Astronomy & Celestial Mechanics
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Concept

Magnetorotational instability — where it appears

The instability of a differentially rotating, magnetised fluid in which the angular velocity falls outward, which is every astrophysical disc. Its maximum growth rate is exactly three quarters of the orbital frequency and does not depend on the field strength; the field sets only the wavelength.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

stars · Accretion 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.

gravitation · Magnetorotational instability 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.

gravitation · Accretion A bead on a rotating wire: when gas is flung off a disc. The effective potential — gravity plus the centrifugal potential in the frame rotating with the footpoint — for gas sliding along a rigid magnetic field line that leaves a Keplerian disc at 20°, 30°, 40° from the vertical, in units of GM over the footpoint radius, against distance along the line in units of that radius. The field line corotates with the disc where it is anchored, like a wire whipped round, and gas on it behaves like a bead. Close to vertical the potential rises away from the disc and the gas is bound to its footpoint. Inclined by more than 30° it falls, and gas is flung outward along the line with nothing but the rotation to drive it. The 30° threshold is exact for a Keplerian disc — the second derivative of the potential at the footpoint changes sign there — and it was the result that made magnetised disc winds a mechanism rather than a speculation.

A disc that throws its spin upward

A disc does not need turbulence to accrete. A magnetic field threading it can fling gas off its surface along field lines tilted more than thirty degrees from the vertical, and each gram flung out leaves with several times the angular momentum it had — so a wind carrying a few per cent of the inflow can drive the rest onto the star. Because it pulls on the disc's faces rather than across its edge, a wind outpulls the turbulence as soon as the field's pressure is a few ten-thousandths of the gas's.

gravitation · Magnetorotational instability How far a turbulent disc bends the field it drags. The tangent of the angle by which an accretion flow bends a vertical magnetic field at the disc's surface, against the magnetic Prandtl number Pm — the ratio of the turbulent viscosity that drives the inflow to the turbulent resistivity that lets the field slip — for discs of aspect ratio 0.03, 0.05, 0.1. In a steady state the inflow drags the field in at exactly the rate it diffuses back, and that balance sets tan i = (3/2) Pm H/R. The dashed line is 30° from the vertical, the inclination a field needs before gas is flung off along it. Turbulence whose own Prandtl number is of order one bends the field by about the disc's aspect ratio, a few degrees; reaching 30° needs Pm of 13, 8, 4. The same turbulence that carries the gas in lets the field out almost as fast, because both are the same eddies.

A field the turbulence will not let in

A disc that accretes through its own turbulence drags its magnetic field inward, and the same eddies let the field slip back out almost as fast. A disc of ordinary thickness bends its field by a few degrees instead of the thirty a wind needs, so a turbulent disc cannot gather the flux that would let it stop being turbulent.

gravitation · Magnetorotational instability A star fed in bursts by a disc that cannot pass its supply on. The accretion rate onto a 0.5 M☉ protostar over 100 thousand years, for a steady supply of 10⁻⁶ M☉ a year into a disc whose active layer can carry only 10⁻⁸ M☉ a year at 1 AU. The excess fills the dead zone until the column inside 1 AU reaches 4.4·10⁴ grams per square centimetre — 0.016 M☉, the column at which it becomes gravitationally unstable and its spiral arms heat it to ignition — and then drains on the hot disc's viscous time of 189 years, at 8.3·10⁻⁵ M☉ a year. The cycle repeats every 16,083 years, 6 times in the span drawn. The star spends 1.2 per cent of its time in outburst and gains 99 per cent of its mass there: a census at a random moment finds almost every protostar faint, while almost all of the mass arrives during events nobody happens to be watching.

A disc that stores its inflow and spends it in a century

Where only a thin ionised skin of a young disc can transport angular momentum, the gas arriving from the cloud cannot be passed on, and it piles up in the dead midplane until gravity stirs it and the whole column ignites. The star then gains most of its mass in bursts of a century every ten thousand years — which is why almost every protostar ever observed is fainter than its growth requires.

gravitation · Magnetorotational instability

Named alongside it

The objects these essays reach for when they reach for this one.

Dead zoneAngular momentum transportPlasma betaProtoplanetary discAccretionAccretion discAlfven radiusAngular momentumDisc windAccretion rateAlfven speedAlpha viscosity

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