Astronomy & Celestial Mechanics
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Gravitation

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.

Assumes Accretion and Pulsars.

16 min read 9 figures Two bodies, and no moreMeasuring the unreachable

The millisecond pulsars are a puzzle stated by their own position in a diagram. They spin hundreds of times a second, their inferred magnetic fields are ten thousand times weaker than an ordinary pulsar’s, and their characteristic ages run to billions of years. A young, rapidly spinning neutron star should have a strong field; these have the spin of youth and the field of age.

The resolution is that they are old stars spun back up, and the agent is a companion. What decides the period they reach is a balance between two radii.

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.
Fig. 1 Two radii against accretion rate for a neutron star with a hundred-million-gauss field. The falling curve is where the field’s stress on the disc matches the inward flux of angular momentum; the flat lines are corotation radii for three spin periods. Where a falling curve crosses a flat line, the star’s spin stops changing.

Where a disc stops

A neutron star’s field falls as the cube of the distance, so its energy density falls as the sixth power. The accreting gas carries a momentum flux that falls far more slowly. There is therefore a radius inside which the field dominates, and inside it the gas cannot continue to orbit freely: it is picked up by the field lines and channelled along them to the magnetic poles.

Setting the magnetic stress equal to the rate at which the accretion flow carries angular momentum inward gives the magnetospheric radius. It depends on the dipole moment to the four sevenths and on the accretion rate to the minus two sevenths, and both exponents are worth reading rather than skipping.

The four sevenths says the radius is not very sensitive to the field: raising the field by a factor of ten moves the boundary out by only a factor of three and a half. The minus two sevenths says it is even less sensitive to the supply: a thousandfold change in the accretion rate moves it by a factor of seven.

Those weak dependences are the same arithmetic as a magnetopause standing off a wind, and for the same reason — a dipole’s steep fall-off means a large change in pressure is a small change in position.

The exponents differ between the two cases because the outside pressure differs. A wind supplies ram pressure, giving a sixth root; a disc supplies an angular momentum flux that itself depends on radius, giving a seventh.

For a neutron star the numbers are small in a way that matters. A field of a hundred million gauss and an accretion rate of a hundredth of the Eddington limit put the boundary at about thirty kilometres — three stellar radii, which is inside the region where general relativity is not a correction. A field of ten to the twelfth gauss at the same rate puts it at five hundred kilometres, which is fifty stellar radii and comfortably Newtonian. The same formula spans both regimes, and only the first requires care.

There is a corollary that decides what the object looks like. Once the disc is truncated, the gas no longer radiates from a boundary layer at the surface but falls along field lines to two polar caps, and the caps are hot spots that rotate with the star. That is why an accreting magnetised neutron star pulses at all, and why an unmagnetised one does not.

Where corotation is

The second radius involves no field at all. It is the radius at which the Keplerian orbital period equals the star’s rotation period — where a particle in orbit and a point on the star keep pace.

For a neutron star spinning once every three milliseconds, that radius is about thirty kilometres, which is three stellar radii. For one spinning once a second it is nearly two thousand kilometres.

The comparison between the two radii decides the direction of the torque, and the argument is one of the cleanest in accretion physics.

If the magnetospheric radius lies inside corotation, the gas at the boundary is orbiting faster than the star turns. The field lines connecting them are dragged forward by the gas, and the tension pulls the star forward: the star is spun up.

If the magnetospheric radius lies outside corotation, the gas is orbiting more slowly than the star turns. The field lines are dragged backward, the tension pulls the star back, and the star is spun down — and worse for the gas, it is being flung outward by a field rotating faster than its own orbit can support, so it may not accrete at all. That is the propeller regime.

How much of a mass can be turned into light, and what decides it. The fraction of rest energy an accreting object releases, against the radius its disc has to stop at. The curve is the Newtonian half-binding-energy GM/2rc², and the whole content of the plot is that the answer is set by the inner edge and by nothing else — not by the mass, which cancels when the radius is measured in GM/c², and not by the accretion rate, which sets the luminosity and not the efficiency. A white dwarf's surface is far out and returns 0.015%; a neutron star's is at a few gravitational radii and returns 11%; a black hole has no surface, so the disc stops at the last stable orbit instead and returns 5.72%. That last number is not on the curve: the Newtonian expression gives 8.3% at the same radius, 1.46 times too much, and the difference is the point at which this picture has to be handed over to the metric. Hydrogen fusion, drawn as the flat line, returns 0.7% — an accreting black hole is an order of magnitude better at converting mass into light than a star is, which is why quasars outshine the galaxies they sit in.
Fig. 2 What the accretion is worth, which decides whether the torque can be delivered at all. The efficiency of turning rest mass into light rises steeply as the inner edge moves inward, and for a neutron star it is a tenth — so the same accretion rate that spins the star up also makes it one of the brightest objects in the sky. The two are not independent: the luminosity is what the disc gave up to move the material inward, and the angular momentum it carried is what ends up in the star.

The equilibrium, and the period it fixes

Between the two regimes is a value of the accretion rate at which the two radii coincide, and there the torque vanishes.

That is an attractor rather than a coincidence. Suppose the star is spinning too slowly for the current accretion rate: corotation lies outside the magnetosphere, the star is spun up, and corotation moves inward. Suppose it is spinning too fast: corotation lies inside, the star is spun down, and corotation moves out. Either way the system moves toward the crossing.

Setting the two radii equal and solving for the period gives the equilibrium: it goes as the dipole moment to the six sevenths and the accretion rate to the minus three sevenths.

Put numbers in. A neutron star with a field of a hundred million gauss accreting at a hundredth of the Eddington rate settles at a few milliseconds. A star with a field of ten to the twelfth gauss — an ordinary young pulsar’s — accreting at the same rate settles at several seconds.

That is the whole explanation of the millisecond pulsars, and it explains the field as well as the period. To be spun up to milliseconds the field has to be weak, so the population of rapidly spinning neutron stars is necessarily a population of weakly magnetised ones. The two anomalies in the diagram are one anomaly.

The amount of matter required is modest and is worth computing, because it is what makes the story plausible. Spinning a neutron star from rest to three milliseconds requires an angular momentum of about ten to the forty-eighth in centimetre-gram-second units. Material arriving at the magnetospheric radius carries the specific angular momentum of a Keplerian orbit there, and dividing gives a few hundredths of a solar mass — a quantity a low-mass companion can transfer comfortably over a billion years.

That number also explains why the recycled population is dominated by systems with low-mass companions. A massive companion evolves in a few million years and transfers its envelope quickly; a low-mass one can lose material for a billion, which is what the arithmetic requires.

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 plane the whole argument lives in. The millisecond pulsars occupy the lower left, at short periods and tiny derivatives; the ordinary pulsars occupy the island above. The gap between the two populations is not empty because nothing lives there but because the recycling process moves a star across it quickly.

Why the field is weak

Spin equilibrium explains why fast spin and weak field go together in the sense that only weakly magnetised stars can be spun up fast. It does not explain why the field became weak, and the honest position is that this is not settled.

The leading account is that accretion itself buries the field. Material channelled onto the polar caps spreads sideways under its own weight, dragging field lines with it and pushing the dipole component under a layer of accreted matter. The prediction is that the surface field should decline with the total accreted mass, which is roughly what the population shows: the more thoroughly recycled systems have the weaker fields.

An alternative is ohmic decay accelerated by the heating that accretion causes, since the crust’s resistivity depends steeply on temperature. Both mechanisms predict a decline and they are difficult to separate observationally.

What is clear is the correlation. Neutron stars with long-lived low-mass companions, which transfer a few tenths of a solar mass over a billion years, end up at a few milliseconds and ten to the eighth gauss. Those with high-mass companions, which transfer far less material over a far shorter time, end up at seconds and ten to the twelfth.

There is a useful way to state the relationship between the two effects. The equilibrium period depends on the field, and the field depends on the accreted mass, so a star sliding down a burial track is chasing a moving equilibrium — its target period falls as its field falls. The end state is therefore not a period fixed by the initial field but by the total mass transferred, which is why the recycled population clusters in period rather than spreading over the range initial fields would give.

The systems where it is watched happening

For thirty years this was a story about two populations and an inferred connection. Then the connection was observed directly.

An accreting millisecond X-ray pulsar is a neutron star in a low-mass binary whose X-ray emission pulses at a period of a few milliseconds — that is, a star being spun up now, with the accretion visible and the spin measurable. The first was found in 1998, and there are now about twenty. They confirm the picture in detail. Their spin frequencies lie in the range the equilibrium argument predicts. Their inferred fields, from the requirement that the magnetosphere sit inside corotation during outburst and outside it in quiescence, are around ten to the eighth gauss. And several of them show the spin-up during outburst and a slower spin-down between outbursts that the propeller regime requires.

A spectrum belonging to no temperature at all. The disc's summed emission, with the individual annuli drawn faintly beneath it. Each ring is a blackbody at its own temperature, and each is drawn at the area it actually has — the outer rings are cool and enormous, the inner ones hot and small. The sum has three parts and only the two ends belong to a temperature: a Rayleigh–Jeans rise of slope 2 from the outermost ring, a Wien cutoff at the hottest, and between them a stretch of slope 0.316, against the 1/3 that comes out of integrating ν²T(r)r dr with T ∝ r^−3/4. That middle section is the observational signature of a disc: no single blackbody produces it, no photosphere produces it, and its width rather than its peak is what says how far in the disc goes. What the figure cannot show is that a real disc's innermost rings are neither thin nor blackbodies, which is where the model's clean edges stop.
Fig. 4 And the spectrum the truncated disc produces, which is how the truncation radius is actually measured. A disc is a stack of blackbodies, so its spectrum is a power law between the temperature of the outer edge and that of the inner one — and cutting the disc off at a magnetospheric radius removes the hottest annuli and takes the high-frequency end away with them. Where the spectrum turns over is where the disc stops, and that is the observable the equilibrium period is inferred from.

The clinching object arrived in 2013: a system observed to switch, over weeks, between behaving as an accreting X-ray pulsar and behaving as a rotation-powered radio pulsar, and back. The transition that the recycling picture said must happen at the end of the accretion phase was caught in progress, repeatedly, in one object.

The mechanism of the switch is the same comparison of radii. When accretion is proceeding the magnetosphere is pushed inside corotation and material reaches the surface; when the supply falls the magnetosphere expands beyond corotation, the propeller ejects what arrives, and the pulsar mechanism — which requires a magnetosphere free of infalling matter — can operate. A system near the boundary flips between the two as the supply fluctuates, which is exactly what the transitional objects do.

Those objects also answer a question that the population argument could not. The recycling picture requires that a spun-up neutron star eventually turns on as a radio pulsar, and that requirement had no direct evidence for it; a system observed to do it, and to do it reversibly, supplies the missing step.

The same argument on a young star

The arrangement is not peculiar to neutron stars, and the version that occurs during star formation is where it was first worked out.

A young low-mass star has a kilogauss field, a disc, and an accretion rate around ten to the minus eight solar masses a year. Its magnetospheric radius comes out at a few stellar radii, which is where the disc is observed to be truncated — the inner edge inferred from the near-infrared excess sits about there, and the accretion is observed to proceed along funnels onto hot spots at high latitude, exactly as the geometry requires. The spin consequence is the interesting part, because young stars have a spin problem: they contract by a large factor while accreting, and conservation of angular momentum would leave them at break-up. They are not at break-up; the ones with discs rotate at about a tenth of it, and the ones without discs rotate faster. Something ties the star’s spin to its disc, and the leading account is the same one — a magnetosphere reaching out to near corotation, exchanging torque with the disc, and holding the star at a period set by the field and the accretion rate.

The observational support is the correlation between rotation rate and the presence of a disc, and it is one of the more robust results in the study of young clusters. The mechanism’s details are as contested there as here, and the geometry is the same. What differs is the field’s origin: a young star’s is generated by a convective dynamo running now, while a neutron star’s is a fossil, which is why the two behave so differently as they age.

Where the simple version fails

The two-radius argument is a sketch and it is known to be one.

The torque is not applied at a single radius. Field lines thread the disc over a range of radii, coupling to material both inside and outside corotation, and the net torque is an integral over that range with contributions of both signs. Whether the coupling extends far or is confined near the boundary depends on how quickly the field lines are twisted and how they open up, and different assumptions give equilibrium periods differing by factors of a few.

The truncation radius itself is defined by a stress balance with a coefficient of order one that nobody can compute from first principles. It is written as a fudge factor of about a half times the Alfvén radius, and the equilibrium period depends on it.

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. 5 The disc outside the boundary, whose structure the argument treats as given. A steady disc’s temperature run follows from energy conservation, and it is the inner edge — here set by a magnetosphere rather than by a last stable orbit — that decides both the spectrum and the torque.

And the accretion rate is not steady. Low-mass X-ray binaries spend most of their time in quiescence and accrete in outbursts lasting weeks, so the magnetosphere moves in and out by large factors and the star sees a torque that alternates in sign. The outbursts themselves are a thermal instability of the disc rather than a change in what the companion is supplying, which means the star’s torque history is set by the disc’s own bookkeeping. What the equilibrium argument gives is an average over that cycle, and averaging a nonlinear relation is not the same as evaluating it at the average.

There is also a class of system in which the observed torque contradicts the picture outright. Several high-mass X-ray binaries show spin-up and spin-down episodes uncorrelated with their X-ray luminosity, and one or two show sustained spin-down while accreting brightly. Those are usually attributed to a retrograde disc, or to accretion from a wind rather than a disc, or to the coupling extending far beyond corotation. None of the explanations is compelling, and they are the standing counter-examples.

What the boundary does to the light

The truncation is not only a torque; it is what an X-ray spectrum is made of, and the two are measured together.

An unmagnetised neutron star’s disc reaches the surface and most of the accretion luminosity emerges from a boundary layer there. A magnetised one’s disc stops early, and the same luminosity emerges instead from two polar caps a kilometre across, at a temperature far higher because the area is far smaller.

A spectrum belonging to no temperature at all. The disc's summed emission, with the individual annuli drawn faintly beneath it. Each ring is a blackbody at its own temperature, and each is drawn at the area it actually has — the outer rings are cool and enormous, the inner ones hot and small. The sum has three parts and only the two ends belong to a temperature: a Rayleigh–Jeans rise of slope 2 from the outermost ring, a Wien cutoff at the hottest, and between them a stretch of slope 0.316, against the 1/3 that comes out of integrating ν²T(r)r dr with T ∝ r^−3/4. That middle section is the observational signature of a disc: no single blackbody produces it, no photosphere produces it, and its width rather than its peak is what says how far in the disc goes. What the figure cannot show is that a real disc's innermost rings are neither thin nor blackbodies, which is where the model's clean edges stop.
Fig. 6 Why the geometry is legible in the spectrum. The same power emerging from a much smaller area emerges much hotter, so the truncation radius shows up as a temperature — and a hard, pulsed component in an X-ray spectrum is the signature of a magnetosphere.

The pulse profile carries geometry too. Its shape depends on where the caps are, how large they are, and how the emission is beamed, and fitting it constrains the star’s compactness because light bending near a neutron star’s surface changes what fraction of a cap is visible. That is how several neutron star radii have been measured, and it is a magnetic measurement in disguise: without the field there would be no cap and no pulse. The radius so obtained feeds directly into what matter is allowed to do at those densities.

There is a further diagnostic in the iron line. Material in the disc fluoresces in the illumination from the central source, and the line’s profile is broadened by orbital motion and by gravitational redshift in a way that depends on how close to the star the disc reaches. Fitting the line gives the inner radius directly, and comparing that with the magnetospheric radius predicted from the field and the accretion rate is the closest thing to a direct test the model has.

The equilibrium is a balance between two radii, and each of them is set by a quantity worth reading at a second value.

Where the disc stops, and the spin that follows from it. Two radii against accretion rate, for a neutron star with a 3·10⁸-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.
Fig. 7 The truncation radius at three times the field. It moves outward as the four-sevenths power of the field strength, so a stronger magnetosphere holds the disc further out, and the corotation condition is then satisfied at a longer period — the equilibrium period is a field strength read through a balance.
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.
Fig. 8 And the temperature run over four decades of radius rather than three. The three-quarters power law holds across the whole of it, because it comes from energy conservation and not from any property of the material — the only thing that changes at the inner edge is where the law stops.

The third quantity in the balance is the one that does not move at all, and it is worth putting beside the other two for exactly that reason.

How much of a mass can be turned into light, and what decides it. The fraction of rest energy an accreting object releases, against the radius its disc has to stop at. The curve is the Newtonian half-binding-energy GM/2rc², and the whole content of the plot is that the answer is set by the inner edge and by nothing else — not by the mass, which cancels when the radius is measured in GM/c², and not by the accretion rate, which sets the luminosity and not the efficiency. A white dwarf's surface is far out and returns 0.015%; a neutron star's is at a few gravitational radii and returns 11%; a black hole has no surface, so the disc stops at the last stable orbit instead and returns 5.72%. That last number is not on the curve: the Newtonian expression gives 8.3% at the same radius, 1.46 times too much, and the difference is the point at which this picture has to be handed over to the metric. Hydrogen fusion, drawn as the flat line, returns 0.7% — an accreting black hole is an order of magnitude better at converting mass into light than a star is, which is why quasars outshine the galaxies they sit in.
Fig. 9 The efficiency for the supermassive case, for comparison with the stellar-mass one above. It is the same curve: the fraction of rest mass released depends on where the inner edge is in gravitational radii, and that is a property of the metric rather than of the mass.

What the argument is worth

Set against those caveats, three things survive that are worth having.

The exponents. Six sevenths in the field and minus three sevenths in the rate are not adjustable, and they predict a relation between three measurable quantities across a population rather than a value for one object. The accreting millisecond pulsars occupy the region of that relation they should. The direction. The torque changes sign at corotation, which is a statement about geometry and requires none of the disputed coefficients. Any system with a magnetosphere inside corotation is spinning up, and that has been checked object by object.

And the closure of the population argument. The millisecond pulsars have companions or the traces of them far more often than ordinary pulsars do, their orbits are nearly circular in the way a long episode of tidal interaction would make them, and the few that are isolated are plausibly ones whose companions were destroyed — several are observed ablating a companion now, which is a wake and a meal in one calculation run in reverse. The exponent that governs their subsequent spin-down is a separate question, and the field they carry into it was set here.

The most economical statement of what this essay is about is that a star’s rotation rate is being set by a boundary condition tens of stellar radii away, negotiated by a field, on material it never touches. It is the same relationship a disc has with the object at its centre — angular momentum moved outward so that mass may move inward — with the star, unusually, on the receiving end.

Where this goes next

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. 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. 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. A vanishing field that changes the answer completely. The growth rate of the magnetorotational instability against wavenumber, both in units the orbital frequency sets: the vertical axis is the growth rate divided by Ω, the horizontal is the wavenumber multiplied by the Alfvén speed and divided by Ω. The maximum is exactly three quarters of the orbital frequency and it does not depend on the field strength at all — it occurs at kv_A = √15Ω/4, so a weaker field simply moves the fastest-growing wavelength to a longer one. The curve vanishes above kv_A = √3Ω, which is the only thing the field decides: modes shorter than that are stabilised by magnetic tension. An unmagnetised Keplerian disc is Rayleigh-stable and would never accrete; a disc with a field a millionth of the strength needed to matter dynamically grows a mode that doubles in about a fifth of an orbit. The limit is discontinuous, which is the reason this was found late and by algebra rather than early and by observation. The weakest field changes the answer A Keplerian disc is stable by every hydrodynamic test there is. Add a magnetic field of any strength whatever — a millionth of what would matter dynamically — and it becomes violently unstable, at a growth rate of three quarters of an orbit per radian that does not depend on the field at all.

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.

AccretionAlfven radiusAngular momentumCorotation radiusMagnetorotational instabilityMagnetospheric radiusMillisecond pulsarPropeller regimeRecyclingSpin equilibriumX-ray binary