Astronomy & Celestial Mechanics
the sky, computed
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 3 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

Named alongside it

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

AccretionAngular momentumAccretion discAlfven radiusAlfven speedAlpha viscosityAngular momentum transportCorotation radiusDead zoneEddington limitInnermost stable orbitIonisation

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