today and is estimated to have started near 20. Work backwards: to be born at 20 milliseconds, the core must have been turning once every 1,800 seconds. Half an hour.

Fast or slow? White dwarfs, the leftover cores of lower-mass stars and the closest analogue we can measure, mostly rotate with periods of hours to days — far slower than half an hour. So the collapsing core spins faster than that, yet vastly slower than the whole-star calculation demanded. The conservation law brackets the truth from both sides and predicts neither. It gives the ratio, not the starting value.

So what set the core's rotation? The core is not an isolated flywheel. Over the star's life, magnetic stresses and internal waves couple it to the envelope and carry angular momentum outward; the envelope then swells enormously and is largely blown away in a wind. Most of the star's spin leaves the star entirely, long before anything collapses.

That is measured, not merely supposed. Asteroseismology with the Kepler telescope read the rotation of red-giant cores directly from their oscillation frequencies and found them turning far more slowly than models with standard angular-momentum transport predicted. Some coupling is stronger than theory can account for; which mechanism supplies it — magnetic torques, internal gravity waves, something else — is an open and actively argued problem.

That leaves the fast pulsars, the second surprise. Millisecond pulsars are old, often billions of years; their fields are around a hundred million gauss rather than the roughly ten-trillion-gauss fields of young pulsars; and nearly all are in binaries or show signs of once having been. The accepted picture is that they were spun up afterwards: matter from a companion spirals in through a disc, every gram arriving with angular momentum, torquing the star for tens or hundreds of millions of years. The weak field is what lets them keep it, since a pulsar brakes itself magnetically and a feeble field brakes feebly. The fastest spins in the universe are not squeezing at all. They are a slow winding-up, paid for by a neighbour.

Three refinements connect the steps. First, nothing above contradicts conservation of angular momentum. The law is fine; it is under-determined. It converts a starting rate into a final rate and is silent about the starting rate, and everything interesting here lives in that silence.

Second, birth periods are inferred rather than watched — from braking indices, remnant ages, and population modelling — and the inferred distribution is broad, with some neutron stars probably born turning only once a second or slower. Treat "a few tens of milliseconds" as typical, not constant.

Third, do not blur the two populations. A young pulsar is a strong-field object slowing fast: the Crab loses about 36 nanoseconds of period per day, and that lost rotational energy is what lights its nebula. A millisecond pulsar is a weak-field object spun back up, now slowing so gently it rivals atomic clocks. Same kind of star, opposite histories — and the clinching evidence for recycling is the handful of transitional systems caught switching between accretion-powered X-ray behaviour and radio pulsing within a few years.

• How pulsar timing became precise enough to detect gravitational waves from supermassive black holes.
• Why magnetars, with fields a thousand times stronger still, turn only once every several seconds.
• What sets the true maximum spin of a neutron star, and what that says about matter at nuclear density.

Every term the collection defines is gathered in the glossary.