Astronomy & Celestial Mechanics
the sky, computed
Stars

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.

Assumes Pulsars, Accretion and Mass transfer.

16 min read 9 figures The sky as a clockTwo bodies, and no more

A pulsar’s spin-down rate says what the star is, and the previous rungs of this anchor read a magnetic field, a characteristic age and a rate of energy loss out of two numbers. Those readings work for the ordinary pulsars, which are born spinning tens of times a second and slow down over a few million years.

They do not work for the millisecond pulsars. A star spinning at 640 times a second with a period derivative of ten to the minus nineteen has a characteristic age longer than the universe and a field of a hundred million gauss, four decades below every young pulsar. Read naively, those are the numbers of an object that was born almost exactly as it is now and has done nothing since.

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. 1 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 field and accretion rate. Eliminating the field between that relation and the dipole formula every point in this diagram is already read through leaves a straight line of slope four thirds. The recycled pulsars all sit below it. None of them was spun up faster than the limit allows.

Why they cannot be young

The argument that a millisecond pulsar is old and was recently spun up rather than young and slowly evolving has three legs, and each is worth stating because together they are conclusive.

The first is the field. A young neutron star has a field of ten to the twelve gauss; a millisecond pulsar has ten to the eight. If millisecond pulsars were a separate population born with weak fields, that population would have to be born weak, and nothing in the collapse that makes a neutron star produces a field four decades below the ordinary one — a field of that magnitude is what flux conservation during the collapse of an ordinary stellar core gives, and there is no version of the calculation that gives ten to the eight.

The second is the companions. Some eighty per cent of millisecond pulsars are in binaries, against a few per cent of ordinary pulsars — and the transfer that made them is the same valve that narrows its own channel. That correlation is not a selection effect: pulsars in binaries are harder to find, not easier, because the orbital Doppler shift smears the periodicity out of a search.

The third is the position of the companions in the Hertzsprung–Russell diagram. They are overwhelmingly white dwarfs of low mass — a class that can only be made by stripping a red giant of its envelope before it finishes burning helium.

The luminosity classes, drawn as what they measure. The HR plane with lines of constant surface gravity across it, for a solar mass. They are straight and parallel because log g = log g⊙ + log M + 4 log T − log L, so a fixed gravity is a fixed offset from a line of slope four — and the classes fall where they do for that reason and no other. At 4300 K a class V dwarf sits at log g ≈ 4.6, a class III giant at 1.7 and a class Ia supergiant at 0.1: one temperature, 4.4 orders of magnitude of gravity — and since a collisional wing's width goes as the square root of the damping constant and that goes as the pressure, line wings differing by a factor of 162. The four marked stars are placed at their catalogued temperature and luminosity and labelled with the gravity that geometry gives; each agrees with the log g their spectra were independently fitted with, which is the check that the two quantities are one quantity. What the figure cannot show is the mass, which enters as its logarithm and is the weakest link in the chain — a factor of two in mass is 0.3 dex in log g, and that is the floor on a spectroscopic distance.
Fig. 2 Where those companions sit. A helium white dwarf below about 0.45 solar masses cannot be made by single-star evolution in the age of the universe, because a star light enough to end at that mass has not finished the main sequence yet. Every one of them is therefore the stripped core of something, and what stripped it is the neutron star it is now in orbit with. The companion is the direct evidence for the transfer, and it is the leg of the argument that does not depend on any dynamics at all.

The torque balance

Material falling toward a magnetised neutron star arrives, as always, with far too much angular momentum to fall straight in, and it is stopped where the magnetic pressure equals the ram pressure of the flow. That radius — the magnetospheric radius — scales as the field to the four sevenths and the accretion rate to the minus two sevenths.

Inside it the material is forced to corotate with the star. If the star’s surface at that radius is turning faster than a Keplerian orbit there, the material is flung outward and the star loses angular momentum: the propeller regime. If it is turning slower, the material is accreted and the star gains. Equilibrium is where the two rates match, which is where the magnetospheric radius equals the corotation radius.

Solving that gives an equilibrium period,

Peq≃1.9 ms (B109 G)6/7(M˙M˙Edd)−3/7,P_{\rm eq} \simeq 1.9\ {\rm ms}\ \left(\frac{B}{10^{9}\,{\rm G}}\right)^{6/7}\left(\frac{\dot M}{\dot M_{\rm Edd}}\right)^{-3/7},

and the numbers in it are what make the mechanism work. A field of ten to the eight gauss and accretion near the Eddington limit give a period of a few tenths of a millisecond — faster than any observed pulsar, and faster than a neutron star can spin without breaking up, which is the real limit on the population — a limit set by the radius a degenerate star settles at and therefore by the equation of state.

The temperature of a disc around a neutron star. Effective temperature against radius, in units of the inner edge, for a neutron star of 1.4 solar masses accreting 10⁻⁹ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 5.33·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.88·10²⁹ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round.
Fig. 3 The disc doing the torquing, drawn for a neutron star of 1.4 solar masses accreting at ten to the minus nine solar masses a year. Its inner edge in a magnetised system is at the magnetospheric radius rather than at the stellar surface, and the temperature there is of order a keV — which is why these systems are X-ray sources during the transfer and radio sources afterwards. The same object is called a low-mass X-ray binary while it is being fed and a millisecond pulsar once it stops.

Eliminating the field

The equilibrium period contains the field, which is not directly observed. What is observed is the period and its derivative, and those give the field through the assumption that the spin-down is magnetic dipole radiation in vacuum.

Setting the two expressions for the field equal to each other eliminates it, and leaves a relation between period and period derivative alone. The exponent works out at four thirds.

That is the spin-up line, and its meaning is a limit rather than a track. A star spun up at the Eddington rate ends on the line; one spun up more slowly ends below and to the right of it. Nothing ends above it, because that would require accretion faster than the material can radiate away its energy on arrival.

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. 4 The same diagram with the field and age contours that the ordinary reading uses. A pulsar’s position gives a field from the product of period and derivative and an age from their ratio, and for a young pulsar both are meaningful. For a recycled one the age is not: the characteristic age of a millisecond pulsar exceeds the age of the universe, because the formula assumes the star was born spinning infinitely fast and has been slowing ever since, and this one was born spinning slowly and was spun up in between.

What happened to the field

The weak field is the part of the story with the least agreement behind it.

Something reduces a neutron star’s field by four decades during the accretion phase, and there are two families of explanation. One is that the accreted material buries the field — a hundredth of a solar mass arriving on the surface is enough to push the flux under the photosphere, where it cannot be seen until it diffuses back out. The other is that the field decays ohmically in a crust whose conductivity is changed by the accretion heating.

The observational discriminant is whether the field recovers after accretion stops. Buried fields should re-emerge on a diffusion timescale; decayed fields should not. Millisecond pulsars have been observed for decades and no field growth has been detected, which favours decay — but the diffusion timescales proposed for burial are longer than the observations, so the test has not been run for long enough.

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. 5 The assumption underneath every field in the diagram. A vacuum dipole radiates with a braking index of exactly three, and the measured indices of young pulsars are between one and a half and three — none of them three. So the field read off any point in this diagram is a scaling rather than a measurement, and the spin-up line’s position carries that same uncertainty. What the line does robustly is separate two populations; where exactly it falls is model-dependent at the level of a factor of two.

The system caught in the act

The strongest evidence is a handful of objects that switch.

Accreting millisecond X-ray pulsars are neutron stars in low-mass binaries whose X-ray flux is pulsed at millisecond periods — so the star is spinning fast, and it is accreting now. That established the spin-up mechanism directly in 1998, forty years after it was proposed.

Better still are the transitional systems, three of which are known, which have been seen to switch between a radio millisecond pulsar state and an accreting X-ray state and back within a few years. In the accreting state the radio pulses vanish; in the pulsar state the disc does. Those are the same object doing both things, observed in both, with the transition timescale measured.

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. 6 What the characteristic age says about the fastest known pulsar, and why it should not be believed. The ratio of period to twice its derivative is an age only if the star was born spinning far faster than it does now and has been braked by a constant torque ever since; for a recycled pulsar neither clause is true, and the number that comes out exceeds the age of the universe. The contour lines are what the assumption implies, and this object’s position among them measures how badly the assumption fails rather than anything about the star.

How much mass it takes

The angular momentum a neutron star needs to reach a millisecond period is a number, and it converts into a mass.

A neutron star of 1.4 solar masses and 12 kilometres radius has a moment of inertia of about ten to the forty-five gram square centimetres. Spinning it at 600 hertz requires an angular momentum of a few times ten to the forty-eight in the same units. Material arriving from the inner edge of a disc brings the Keplerian specific angular momentum there, which for a weakly magnetised star is close to the value at the stellar surface.

Dividing gives a required accreted mass of a few hundredths to a tenth of a solar mass. That is a small number, and it matters that it is small: a tenth of a solar mass transferred at the Eddington rate takes ten million years, which is short compared with the lifetime of a low-mass companion and therefore easy to arrange.

It is also large enough to be visible in the masses. A recycled pulsar should be heavier than a young one by that amount, and the measured mass distributions do differ in that direction — recycled pulsars average about 1.5 solar masses against 1.35 for the young ones, with the heaviest known objects, at two solar masses, all recycled.

The population, and the objects that should not exist

Two features of the millisecond pulsar population are harder to account for than the mechanism itself.

The first is the isolated ones. About a fifth of millisecond pulsars have no companion, and the mechanism requires one. The standard resolution is that the companion was destroyed — ablated by the pulsar’s own wind after the accretion stopped — and there are systems caught partway through, with companions of a few hundredths of a solar mass being visibly eroded.

The second is the concentration in globular clusters. A cluster containing a millionth of the Galaxy’s stars can contain a substantial fraction of its known millisecond pulsars, which is a thousandfold overdensity. That is not because recycling works better there; it is because encounters in a dense core exchange companions, so a neutron star that had no companion can acquire one, and one whose companion is exhausted can acquire another.

The companions that are being eaten

The mechanism that produces the isolated millisecond pulsars can be watched, and the systems in which it is happening have turned out to be the most productive and the most treacherous places to measure a neutron-star mass.

They come in two kinds, separated by companion mass. In one, the companion is a few hundredths of a solar mass — a degenerate remnant of something much larger, being ablated by the pulsar’s wind and losing what is left of itself. In the other, the companion is a few tenths of a solar mass and is not degenerate; it is a bloated, irradiated star that periodically fills its own Roche lobe. Both eclipse the radio pulses for a substantial fraction of the orbit, and the eclipses are not the companion’s own disc blocking the beam but ionised material driven off it, which absorbs and disperses the radio signal over a region far larger than the star.

The eclipses are the reason the systems are useful. Their duration and their frequency dependence measure the density of the outflowing material, and therefore the rate at which the companion is being destroyed. Several of these systems are losing mass fast enough to be gone within a few hundred million years, which is short compared with the pulsar’s remaining life and is what the isolated population requires.

They are also the reason the masses have to be treated carefully. A companion mass comes from a radial-velocity curve of the companion’s own spectral lines, and those lines are formed on a star that is being heated on one side only. The photometric and spectroscopic centre of such a star is displaced towards the irradiated face, which is the face nearest the pulsar, so the measured velocity amplitude is larger than the true centre-of-mass amplitude and the inferred pulsar mass comes out too high.

That is not a small correction in the systems where it matters most. Several of the highest neutron-star masses ever claimed came from irradiated companions, and the corrections applied to them — modelling the temperature distribution across the companion’s surface and re-deriving the velocity of its centre of mass — have moved individual results by more than their originally quoted uncertainties. The objects that best test the equation of state are the objects whose masses are hardest to measure, and the reason is the same wind that makes them interesting.

The clocks this produces

A recycled pulsar is the most stable clock in nature. Its period derivative is ten to the minus twenty, so it loses a second in three hundred million years, and the pulse arrival times can be predicted over decades to a fraction of a microsecond.

That stability is what makes them instruments. Timing a pulsar in a binary gives the orbit to a precision no optical measurement approaches, and timing an array of them across the sky is a gravitational wave detector the size of the Galaxy.

Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 8 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 0.9 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 0.999 — with amplitude (a/c)·δθ·cos β = 2.0 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 1.4 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 8 years as at 4. 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 1e-12 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. 7 What a timing model has to contain before a residual means anything: the spin period and its derivative, the position and proper motion, the parallax, the orbit if there is one, and the solar system’s own motion. Each term has a distinctive signature in the residuals, and fitting them is how the position of a pulsar is measured to sub-milliarcsecond precision from timing alone — with no image and no interferometer.

How the population was actually found

The argument above is about a population, and it is worth recording that most of that population was found by an instrument that cannot detect a radio pulse.

A millisecond pulsar is a hard target for a blind radio survey. The dispersion of the interstellar medium smears a millisecond pulse unless it is removed with fine frequency resolution, which multiplies the computation; and a pulsar in a binary has its apparent period modulated by the orbit, so a coherent search has to be run at many trial orbital accelerations, which multiplies it again. The cost of a blind survey rises steeply as the periods searched get shorter.

What changed the arithmetic was gamma rays. A millisecond pulsar’s rotation-powered emission includes a gamma-ray component that is beamed much more broadly than the radio, so a large fraction of them are detectable above a hundred megaelectronvolts even when their radio beam misses the Earth. An all-sky gamma-ray survey therefore produces a catalogue of point sources, many of which have no counterpart at any other wavelength and the spectral shape characteristic of a pulsar — a power law with an exponential cut-off at a few gigaelectronvolts.

Pointing a radio telescope at each of those, and searching only that position, converts a blind survey into a targeted one and reduces the cost by orders of magnitude. The known millisecond pulsar population has more than tripled that way, and the systems described in the previous section were found almost entirely by it: an ablating companion produces an X-ray and optical variable at the same position, which makes the identification unambiguous before any pulse is detected.

The same population has a further consequence that is still argued about. The centre of the Galaxy shows an excess of gamma rays above what the known sources account for, and its spectrum resembles the summed spectrum of millisecond pulsars. Whether it is an unresolved population of them or something else entirely is a question about whether a bulge that old should contain that many, which returns to the recycling arithmetic of this essay: how many low-mass binaries the bulge formed, and how many of them survived long enough to deliver a tenth of a solar mass.

What was actually measured

Four quantities, in decreasing order of directness.

The period is measured to fifteen significant figures by counting pulses over years. It is the best-measured quantity in astronomy.

The period derivative is measured from the drift in arrival times, and requires removing everything else that produces a drift — the Earth’s motion, the pulsar’s own proper motion, the interstellar dispersion. For a millisecond pulsar the derivative is so small that the correction for the star’s transverse velocity is a substantial fraction of it, and for a few objects it is larger than the intrinsic derivative, giving an apparently negative value.

The field and the characteristic age are not measured. They are the period and its derivative combined under an assumption about the braking mechanism that the observed braking indices contradict.

The masses are measured, in the systems where relativistic effects in the orbit are detectable, and they are the reason these objects matter beyond their own dynamics: a two-solar-mass neutron star deletes an equation of state, and the heaviest known neutron stars are all recycled ones, because recycling is what added the mass.

A last remark about what the diagram is and is not. It looks like an evolutionary track and it is not one: no star moves along the spin-up line, and no star moves along the age contours either. A young pulsar drifts to the right and downward, roughly parallel to the field contours, as it loses rotational energy. A recycled one arrives at the bottom left in a single episode lasting a hundred million years and then barely moves at all for ten billion. The two populations are therefore separated by a process rather than by a stage, and the empty region between them is empty because nothing spends time there — an object crossing it is doing so during the accretion phase, when it is an X-ray source rather than a radio pulsar and is not in this catalogue at all. Reading the diagram as a sequence is the commonest mistake made with it, and the transitional systems are the proof that the crossing happens too fast to populate.

One more reading applies the same age arithmetic to the one object whose true age is known.

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. 8 The characteristic age against the true age for the Crab pulsar. The two differ by about a fifth, which is the only direct calibration the characteristic age has ever received — and it is at the young end of a scale used over six orders of magnitude.

One more record shows the discontinuities that the smooth spin-down history hides.

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 its glitches. Each step is a sudden transfer of angular momentum from a decoupled interior to the crust, and the recovery afterwards is the timescale on which the two recouple — neither of which appears anywhere in the period–period-derivative plane.

That is the shape of every quantity read off this plane: two measured axes, and a family of contours computed from a model whose assumptions are not measured at all. The corner is earned by the object and the interpretation is supplied by the model.

Where the ladder goes

The next rung is the mass the recycling added, which is the observable that turns the spin-up story into a constraint on dense matter: how much mass a neutron star must accrete to reach a millisecond period, and whether that is consistent with the companion masses observed.

The other direction is what happens when the companion is destroyed. Some millisecond pulsars are ablating their companions with their own wind, and the systems where the companion is nearly gone are the plausible progenitors of the isolated millisecond pulsars — which are otherwise a puzzle, since the mechanism that made them requires a companion that is not there.

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. Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through. The flow that narrows its own channel Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand. The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round. The disc that has to throw angular momentum away Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for.

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.

AccretionAngular momentumBinary pulsarEddington limitMagnetic fieldMagnetosphereMass transferMillisecond pulsarNeutron starPulsarSpin-down