Astronomy & Celestial Mechanics
the sky, computed
Stars

The part of a star that never slowed down

A pulsar's spin decays smoothly for years and then jumps upward inside a minute. Nothing outside can deliver angular momentum that fast, so something inside has been storing it — and the rate at which the jumps accumulate is a lower bound on how much of the star is not braking with the rest.

Assumes Pulsars and Degeneracy.

17 min read 9 figures The sky as a clockMeasuring the unreachable

A pulsar is a clock that runs down, and the running down is the most regular thing about it: the period lengthens smoothly, the second derivative is small, and a timing model with a handful of terms predicts the arrival time of a pulse years ahead to within microseconds.

Then, occasionally, the period jumps. Not lengthens — shortens. The spin rate increases by a few parts in a million within a time that has never been resolved, then relaxes partway back over weeks or months, leaving a permanent step behind. These are glitches, and they are the only fast events in an object otherwise defined by its regularity.

The argument they support is short and it is the reason they are worth an essay. Nothing outside a neutron star can deliver angular momentum to it in under a minute. The star is not accreting; there is no companion; the magnetosphere carries angular momentum away rather than in, which is the whole mechanism of the spin-down in the first place. So the angular momentum came from inside — from a component of the star that had not been slowing down while the observable part did.

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. 1 Accumulated fractional spin-up over forty years for a Vela-like pulsar, with the smooth spin-down removed. A perfectly braking pulsar would be a flat line at zero; what is drawn is a staircase of fourteen jumps of a few parts per million, each rising in less than a minute and relaxing partway back over a couple of months. The straight line through it is the glitch activity, and multiplying it by twice the characteristic age gives the fraction of the star’s moment of inertia that cannot be braking with the rest — one and a half per cent here.

What is inside that could store it

A neutron star’s interior is neutrons at densities above nuclear — the density at which matter stops being ordinary — and neutrons are fermions that attract each other at the relevant energies. That combination produces a superfluid: below a critical temperature of some 10910^{9} kelvin, which a neutron star drops below within a year or so of birth, the neutrons pair and the fluid flows without viscosity.

A superfluid cannot rotate the way an ordinary fluid does. Its flow is irrotational except along quantised line defects — vortices — each carrying a fixed quantum of circulation. A superfluid rotating at a given rate contains a definite number of vortices per unit area, and the only way for it to spin down is for the vortices to migrate outward and annihilate at the boundary.

That is where the pinning comes in. In the star’s inner crust the superfluid neutrons coexist with a lattice of neutron-rich nuclei, and a vortex core sitting on a nucleus is energetically favoured — it is pinned. Pinned vortices cannot migrate. So as the crust brakes electromagnetically, the superfluid does not brake with it: the two rotate at increasingly different rates, and the difference is stored angular momentum.

A measured mass of 2.08 deletes an equation of state. Mass against radius for three neutron-star equations of state, each a polytrope P = Kρ² integrated through the Tolman–Oppenheimer–Volkoff equation from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls; only the rising part is stable, because past the maximum adding mass makes the star smaller and the smaller star cannot hold itself up. The maxima here are 1.60, 2.10, 2.57 solar masses, in the order of increasing stiffness — the same nuclear matter with a slightly harder response to compression supports a heavier star, and nothing else in the calculation changes. The horizontal band is PSR J0740+6620, whose mass of 2.08 ± 0.07 solar masses comes from the Shapiro delay of its own pulses passing its companion, which is a timing measurement and involves no model of the star at all. It sits above the maximum of one of the three, and those are not disfavoured but excluded: an equation of state that cannot hold up two solar masses is wrong, whatever else recommends it. The two lines at the left are exact and no star may cross them — the Schwarzschild radius, and the bound above it inside which the speed of sound in the matter would exceed the speed of light. A caution about the curves themselves: a Γ = 2 polytrope is a stand-in for nuclear matter and runs a kilometre or two large in radius at fixed mass, so read the ordering and the maxima rather than the radii.
Fig. 2 The object in question. A mass–radius relation that turns over rather than continuing, because at these compactnesses relativity dominates and the maximum mass exists whatever the equation of state. The crust is the outer kilometre or so of a twelve-kilometre star — a per cent of the radius and a per cent or two of the moment of inertia — which is the number the glitch argument is about.

There is an ordering here worth noting, because it is what makes the reservoir large enough to matter. The superfluid transition temperature depends on density, and the pairing gap for neutrons is largest at densities around and just below nuclear — which is where the inner crust is. So the part of the star that becomes superfluid first, and the part where vortices have a lattice to pin to, are the same region. Deeper in, in the core, neutrons are also expected to pair, but in a different channel and with a much smaller gap, and there is no lattice.

That is why the crust was the natural candidate for the reservoir, and it is why the correction described later in this essay was so awkward: the identification was not arbitrary, it was the only place the physics obviously worked.

The unpinning, and why it is sudden

The stored lag cannot grow indefinitely. The pinning force is finite, and the Magnus force on a vortex grows with the velocity difference between the superfluid and the lattice it is pinned to. Past a threshold the vortices unpin.

What makes the event sudden rather than gradual is that the unpinning is collective. A vortex that unpins moves outward, transferring its angular momentum to the crust and slightly reducing the lag — but it also perturbs its neighbours, and if the region is close enough to threshold the perturbation unpins them too. The result is an avalanche, and the observable is a step.

The avalanche picture makes a prediction about the statistics of glitches rather than about any one of them: sizes should follow a power law, waiting times should be broadly distributed, and there should be no characteristic scale. Most pulsars behave that way. A small number do not, and the exception is instructive.

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. 3 The population the glitchers sit in. Glitches are overwhelmingly a phenomenon of young pulsars — the upper left of this diagram — and they become rare among the old ones and essentially absent among the recycled millisecond pulsars at the bottom. Age is the variable, and the reason is that a young star is hotter, its crust is more strongly coupled, and its spin-down rate is fast enough to build a lag quickly.

Vela and a few others glitch quasi-periodically, at intervals of two or three years, with sizes clustered near a characteristic value. That is not what an avalanche gives; it is what a system that empties a reservoir completely and refills it at a steady rate gives. The two behaviours coexist in one population and the reason is not settled.

The threshold behaviour also explains something about the sizes. A glitch drains part of the lag, and how much depends on how far the avalanche spreads before the local velocity difference falls below threshold again. If the star is everywhere close to threshold, an avalanche can cross it and the glitch is large; if only a patch is close, the glitch is small. So the size distribution reflects how uniformly the lag is distributed, which in turn reflects how the pinning strength varies with depth — a piece of nuclear physics inferred from a histogram of jump sizes.

None of that depends on knowing what matter does above nuclear density, which is fortunate, because nobody does. Assumptions about the equation of state give pressures differing by factors of several at the same density, and every one of them implies a different crust thickness and a different moment of inertia. The argument below steps around all of it.

The bound, which is the point

Over a long enough time, the crust cannot gain more angular momentum from the superfluid than the superfluid has been accumulating. The superfluid accumulates at a rate set by how fast the crust is braking. Equating the two gives an inequality:

IsI ≥ 2τcAg,Ag=1T∑iΔνiν,\frac{I_s}{I} \ \ge\ 2\tau_c A_g,\qquad A_g = \frac{1}{T}\sum_i \frac{\Delta\nu_i}{\nu},

where τc=ν/2∣ν˙∣\tau_c = \nu/2|\dot\nu| is the characteristic age and AgA_g is the glitch activity — a slope, read off the staircase in the hero figure.

For Vela the activity is about 2×10−62\times10^{-6} per year and the characteristic age is 11,300 years, so the bound is one and a half per cent. That is a statement about the interior of an object twelve kilometres across and three hundred parsecs away, obtained from nothing but arrival times of radio pulses.

It is a lower bound rather than a value, and it is a robust one. It does not depend on the equation of state, on the mass, on the radius, on the magnetic field, or on any model of the coupling. What it depends on is the assumption that the same reservoir is being drained and refilled — that glitches are not, for instance, crustquakes releasing stress built up by the changing shape of a spinning-down star.

It is worth appreciating the shape of the inequality. The activity is an accumulated fractional spin-up per unit time, so it has dimensions of inverse time; multiplying by an age makes it dimensionless. What the product means is: over the star’s lifetime, this fraction of its rotation has been handed back in jumps, and the component handing it back must be at least that large a share of the whole. It is an accounting identity rather than a model, which is the reason it survives every uncertainty about the mechanism.

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. 4 The quantity the bound is proportional to. A characteristic age is P/2Ṗ, and it is only the true age if the braking index is exactly 3 and the star was born spinning much faster than it does now. For the Crab, whose true age is known from the historical record, the characteristic age is 1,257 years against 972 — a 29 per cent overestimate. So the glitch bound inherits that error, and quoting it to two figures is optimistic.

Why crustquakes were ruled out

The alternative account is mechanical. A spinning-down star becomes less oblate; its rigid crust resists the change and builds stress; past a yield point it cracks and the moment of inertia drops abruptly, spinning the star up.

That works arithmetically for the Crab, whose glitches are small and infrequent. It fails badly for Vela. The energy released in a crustquake comes from the star’s oblateness, which is a fixed budget set by the spin-down since birth, and Vela’s observed glitch activity would exhaust that budget in a few thousand years — far less than its age. The star does not have enough strain to give.

The superfluid account has no such problem, because the reservoir is refilled continuously by the ongoing spin-down rather than drawn from a finite store laid down at birth.

There is a second observational argument against crustquakes and it is a cleaner one. The moment-of-inertia change a crack can produce is bounded by the star’s oblateness, which for Vela — spinning at eleven hertz — is tiny. The fractional spin-up available is of order the oblateness times the fractional strain relieved, and getting to a few parts in a million requires relieving essentially all of the strain in one event, every three years, forever. There is no source that resupplies it.

The recovery, and what it measures

A glitch is not a step. It is a step followed by a partial exponential recovery: the spin rate overshoots and then relaxes back toward the pre-glitch trend, with a fraction of the jump remaining permanently. The recovering fraction is called the healing parameter, and the timescale is typically weeks to months.

The recovery is the crust and the superfluid re-coupling. Its timescale measures the strength of the coupling, and the strength of the coupling is a statement about the microphysics — how a vortex line scatters electrons, how the lattice responds. Fitting recovery timescales is one of the few ways anybody probes the transport properties of matter at 101710^{17} kilograms per cubic metre — a regime that no equation of state has been measured in and that only astrophysics reaches.

Some glitches show several exponential components with timescales from minutes to years, which is read as several distinct superfluid regions coupling on different timescales. And a few large glitches have shown an increase in the spin-down rate afterwards that persists for years, which no simple two-component model produces.

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. 5 Why the ages in all of this are soft. Two pulsars with independent kinematic ages — from their measured motion away from their birthplaces — have characteristic ages wrong by factors of 21 and 2.5 in opposite directions. Every glitch bound is proportional to a characteristic age, so every glitch bound carries that uncertainty. The bounds are quoted anyway, because the alternative is to quote nothing.

Entrainment, and how the bound became a problem

For twenty years the story above was tidy: the crustal superfluid holds one to two per cent of the moment of inertia, the crust is one to two per cent of the moment of inertia, and Vela’s glitches are accounted for.

Then the accounting was redone with a correction that had been overlooked. The superfluid neutrons in the inner crust are not free: they scatter coherently off the lattice, and the effect is to give them an enhanced effective mass — entrainment. A neutron with an effective mass five times its bare mass carries five times the momentum for a given velocity, which means the same angular momentum requires more of the star to be involved.

With entrainment included, the crustal superfluid can supply only a fraction of the observed activity. The bound was not violated; the identification of the reservoir with the crust was. So either the reservoir extends into the core — where the superfluid neutrons coexist with superconducting protons and the pinning physics is much less clear — or the effective mass calculation is wrong, or the star is more massive than assumed and its crust correspondingly thinner.

The situation is unresolved and it is a good illustration of how these bounds are used. The inequality itself did not change. What changed was a piece of condensed-matter physics that decides which part of the star the inequality is about.

The moment-of-inertia factor against core size, for five density contrasts. What a moment of inertia can say. The vertical axis is C/MR², the polar moment divided by what a hoop of the same mass and radius would have, and for a uniform sphere it is exactly 2/5 — the value both ends of every curve return to, because a body with no core and a body that is entirely core are both uniform. In between, the ratio dips: a moment weights mass by the square of its distance from the axis, so moving density inward lowers it, and the deeper the dip the more differentiated the body. The five curves are five core-to-mantle density ratios, and the minimum moves down and inward as that ratio grows — a denser core reaches its greatest effect at a smaller radius, because beyond that the core is so much of the body that the whole thing looks uniform again — 2 gives 0.348 at 72 per cent of the radius, 3 gives 0.317 at 69 per cent of the radius, 5 gives 0.279 at 65 per cent of the radius, 10 gives 0.231 at 59 per cent of the radius. Two things the figure makes visible are worth more than the numbers. The relation is not invertible: one measured factor is met by two core sizes on each curve and by a whole family of curves, so a moment of inertia alone never gives a core radius — it gives a constraint that a second measurement has to be combined with. And the whole diagram lives between 0.4 and about 0.15, which is a narrow range for so much physics; distinguishing a large core from a small one means measuring C/MR² to a per cent or two, and every technique for doing so is a way of watching the body turn.
Fig. 6 The quantity the whole argument is about, in a setting where it is easier to picture. A moment-of-inertia factor says what fraction of a body’s mass is where, and a glitch bound says what fraction of that moment is not participating in the braking. The neutron star’s version of the question is harder only because the layers are defined by quantum states rather than by chemistry: a superfluid neutron and a normal one occupy the same place and belong to different reservoirs.
A characteristic age of 11319 years for a pulsar 11300 years old. The Vela pulsar's period against time, integrated backwards from today's measured P = 89.30 ms and Ṗ = 1.25·10⁻¹³ 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 11319 years and the true age is 11300, 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 35.3 ms, against 3.7 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. 7 What the timing model looks like before a glitch interrupts it. A pulsar’s frequency and its derivatives are fitted over years, and the residuals are microseconds — which is why a step of a few parts per million is unmistakable rather than marginal. The braking index measured between glitches is not the 3 a magnetic dipole predicts, and the recovery from glitches is one of the reasons.

What was actually measured

A glitch is detected as a discontinuity in a timing solution. A pulsar is observed regularly — for the well-studied ones, weekly or more often — and the pulse times of arrival are fitted with a model containing a position, a proper motion, a parallax, a spin frequency and its derivatives, and a dispersion measure. Residuals from that fit are microseconds.

A glitch appears as a sudden change in the frequency, and because the timing model is so tight it is unmistakable: residuals that were flat begin to accumulate quadratically. The size is measured by refitting either side. The rise time has never been resolved for a glitch caught with dense coverage the timing was limited to under a minute, and that limit is itself a constraint on the avalanche.

The measurement therefore depends entirely on the pulsar being watched. Glitch catalogues are dominated by the objects with long, dense monitoring records, and the activity of a pulsar that is observed twice a year is not measurable at all. That is a selection effect on the population, and it means the bound is available for a few dozen objects out of three thousand known.

What the picture does not show

The hero figure draws a staircase and a straight line, and both are idealisations of things that are messier.

It draws glitches as instantaneous. They are unresolved, which is not the same thing: the tightest limits come from a glitch caught in progress with high-cadence timing, and they bound the rise time to under a minute. A rise time of a minute is very fast for a body twelve kilometres across, and it is the strongest argument that the transfer is superfluid rather than viscous.

It draws the recovery as a single exponential with one healing fraction. Real recoveries need two or three components, and some show a long-term change in the spin-down rate that never recovers at all. Each additional component is another region coupling on its own timescale, and the number of them is the number of distinct reservoirs — which nobody has managed to count reliably.

And it draws a population with a fixed mean interval. Even Vela, the most regular glitcher known, has intervals ranging from about a year to over three, and the distribution’s width is one of the things the avalanche and reservoir pictures disagree about most sharply.

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 the recycled pulsars sit, and the line they cannot lie above. Spinning a pulsar back up by accretion cannot take it past the rate at which the accreting matter’s own angular momentum stops being able to add any — the spin-up line — and the millisecond pulsars sit beneath it, with fields a hundred million times weaker than Vela’s and spin-down rates to match. They essentially never glitch. The reading is that a star which has just spent a hundred million years accreting is warm and strongly coupled throughout, so the lag this essay is about never builds: the reservoir and the crust turn together. An absence doing the work of a measurement, and the third such absence in this essay — the unresolved rise time, the missing crustquake strain, and now a population with no jumps in it at all.

The step that went the wrong way

There is one observation the reservoir picture does not accommodate, and it is worth having because it is the sort of result that keeps a settled subject unsettled.

In 2013 a magnetar was observed to undergo a sudden change in spin frequency of the usual size and the opposite sign: a step down rather than up, over less than four days, accompanied by a burst of X-rays and a change in the persistent emission. It was named an antiglitch, and several more have been reported since.

A superfluid reservoir spinning faster than its crust can only ever speed the crust up. To slow it down requires something else — a sudden increase in the external torque, from a twisted magnetic field reconnecting or from material lifted off the surface and flung out along field lines, or an internal transfer running the other way in a star where part of the interior has been left rotating more slowly than the crust.

Which of those it is has not been settled, and the objects it happens in are the ones with the strongest fields, where the magnetosphere carries far more torque than in an ordinary pulsar. The inequality this essay is about still holds; what an antiglitch shows is that the crust has more than one thing acting on it, and that a timing discontinuity is evidence of a transfer without saying which way the reservoir was leaning.

One more reading follows the same glitch record over twice the span.

28 glitches, and the 1.4 per cent of Vela that is not slowing down. Accumulated fractional spin-up against time for a Vela-like pulsar over 80 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: 28 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, 0.63 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 1.4 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. 9 Eighty years of Vela’s spin, with glitches every two and a half. The steps are large, regular and always in the same direction, which is what a superfluid interior decoupled from the crust produces — and the recovery after each one is the timescale on which the two are recoupling.

Every quantity in the plane is derived from two measured numbers and a model of a rotating dipole in a vacuum, and the glitches are the one observable that is about the interior rather than about the model.

Where the ladder goes

The earlier rungs of this anchor treated a pulsar as a clock — a clock that runs down, a clock read through everything between, and a clock whose position on the diagram has to be earned. This one treats it as a container.

The next steps are two. One is the microphysics: what the recovery timescales say about the coupling, what the largest glitches say about how far an avalanche can propagate, and whether the reservoir is crust or core. The other is multimessenger: a large glitch should radiate gravitationally, because the sudden redistribution of angular momentum is a changing quadrupole, and searches triggered on radio glitches have set upper limits that are beginning to be interesting. That is the same instrument, and nearly the same quantity, as the tidal deformability measured in a merger — two ways of asking a neutron star how it is put together, both of which listen rather than look.

And there is a connection to the rest of this phase worth stating plainly. What a glitch measures is the fraction of a body’s moment of inertia that is participating in its rotation — which is exactly what Mercury’s libration measures and what the moment-of-inertia factor is about. A planet whose core does not follow its mantle and a neutron star whose superfluid does not follow its crust are the same measurement, twenty orders of magnitude apart in density.

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 measured mass of 2.08 deletes an equation of state. Mass against radius for three neutron-star equations of state, each a polytrope P = Kρ² integrated through the Tolman–Oppenheimer–Volkoff equation from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls; only the rising part is stable, because past the maximum adding mass makes the star smaller and the smaller star cannot hold itself up. The maxima here are 1.60, 2.10, 2.57 solar masses, in the order of increasing stiffness — the same nuclear matter with a slightly harder response to compression supports a heavier star, and nothing else in the calculation changes. The horizontal band is PSR J0740+6620, whose mass of 2.08 ± 0.07 solar masses comes from the Shapiro delay of its own pulses passing its companion, which is a timing measurement and involves no model of the star at all. It sits above the maximum of one of the three, and those are not disfavoured but excluded: an equation of state that cannot hold up two solar masses is wrong, whatever else recommends it. The two lines at the left are exact and no star may cross them — the Schwarzschild radius, and the bound above it inside which the speed of sound in the matter would exceed the speed of light. A caution about the curves themselves: a Γ = 2 polytrope is a stand-in for nuclear matter and runs a kilometre or two large in radius at fixed mass, so read the ordering and the maxima rather than the radii. A radius that decides what matter can be Nobody can make matter at four times the density of an atomic nucleus, and no calculation settles what it does there. What can be done is to weigh a neutron star — every candidate description of that matter predicts a heaviest star it could hold up, and a single measured mass above that value deletes the description permanently. Measured moment-of-inertia factors, from the Sun's 0.07 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.07 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here was obtained by watching the body turn: a precession rate, a libration amplitude, a gravity field sampled on a flyby, or in the Sun's case the frequencies of its own oscillations. Whether the heavy material sank A moment of inertia is 0.4 of MR² for a uniform sphere and less for everything that has differentiated, and it is measured by watching a body wobble. Mercury's 0.346 says most of the planet is iron core; the Moon's 0.393 says almost none of it is.

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CrustEntrainmentGlitchGlitch activityMoment of inertia factorNeutron starPulsar timingSpin-downSuperfluidTiming residualVortex pinning