Astronomy & Celestial Mechanics
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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.

Assumes Pulsars and Magnetic braking.

16 min read 9 figures Measuring the unreachableThe sky as a clock

A pulsar catalogue lists, for each object, a period and a period derivative — two measured numbers, both of them extraordinarily precise. Beside them it lists a magnetic field strength, a characteristic age and a spin-down luminosity, which are not measured at all. The clock runs down and is read as saying what it is, and this essay is about the one prediction of that reading which the same data can test. They are the two measurements passed through a model.

The model is that the star is a magnetic dipole rotating in a vacuum, radiating electromagnetically at its rotation frequency. It gives a field of three times ten to the nineteenth gauss times the square root of the period times its derivative, an age of the period over twice its derivative, and it makes one further prediction that the same data can check.

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. 1 Above: the braking index measured for the pulsars whose spin-down has been followed long enough to yield a second derivative, against the value a vacuum dipole requires. Not one reaches it, and all fall short in the same direction. Below: what that costs — the true age is the characteristic age times two over the index minus one.

The exponent, and why it is exactly three

A rotating magnetic dipole radiates. The power goes as the square of the dipole moment times the fourth power of the rotation frequency, which is standard electrodynamics and has nothing specific to neutron stars in it.

That power comes out of the rotational kinetic energy, which is half the moment of inertia times the square of the frequency. Differentiating, the rate of change of frequency is proportional to the cube of the frequency.

Write the spin-down as the frequency raised to some power n, and the vacuum dipole gives n equal to three exactly. Different mechanisms give different exponents: a wind of particles carrying away angular momentum gives one; gravitational radiation from a deformed star gives five. So n is a diagnostic of what is doing the braking.

Measuring it requires a second derivative of the frequency, because n is the frequency times the second derivative divided by the square of the first. That is a demanding measurement — the second derivative for a typical pulsar is a part in ten to the twentieth per second squared — and it is possible only for young pulsars, whose spin-down is fast, and only when their timing is clean enough to separate it from noise.

The technique that makes it possible at all is pulsar timing, and it is worth a sentence because nothing else in astronomy is quite like it. Every pulse is counted. A model of the rotation predicts the arrival time of every pulse over decades, and the residuals — the differences between predicted and observed arrival times — are measured in microseconds against a rotation that has occurred a billion times. A second derivative is a term in that model, and it is detectable because the residuals accumulate as the cube of the elapsed time.

That also explains why the measurement is restricted to young objects in a different way than it first appears. It is not sensitivity that limits it but noise: ordinary pulsars show low-frequency wander in their rotation, timing noise, whose spectrum is steep enough to masquerade as a second derivative. Only where the genuine second derivative is large does it stand clear of the wander, and that means young.

What is measured

About a dozen pulsars have a braking index that most people would accept, and they cluster below three.

The Crab is at 2.51, measured over decades. PSR B1509−58 is at 2.84. PSR J1119−6127 is at 2.68. B0540−69 is at 2.14. The Vela pulsar, which glitches frequently and requires careful handling, comes out near 1.4.

Two features of that list matter more than any individual value. The scatter is large — from 1.4 to 2.84 — so this is not one systematic offset to be absorbed by a correction. And every value is below three, so whatever is happening is not random.

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. 2 The plane in which every catalogued field and age is assigned. Only the two axes are measured; the contours of constant field and constant age are the model this essay is about, and they cross the population at right angles, which is why one dot appears to fix both.

There is a third feature, less often stated. The measurements are of the instantaneous index, over a few decades of a life lasting thousands of years, and there is no guarantee that the value is constant. A pulsar could have a long-term index of three and a short-term one of two, if something varies on an intermediate timescale.

There is one way to get at the long-term value, and it has been tried. If a pulsar’s true age is known independently, and its birth period can be argued for, then the average index over its whole life follows from the integrated spin-down. For the Crab that exercise gives a long-term index near 2.5 as well, consistent with the instantaneous one — which is mild evidence that the exponent is a property of the star rather than of the epoch.

There is also a whole population where the exponent is not measurable and is measured anyway, statistically. The distribution of pulsars in the period–period-derivative plane depends on the birth distribution, the braking law and the death line, and fitting a population synthesis to the observed distribution constrains the mean index. Those fits prefer values below three too, at around 2.5 to 3, with wide uncertainty.

What it costs the catalogue

The characteristic age assumes n equals three and assumes the star was born spinning much faster than it is now. Relaxing the first, the true age is the characteristic age times two over n minus one.

At n equal to 2.51 that is a factor of 1.32; at 1.4 it is a factor of five. So the ages in the catalogue are systematically too small, by a factor between one and a few, and the factor is different for each object.

The one place this is checkable is where a pulsar has an independently known age — a historical supernova, or an association with a remnant whose expansion can be measured. The Crab is the standing case: born in 1054, so 972 years old, against a characteristic age of 1240. The correction from its measured index of 2.51 gives 1320, which is further from the truth rather than closer.

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. 3 The Crab’s spin history under two exponents, against its known age. The discrepancy that survives the index correction is absorbed into the birth period: the star was not born spinning arbitrarily fast, and a birth period of nineteen milliseconds reconciles the numbers.

That is the resolution, and it is worth noticing that it uses the age to measure the birth period rather than the other way round. For the handful of pulsars with known ages, the inferred birth periods are tens of milliseconds — which is an important constraint on what happens in a core collapse and is far slower than an untransported core would give, sitting alongside the field strengths inferred for the same objects as the two things a newborn neutron star has to be given by its progenitor.

The field strength is affected too, though less. It depends on the assumed geometry and on the moment of inertia as well as the exponent, and the accumulated uncertainty is perhaps a factor of three. What survives is the ordering: the magnetars really do have stronger fields than the ordinary pulsars, and the millisecond pulsars really do have weaker ones, by many orders of magnitude in each case.

It is worth being explicit about what the quoted field even refers to, since it is not the field at the surface in any simple sense. The formula returns the dipole moment required to produce the observed torque, divided by the cube of an assumed radius, at an assumed inclination, for an assumed moment of inertia. Higher multipoles, which observations of thermal hot spots suggest are substantial near the surface, contribute almost nothing to the torque and are therefore invisible to it. So a pulsar’s catalogued field is a statement about its far field and says little about the field where the emission is generated.

Why the exponent is low

Several mechanisms push it below three and none is established.

The most economical is that the magnetosphere is not a vacuum. A rotating magnetised neutron star cannot maintain a vacuum around it — the induced electric field is strong enough to pull charges off the surface — so the star is surrounded by a plasma, and the spin-down is a combination of dipole radiation and a particle wind. A wind alone gives an index of one, and a mixture gives something between one and three.

That account is attractive and it makes a prediction that is not obviously satisfied: the index should correlate with something about the wind, and no such correlation has been found in a sample of a dozen.

A second is that the magnetic inclination angle changes. The dipole formula’s coefficient depends on the angle between the magnetic and rotation axes, and if that angle evolves the effective index shifts. An increasing inclination lowers the index, and there is independent evidence from pulse profile changes that the Crab’s inclination is increasing by about a degree per century — enough, roughly, to account for its index. A third is that the field itself decays or grows on a timescale of thousands of years. Field decay raises the index; field growth lowers it. There is no independent evidence for either in ordinary pulsars, though field decay is central to how magnetars are understood.

A fourth, applying particularly to Vela, is that the measurement is contaminated. A pulsar that glitches — suddenly spins up, then relaxes — has a recovery whose long-term effect on the second derivative is not zero, and extracting an index from a glitching pulsar requires modelling the recovery.

It is worth noticing what kind of disagreement this is. The vacuum dipole’s index of three is a derivation with nothing adjustable in it; every proposed explanation of the shortfall adds a parameter — a wind fraction, an inclination drift rate, a field decay timescale, a glitch recovery model — and with a dozen measurements and four candidate mechanisms, the data cannot select. That is the ordinary condition of a subject where the object is thirty kilometres across and ten thousand light years away, and it is why the observational effort goes into finding more pulsars with measurable indices rather than into refining the ones that exist.

Glitches, and what they say about the interior

The glitches deserve their own treatment because they are the reason several indices are contested and because they are one of the few probes of the neutron star’s inside — the others being its mass, and the radius its equation of state permits.

A glitch is a sudden fractional increase in the rotation rate, of a part in a million or less, followed by a partial relaxation over days to years. The standard account is that a superfluid component inside the star rotates faster than the crust, is pinned to the crustal lattice, and occasionally unpins — transferring angular momentum abruptly.

That picture is quantitative in one respect. The total angular momentum available for glitches over time constrains the fraction of the star’s moment of inertia that is in the decoupled superfluid, and the observed glitch activity in Vela requires at least a per cent or so. Whether the crust alone can supply that, once the neutrons’ effective mass is accounted for, is an open question that touches on the equation of state.

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.
Fig. 4 The structure the glitch argument constrains. What fraction of a neutron star’s moment of inertia lies in the crust depends on the equation of state, so an observation of how much angular momentum a star can dump in a glitch bears on the same physics that fixes its radius.

For the braking index the glitches are simply a nuisance, and dealing with them is why Vela’s index is quoted with far less confidence than the Crab’s.

What a second measured age would be worth

The whole difficulty is that two numbers are measured and three are wanted, and the third — the age — is available for almost nothing.

There are perhaps five pulsars with an age from something other than their own spin-down: the Crab and one or two others from historical records, a handful from an association with a remnant whose expansion gives a kinematic age, and a couple from a transverse velocity and a plausible birthplace. Each one is worth more than a hundred more period measurements, because it is the only quantity that breaks the model open — the same asymmetry that makes a single independently measured distance worth more than a survey. Those five disagree with their characteristic ages in both directions. PSR J0538+2817’s timing age is twenty times its kinematic one; B1757−24’s is less than half. Neither discrepancy is explained by a braking index, because no index in the plausible range produces a factor of twenty.

The escape in both cases is the birth period. A pulsar born spinning at nearly its present rate has a characteristic age that means nothing at all, because the formula assumes the star has slowed by a large factor. That is a second unmeasured quantity entering the same expression, and it is why a characteristic age should be treated as an upper bound rather than an estimate.

The one case with an unambiguous answer

There is a pulsar whose index is not merely below three but negative, and it is instructive.

PSR B0540−69 changed its spin-down rate abruptly in 2011, by about thirty-six per cent, with no accompanying glitch. Its X-ray pulsations continued unchanged. The straightforward reading is that something about the magnetosphere reconfigured — that the torque on the star is not a fixed function of its spin at all, but can switch between states.

That behaviour is seen more mildly elsewhere: a class of pulsars switches between two spin-down rates correlated with two pulse profile shapes, on timescales of days to years. The torque and the emission change together, which says both are controlled by the magnetospheric state rather than by the star.

If the torque is a switchable quantity, then a braking index measured over a few decades is a property of the current state and not of the star, and the whole exercise of measuring an exponent has a different meaning. That is a live possibility and it is the reason the dozen measured indices have not settled the question.

The magnetars, where the model fails differently

There is a corner of the population where the vacuum dipole is not merely imprecise but wrong in a way that is visible without any index measurement.

Magnetars have periods of a few seconds, large period derivatives, and therefore inferred fields of ten to the fourteenth or fifteenth gauss. They also radiate in X-rays at a rate that exceeds their spin-down luminosity by one or two orders of magnitude, which is the diagnostic: whatever is powering the emission, it is not rotation.

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. 5 Where they sit. The magnetars occupy the upper right of the plane, at long periods and large derivatives, well above the ten-to-the-fourteenth-gauss contour — which is what the model says and what their behaviour independently requires.

The energy source is generally taken to be the field itself, decaying and stressing the crust, and the evidence is the bursts: sudden releases of up to ten to the forty-fifth ergs, accompanied in at least one case by a global oscillation of the star. A field of ten to the fifteenth gauss stores about that much energy in a neutron star’s volume, and nothing else available does. Some magnetars have measured braking indices, and they are erratic — varying between outbursts, occasionally exceeding three. That is consistent with a torque controlled by a variable magnetospheric twist rather than by a static dipole, and it is the strongest evidence that the torque is a property of the magnetosphere’s state rather than of the star, in the way that a wind’s lever arm rather than its mass carries a star’s spin.

Two of the three things the essay measures are worth reading at other settings, since each of them is a number quoted without an error bar throughout the literature.

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. 6 The measured braking indices over the range they actually occupy. Not one reaches three, the spread is more than a factor of two, and the dipole prediction sits outside the whole distribution — which is a stronger statement than any single measurement makes.
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. 7 And the characteristic age computed with a moment of inertia at the upper end of what the equation of state allows. The age moves by more than an order of magnitude in each direction, from a quantity that is never quoted with the age it produces.

What the catalogue is still good for

It would be easy to read all this as saying that the pulsar catalogue’s derived quantities are worthless, and that would be wrong.

The measurements are the period and its derivative, and both are superb. Everything derived from them shares one model, so the derived quantities are wrong together — which means comparisons between pulsars are far more robust than absolute values, and almost all the astrophysics is comparative.

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. 8 The comparison that survives. Whatever the exponent, the millisecond pulsars occupy the lower left of the plane and the magnetars the upper right, and the separation between them is many orders of magnitude in inferred field — a gap no plausible correction to the model can close.

The population’s structure — a main island of ordinary pulsars, a spur of magnetars, a separate clump of recycled millisecond pulsars, a death line below which radio emission stops — is entirely a statement about the two measured quantities, dressed in the model’s coordinates. The recycled population’s location is explained by an accretion equilibrium that predicts where in the plane a spun-up star should land, and it lands there.

And the exponent itself, once measured, is a genuine physical result. A star spinning down at an index of 2.5 is losing angular momentum by a mechanism that is not pure dipole radiation, and identifying that mechanism is a real question about what a pulsar magnetosphere does — the same magnetosphere whose radius, in an accreting system, decides the period the star settles at. The value of the measurement is not that it corrects a number in a table. It is that it refutes, using the same data the table is built from, the model the table assumes.

A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 2500, 10000 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.30: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.
Fig. 9 The same exercise on ordinary stars, where it works. A spin-down law calibrated against clusters of known age turns a rotation period into an age reliably, because the calibration is empirical rather than derived — which is exactly what the pulsar case lacks, and what a larger sample of independently aged pulsars would supply.

That is a healthy state for a model to be in. A model that could not be checked would be worse, and one that passed would be less interesting; what exists is a derivation with no free parameters, a measurement of the quantity it predicts, and a discrepancy large enough to demand a mechanism and small enough that the derived numbers remain useful.

The general shape is worth extracting because this collection meets it repeatedly. A model with no adjustable constants makes a sharp prediction; the prediction is measured; it is wrong by tens of per cent rather than by orders of magnitude; and every candidate repair adds a parameter that the data cannot constrain. The result is a subject that is neither settled nor open — the model is known to be incomplete and is used anyway, because nothing better is available and because the quantities it produces are used comparatively.

That is a perfectly respectable state for a measurement to be in, provided it is stated. What is not respectable is the alternative that catalogues drift toward, which is to print a field strength in gauss beside a period in seconds, with the same number of significant figures and no indication that one was measured and the other computed from a model the same table refutes. The catalogues do label the derived columns, and the labels are read about as often as any other footnote.

The practical consequence is visible in how pulsar fields are used downstream. Population syntheses, magnetar formation arguments and estimates of how many neutron stars the galaxy has all take the catalogued fields as data, propagate them, and report conclusions whose stated uncertainties do not include the factor of a few this essay has been about. Those conclusions are mostly comparative and therefore mostly survive, which is fortunate rather than principled. The ones that do not survive are the absolute ages, and those are quoted freely. A characteristic age is an upper bound with a factor of a few of slack in it, and it is printed as a number.

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 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. A corner of the diagram that has to be earned A pulsar spinning a thousand times a second cannot have been born that way and stayed that way, because its own radiation would have slowed it in a few million years. It got there by being fed, and the line it cannot lie above is where the accretion torque balances the magnetic one. Where the disc stops, and the spin that follows from it. Two radii against accretion rate, for a neutron star with a 10⁸-gauss field. The falling curve is the magnetospheric radius, where the field's stress on the disc matches the rate at which the flow carries angular momentum inward; it goes as the accretion rate to the minus two sevenths, which is a weak enough dependence that the factor of 1000 in supply drawn here moves the boundary by a factor of 7.2. The horizontal lines are corotation radii for three spin periods — the radius at which the disc orbits as fast as the star turns. Above corotation the field is spinning the gas faster than it wants to go and flings it out; below, the gas is faster and spins the star up. So the crossing is an attractor, and a star accreting steadily walks to the period where the two coincide. That period goes as the field to the six sevenths and the rate to the minus three sevenths — which is why a neutron star that has swallowed a tenth of a solar mass from a companion comes out at a few milliseconds, and why the millisecond pulsars have fields ten thousand times weaker than the young ones. The period a star is pulled towards A magnetised neutron star does not let a disc reach it. The disc stops where the field's stress wins, and if that radius lies inside the corotation radius the star is spun up while outside it the star is spun down — so there is one period at which nothing changes, and an accreting star walks to it.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Braking indexCharacteristic ageGlitchMagnetarMagnetic dipole radiationMoment of inertiaNeutron starPulsar magnetospherePulsar windSpin-downTiming noise