Astronomy & Celestial Mechanics
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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.

Assumes Supernovae, Pulsars and Internal rotation.

19 min read 10 figures Measuring the unreachableScales that defeat intuition

The interior of a star is the least accessible place in this collection, and the rotation of that interior is the least accessible thing about it. Two independent measurements now reach it. One is the splitting of oscillation frequencies in red giants. The other is a corpse.

When an iron core collapses — and most of what follows never reaches the surface — its angular momentum goes with it — there is nothing to take any away in the fraction of a second the collapse takes. So the spin of the object left behind is a fossil of the spin of the core immediately beforehand, magnified by the enormous contraction. That magnification is the useful part and it is also the problem.

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.
Fig. 1 The diagram every pulsar is placed on. Two measured quantities — the period and its rate of change — and everything else on the page derived from them: a characteristic age, an inferred magnetic field, a spin-down luminosity. Young pulsars sit at the top left with periods of tens to hundreds of milliseconds. Nothing sits where naive conservation of a stellar core’s angular momentum would put a newborn neutron star.

The number conservation alone gives

The arithmetic is short enough to do in one line and wrong enough to be interesting.

An iron core just before collapse has a mass of about 1.4 solar masses and a radius of a few thousand kilometres — comparable to the Earth. It is already turning far faster than the star around it, because it has been contracting throughout the star’s life: from a main-sequence core a thousand times larger, which at fixed angular momentum is a million in rotation rate. A model that transports no angular momentum between the core and the envelope delivers it to collapse with a period of tens of seconds.

Collapse that to a neutron star of twelve kilometres. Angular momentum is conserved through the second or so the collapse takes, so

Pfinal=Pinitial kf2Rf2ki2Ri2≈20 s×0.35×1220.10×30002,P_{\rm final} = P_{\rm initial}\,\frac{k_f^2 R_f^2}{k_i^2 R_i^2} \approx 20\ \text{s}\times\frac{0.35\times 12^2}{0.10\times 3000^2},

which is about a millisecond — the break-up limit for a neutron star, where the equator is moving at a substantial fraction of the speed of light.

Newborn pulsars are not observed to be doing this. The best estimates of birth periods for young objects with independent ages are in the range of tens to a few hundred milliseconds.

Reading a birth period off a present one

No pulsar has been watched being born. A birth period is an extrapolation, and it has to be done carefully.

A pulsar slows under a torque that goes as some power of its rotation rate,

Ω˙=−k Ω n,\dot\Omega = -k\,\Omega^{\,n},

with nn the braking index. Magnetic dipole radiation gives n=3n = 3. Integrating backwards from the present period and period derivative gives the birth period, provided the index and the constant have not changed.

A characteristic age of 1257 years for a pulsar 972 years old. The Crab pulsar's period against time, integrated backwards from today's measured P = 33.39 ms and Ṗ = 4.21·10⁻¹³ under ν̇ = −kν^n, at n = 3 and n = 2.51. Differencing each drawn curve returns 3.000 and 2.510, so the curves really are solutions of the law they are labelled with. The characteristic age P/2Ṗ is 1257 years and the true age is 972, because the supernova was seen and recorded; the discrepancy is not an error in the timing but the assumption buried in τ_c, which is that the pulsar was born spinning infinitely fast under a pure dipole. Feeding the measured index of 2.51 and the known age into the spin-down integral instead gives a birth period of 18.7 ms, against 15.9 ms for a vacuum dipole. Three measurements — a period, its derivative, and a date in a chronicle — produce a fourth that nothing observed directly.
Fig. 2 The law being extrapolated, and the one place it is measured rather than assumed. For a handful of young pulsars the second derivative of the period is measurable, which delivers the braking index directly. The measured values cluster between 2 and 3 rather than sitting at the dipole value of exactly 3 — so the assumption underlying every characteristic age in the previous figure is known to be approximately wrong, in a direction that makes objects older than they look.
The same age, wrong by 21× one way and 2.5× the other. The 7 pulsars whose age is known from something other than their own timing — a supernova seen from Earth in the Crab's case, a remnant's expansion for three, and a transverse velocity carrying the pulsar away from its birthplace for the last two — with the age their timing gives on the vertical axis and the independent one on the horizontal. The diagonal is where the model would be right. Nothing is on it. τ_c = P/2Ṗ assumes a birth period of zero and a braking index of exactly 3, and both push the estimate upwards — the Crab, whose true age is a date in a chronicle rather than a model, sits 1.29 times high. But the scatter runs the other way too: J0538+2817's timing age is 21 times its kinematic one, and B1757−24's is 0.40 of it. So a characteristic age is not an upper bound with a known sign; it is an order-of-magnitude estimate whose error is not even one-sided, and it is the only age available for the ninety-nine per cent of pulsars with no remnant left to date them by.
Fig. 3 What the extrapolation is worth when there is something to check it against. A characteristic age is the present period divided by twice its derivative, which is exact only if the braking index is 3 and the birth period was much shorter than the present one. Where a supernova remnant or a historical record supplies an independent age, the characteristic age is wrong by factors of several in both directions — and the two assumptions that break it are the same two a birth period needs.

The Crab pulsar is the best case. Its supernova was recorded in 1054, its braking index is measured at 2.51, and integrating back with that index gives a birth period near 19 milliseconds. That is the cleanest birth period there is, and it is twenty times slower than what an untransported core would give.

For the rest, birth periods are inferred statistically: a population of young pulsars with remnant-based ages is fitted with a distribution of birth periods, and the answer is broad — most between about 10 and 300 milliseconds, with a tail.

There is a selection problem inside that fit which is worth naming. A pulsar is detected because it beams, and the beaming fraction depends on the period: a slow rotator sweeps a narrower cone and is less likely to point at the Earth. So the observed sample is biased towards fast rotators, and correcting for it requires a model of the beam geometry that is itself uncertain. The correction goes in the direction of making the true birth distribution slower still, which strengthens the conclusion rather than weakening it — but it means the width of the distribution is far less secure than its centre.

A second and larger population is invisible to the argument entirely. A core-collapse event that leaves a black hole leaves nothing that pulses, so the birth spins of stellar-mass black holes are inaccessible by this route. What is known about them comes from the spins measured in merging pairs, and those are consistent with slow rotation too — which is the same conclusion from a third direction, and the only one that reaches the most massive progenitors.

What that says about the core

Run the arithmetic backwards. A twenty-millisecond birth period corresponds to a specific angular momentum of about 101410^{14} square centimetres a second — the number the literature quotes, and the one every model is compared against. Spread that over an iron core of a few thousand kilometres and the pre-collapse period is around six minutes rather than twenty seconds.

That is a factor of about eighteen, which means the core arrived at collapse with roughly a twentieth of the angular momentum an untransported model gives it. Something removed nineteen-twentieths of it during the star’s life.

That is the same conclusion the asteroseismology of low-mass giants reaches, from a completely different object. The two agree in kind, which is the point. Both say that a contracting stellar core does not keep its angular momentum, and that a mechanism is carrying it out to the envelope during the star’s life. Neither says what the mechanism is.

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.
Fig. 4 The assumption underneath every number in this essay, measured. A magnetic dipole rotating in vacuum brakes with an index of exactly 3, and the four pulsars whose spin-down has been followed long enough to give a second derivative all return something less. Not one of them is 3. Every characteristic age and every catalogued field strength on this site assumes the value no pulsar has, which is why the birth periods in the previous section are quoted as bounds rather than as measurements, and why the argument is built on the ones that are least sensitive to it.

The reservoir that has to be drained, and where it can go

It is worth being explicit about the size of the discrepancy in the currency it is actually spent in, because a factor of thirty in period sounds smaller than it is.

Angular momentum goes as the reciprocal of the period, so a birth period twenty times longer than conservation alone allows means the core arrived at collapse with a twentieth of the angular momentum it would otherwise have had. Over the star’s main-sequence life and its subsequent contraction, ninety-five per cent of the core’s spin was handed to the envelope.

There are two places for it to go and they leave different fingerprints. It can be transported outwards inside the star, to the envelope, which then loses it to a wind — in which case the mass lost carries away angular momentum on a long lever and the core’s slowness is coupled to the star’s mass loss rate. Or it can be transported outwards and stay in the envelope, in which case the star’s surface should be rotating faster than a non-transporting model predicts.

Both signatures are looked for and neither is clean. Surface rotation rates of massive stars are measured as a projected quantity with the usual sine ambiguity; mass loss rates are uncertain by factors of several because they depend on how clumpy the wind is. The core’s rotation, measured from the corpse, is currently the better-determined half of the comparison — which is an unusual position for an interior quantity to be in.

A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions.
Fig. 5 The other side of the same transaction, in a star where it can be watched. A surface that expands without a torque slows as the reciprocal of its radius, so a hundredfold expansion is a four-order slowdown — and observed giants are almost all slower than two kilometres a second, as required. The few that are not have to have been given angular momentum from outside. In the massive-star case the transaction runs the other way: the core is losing what the envelope receives, and the envelope is large enough that the gain is hard to see.
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.
Fig. 6 The same plane with Vela marked rather than the Crab. Nothing has moved: the twenty-five pulsars are at their catalogued periods and period derivatives, and the three families of contour are the same models drawn from the same two axes. What the mark does is fix which star the surrounding argument is about — Vela is ten times older than the Crab and sits an order of magnitude down and to the right, on a field contour near the same 101210^{12} gauss. Two pulsars of very different age sharing a field strength is what makes the field look like a birth property.

Dating the corpse

The statistical birth-period distribution rests on having independent ages for young pulsars, and those ages are harder to come by than the phrase suggests. It is worth setting out the three routes, because their weaknesses are what set the width of the answer.

A historical record. A supernova seen and written down gives a date to the year, and there are a handful of such cases within the Galaxy over the last two millennia. It is the best kind of age there is and it is available for almost nothing: most pulsars are far older than any record, and most historical records cannot be matched to a surviving remnant with confidence.

Remnant expansion. A supernova remnant’s shell is expanding, so its radius divided by its expansion speed is an age. The difficulty is that the expansion is not free: the shell sweeps up interstellar material and decelerates, so the ratio overestimates the age by a factor that depends on the density of the surroundings and on how much of the shell’s history has been in the decelerating phase. Correcting for it requires a model of the surrounding medium, which is exactly what is not known.

Association with a birthplace. A pulsar’s proper motion, projected backwards, may intersect a star cluster or an association young enough to have produced it. That gives an age with no rotational or hydrodynamic model in it, and it needs a distance, a proper motion and an unambiguous identification of the birthplace — three quantities that are individually uncertain and whose errors combine badly.

None of the three is available for more than a small minority of the young population, and the objects for which one is available are not a random sample: a pulsar with a visible remnant is a young one in a dense environment, and a pulsar traceable to a cluster is a nearby one with a large proper motion.

So the birth-period distribution is fitted on a sample selected by the availability of a clock, and the direction of that selection is not established. That is the honest reason the quoted range spans more than an order of magnitude for a quantity whose central value is fairly secure.

14 glitches, and the 1.5 per cent of Vela that is not slowing down. Accumulated fractional spin-up against time for a Vela-like pulsar over 40 years, in parts per million. The underlying spin-down has been removed, so a perfectly braking pulsar would be a flat line at zero. What is drawn instead is a staircase: 14 sudden jumps of a few parts per million, each rising in less than a minute and then relaxing partway back over a couple of months, leaving a permanent step behind. Nothing outside the star can deliver angular momentum on a timescale of seconds, so the source is internal, and the only internal component that could have any to give is one that has not been slowing down with the rest. That is a neutron superfluid: it carries its rotation in quantised vortices, the vortices pin to the crustal lattice and cannot migrate outward, and so the superfluid keeps the spin it had while the crust brakes past it. The reservoir grows until the pinning fails somewhere, and a glitch is the unpinning. The straight line through the staircase is the glitch activity, 0.68 parts per million per year, and it converts directly into an interior measurement: the crust cannot on average take more than the superfluid stores, so the decoupled component must hold at least 2τ_c times the activity of the star's moment of inertia, which here is 1.5 per cent. It is a lower bound rather than a value, and it is one of the very few quantitative statements about the inside of a neutron star that needs no equation of state at all. The individual glitch times and sizes here are drawn from a seeded generator rather than a catalogue; what is real is the staircase's shape, the partial healing, and the arithmetic that turns a slope into a fraction.
Fig. 7 What the spin-down of a real pulsar looks like once the smooth part is subtracted. Fourteen sudden spin-ups over forty years, each a few parts per million, accumulating into a staircase — and between them the star recovers, but not all the way. About one and a half per cent of Vela is not slowing down with the rest of it, which is the superfluid interior returning angular momentum it had held back. The birth period inferred from a spin-down history is inferred from the smooth part; this is the size of what was removed to get it.

The exceptions, and what they need

Not every collapse produces a slow rotator, and the exceptions are the reason the subject is not closed.

None of this concerns the millisecond pulsars, which sit in a corner of the diagram that has to be earned by being fed rather than by being born there.

Magnetars. A small fraction of young neutron stars have inferred magnetic fields a thousand times higher than ordinary pulsars, and the standard explanation is a dynamo operating in the first seconds after collapse. That dynamo requires the proto-neutron star to be turning in a millisecond or so — which is the naive answer rather than the observed one. So the magnetar population, which is a few per cent of the whole, needs progenitor cores that did keep their angular momentum.

Long gamma-ray bursts. The collapsar model requires the collapsing core to have enough angular momentum to form a disc around the newborn black hole, which needs specific angular momentum comparable to that at the last stable orbit. That is again the fast case, and long bursts are rare and preferentially occur in low-metallicity galaxies — where line-driven winds are weaker, so a massive star loses less mass and less angular momentum during its life.

A line nothing spun up by accretion can lie above, and the millisecond pulsars beneath it. The period–period-derivative diagram with the spin-up line drawn on it. An accreting neutron star is torqued by the disc until its magnetosphere turns at the same rate as the material arriving there, which fixes an equilibrium period as a function of the magnetic field and the accretion rate. Eliminating the field between that relation and the dipole formula that every point in this diagram is already read through leaves a straight line of slope 1.33, drawn here for accretion at the Eddington rate — the fastest a star can be pushed. The 7 recycled pulsars all sit below it, which is what the figure is for: none of them was spun up faster than the limit allows, and their positions are a record of how much mass each one received rather than of how old it is. The young pulsars are in the opposite corner, above the line and to the right, spinning down from birth. The two populations are not two stages of one life. A star that reaches the bottom left has been fed by a companion for a hundred million years, which is why almost every millisecond pulsar has one and almost no young pulsar does.
Fig. 8 Where a star that was born turning too slowly can still end up. Accretion from a companion spins a neutron star back up, and the line it cannot cross is set by the rate at which arriving material stops being able to add angular momentum — so the millisecond pulsars sit beneath it regardless of what they were doing at birth. That is the escape route from the problem this essay is about: the initial period is forgotten, not by braking, but by a later episode that resets it entirely.

So the picture that fits both is a distribution rather than a rule: most cores are efficiently braked and produce ordinary slow pulsars, and a small tail of cases — in stars that lost less, or were spun up by a binary companion — retains enough to make a magnetar or a burst.

What a fast core would have done to the explosion

The exceptions above were described as needing progenitors that kept their angular momentum. There is a converse worth stating, because it turns the birth-period distribution into a constraint on the explosion mechanism rather than only on the progenitor.

A proto-neutron star rotating in a millisecond has an enormous reservoir of rotational energy — of order 105210^{52} ergs, which is more than the kinetic energy of an ordinary supernova by a factor of ten. If any substantial fraction of that can be extracted on the timescale of the explosion, the explosion is not the ordinary neutrino-driven one at all: it is powered by the spin, through a magnetic field wound up by the differential rotation, and it produces a bipolar outflow rather than a roughly spherical one.

That is the millisecond-magnetar central engine, and it is the leading model for the small class of supernovae that are ten to a hundred times more luminous than the ordinary kind. Their light curves are too broad and too bright to be powered by radioactive nickel in the usual amounts, and they are well fitted by an input of energy from a spinning-down neutron star with a field of 101410^{14} gauss and a birth period of a few milliseconds.

The argument closes on itself in a satisfying way. Those events are rare — a fraction of a per cent of core collapses — and the birth-period distribution says that fast rotators are rare, by roughly the same factor. Two quite different observations, one a population of light curves and one a population of pulsars, agree about how often a stellar core arrives at collapse still spinning fast.

What that does not settle is whether the rarity has the same cause in both. The light curves select on what the explosion did, and the pulsar periods select on what survived; a fast core that makes a black hole appears in neither. The two samples agree on a number and are not sampling the same thing, and closing that gap needs the spin distribution of stellar-mass black holes, which is now beginning to be measured by an instrument nobody had when the question was posed.

There is a second and quieter consequence of the same reservoir, and it applies to the ordinary cases rather than to the exceptional ones. Even a birth period of twenty milliseconds corresponds to a rotational energy of about 104910^{49} ergs — a per cent of a supernova’s kinetic energy, and comparable to the entire radiated output of the event. That is not enough to power the explosion and it is enough to matter for what happens afterwards.

It goes into the surrounding remnant over the pulsar’s first few thousand years, through the same spin-down torque that the period derivative measures, and it inflates a wind nebula inside the expanding debris. The Crab is the standing example: a synchrotron-emitting bubble whose luminosity is supplied by a rotating star and whose energy content is a straightforward integral of the spin-down history. So the birth period is measurable, in principle, from the nebula’s energy content as well as from the extrapolation — a third route, on the one object where it has been tried it agrees, and it has been tried on one object.

Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 12 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 12 years as at 6. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 3e-13 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find.
Fig. 9 Three ways a timing model can be wrong, over twelve years rather than six. Each has its own shape and the shapes are what tell them apart: a position error is a sinusoid of exactly one year, an unmodelled proper motion grows as the year times the sinusoid, and an error in the period derivative is a parabola. Doubling the baseline separates them further, because the three grow at different rates. A residual is not noise until it has been checked against the three curves it could be, and the birth period this essay is about is read off the third of them.

What the measurement cannot pin down

Three things stand between the observed periods and a clean statement about pre-collapse cores.

The moment of inertia of a neutron star is not measured. It depends on the equation of state of matter at several times nuclear density, and the plausible range spans about thirty per cent — which propagates directly into every angular momentum inferred from a period. The collapse itself may not conserve the core’s angular momentum in a simple way. It is not even certain that the whole star is braking together, since part of a pulsar’s interior can decouple and store angular momentum. The proto-neutron star is not a rigid body: it is convectively unstable for the first seconds, it is coupled to the infalling material above by neutrinos and by magnetic fields, and there is time for a torque to act.

And the natal kick — the observed fact that pulsars move at hundreds of kilometres a second, far faster than their progenitors did — is evidence that the explosion is asymmetric, and an asymmetric explosion can exert a torque as easily as a force. Whether kicks and spins are correlated is measurable in principle, from the alignment between a pulsar’s proper motion and its spin axis, and the current evidence is that they are somewhat aligned.

That alignment is the one piece of the argument that pushes back on the whole framework, and it is worth seeing why. A spin inherited from a slowly rotating pre-collapse core is a spin whose direction was set long before the explosion; a kick imparted in the explosion’s first second has a direction set by whatever hydrodynamic asymmetry happened to grow. There is no reason for the two to agree, and they do. The usual reading is that the kick is not a single impulse but the sum of many, delivered over a few hundred milliseconds while the star is already turning — so the transverse components average away and what survives points along the rotation axis. If that is right, then the spin observed at birth is partly made in the explosion rather than merely inherited from the core, and the factor of thirty this essay opened with is an upper limit on how much braking the progenitor had to do.

It is worth returning to the diagram the essay opened with, this time with the one object whose birth is not inferred.

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.
Fig. 10 The same plane with the Crab pulsar marked. Its characteristic age is 1,240 years and its true age is 972, known because the explosion was recorded in 1054 — the single calibration point for a timescale used on two thousand other objects. The nine per cent by which the two disagree is the whole empirical constraint on how far a characteristic age can be trusted.

That one object does a disproportionate amount of work, and it is worth being clear about what it does and does not establish. It shows that the characteristic age is the right order of magnitude for a young pulsar whose braking index is close to the dipole value. It does not show anything about an old pulsar, whose field may have decayed, whose braking index has never been measured, and whose birth period was assumed rather than derived. The calibration is at one end of a scale spanning six orders of magnitude.

There are a handful of other pulsars in historical remnants, and every one of them tightens the same end of the scale. What would test the other end is a pulsar with an independently dated companion — a white dwarf whose cooling age can be computed, in a binary with a recycled pulsar — and those systems exist and give ages that disagree with the characteristic ages by factors of several. The clock is good where it has been checked and unchecked where it is most used. That asymmetry is worth carrying, because characteristic ages are quoted throughout the literature without error bars, and the number of independent checks on them is small enough to count. Two of the three assumptions behind the formula — a constant field and a braking index of three — are known to fail for at least some objects, and the third, a birth period far shorter than the present one, is the one this essay exists to question.

Where the ladder goes

The direct extension is towards a birth period distribution measured rather than inferred: a population of neutron stars with independent ages, from remnant expansion or from association with a cluster, is what would replace the extrapolation. That work is limited by how few young pulsars have a reliable age at all.

The second thread runs back into the star. The transport mechanism that slows a stellar core is the same one that has to be explained for the Sun’s rigidly rotating interior and for the red giants, and pulsars are the only probe of it in massive stars, where the seismology is not available. Every constraint on it therefore comes from measurements at the two extremes — a nearby star’s oscillations, and the corpse of a distant one.

Where this goes next

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. A furnace with the thermostat taken out. Left, the two pressures in a red giant's helium core at 10⁶ g/cm³. The ideal-gas pressure rises with temperature, as the whole regulation of a star depends on it doing; the degenerate electron pressure is a horizontal line, because the exclusion principle does not know the temperature. At the ignition point near 10⁸ K the degenerate term is 5.1 times the gas term, so a temperature rise there raises the total pressure by almost nothing and the core does not expand. Right, what that costs. Both curves start at the same temperature with the same triple-alpha rate, whose logarithmic slope is 41 — the famous T⁴⁰, differentiated here rather than quoted. The regulated core settles; the degenerate one, with nowhere to put the extra energy, goes vertical. The flash reaches 10¹⁰ solar luminosities for a few seconds and not one photon of it is seen: every erg goes into lifting the degeneracy, and the star's visible response is to become fainter and settle on the horizontal branch. The explosion that never reaches the surface When helium ignites in a degenerate core the thermostat every other burning stage runs on is disconnected. The runaway reaches ten billion solar luminosities for a few seconds, and the star's visible response is to get fainter. The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives. A shear layer that should have spread The Sun's convection zone turns differentially — its equator laps its poles about once every three months — and the radiative interior below turns as one rigid piece. Between them is a transition four per cent of the radius thick. Nothing in hydrodynamics keeps a velocity discontinuity that thin for four and a half billion years.

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Angular momentumAngular momentum transportAsteroseismologyBirth periodBraking indexCharacteristic ageCollapsarCore-collapseIron coreMagnetarMoment of inertiaNatal kickNeutron starSpecific angular momentum