Astronomy & Celestial Mechanics
the sky, computed
Concept

Disc wind — where it appears

Material lifted off an accretion disc and accelerated along magnetic field lines, carrying angular momentum away entirely rather than moving it further out. It removes a small mass and a large amount of spin, and it competes with internal transport as the answer to how discs accrete.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.

Ninety-nine per cent of the mass and none of the spin

The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

orbits · Angular momentum A collapse that stops 29 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 2.4·10¹⁶ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 4.27·10¹² metres — 29 astronomical units, or 6135 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 1.7·10⁴ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse.

A cloud that cannot become a star

A molecular cloud core turning once every twenty million years sounds like a body at rest. Conserve its angular momentum through a collapse by a factor of ten thousand and it stops at the orbit of Neptune, spinning, having failed to make anything at all.

galaxies · Star formation 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

Named alongside it

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

Angular momentum transportProtoplanetary discDead zoneMagnetic brakingMagnetorotational instabilityPlasma betaSpecific angular momentumAccretion discAlfven radiusAmbipolar diffusionAngular momentumAngular momentum problem

All concepts