Astronomy & Celestial Mechanics
the sky, computed
Concept

Alfven radius — where it appears

The distance at which a star's magnetic field stops enforcing corotation on its wind, because the outflow's kinetic energy density there matches the field's. Inside it the gas is carried round with the star; outside it the gas goes straight and drags the field into a spiral. It is the lever arm on which a wind removes angular momentum.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.

A wind that takes no mass and all the spin

The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left.

stars · Magnetic braking A radial wind from a rotating star draws a spiral. The interplanetary field out to 5 astronomical units, drawn as the Archimedean spiral it is. Nothing here rotates: the plasma moves radially outward at 400 kilometres a second and the field is frozen into it, so each parcel remembers the longitude it left from and the pattern winds up while the material does not. The pitch angle is arctan(Ωr/v), which is 47° at one astronomical unit — the radius where the star's rotation has carried the footpoint through one radian in the time the wind takes to arrive. The practical consequence is a matter of hours: a flare's particles follow the field rather than the line of sight, so the ones that reach a given planet left a longitude about 61° to the west of it. A magnetically well-connected flare on the western limb delivers a particle storm and a larger one at disc centre does not, and the difference is this geometry rather than anything about the flare.

The flare that arrives from somewhere else

The solar wind blows radially outward and the Sun rotates, so the field frozen into the wind is wound into a spiral making forty-five degrees to the radius at the Earth. Energetic particles follow the field rather than the line of sight, which is why the flares that deliver particle storms are the ones on the western limb rather than the ones facing the planet.

sky · Stellar winds 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 In a mixed field the torque follows the dipole and ignores the rest. The torque of a field that is part dipole and part quadrupole, with the total surface strength fixed, as a fraction of a pure dipole's, against the fraction of the field strength in the dipole, for wind magnetisations of 300, 3,000, 30,000, using the broken power law fitted to simulated mixed-field winds. Above a dipole fraction of 0.39, 0.24, 0.15 respectively, the torque is exactly what the dipole component alone would give: the quadrupole has fallen below the dipole long before the wind reaches its Alfvén radius, and contributes nothing to the lever. Below it, the quadrupole sets a floor. A star whose surface is mostly quadrupole by strength can therefore be braked almost entirely by its small dipole — and a measurement of the total field says little about the torque unless it says how much of the field is dipolar.

A lever set by the dipole and not the field

A star's wind carries away its spin because the magnetic field holds the gas in step with the surface out to the Alfvén radius, and the torque goes as the square of that radius. How far out the field can hold depends less on how strong it is than on how it is arranged: a quadrupole of the same surface strength holds the Sun's wind to half the distance and brakes it at a fifth of the rate. In a mixed field only the dipole reaches far enough to matter — so the field that brakes a star is the part of it that is hardest to measure.

stars · Magnetic braking

Named alongside it

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

Angular momentumAngular momentum transportMagnetic brakingMagnetorotational instabilitySolar windStellar windAccretionConvective envelopeCorotating interaction regionCorotationCorotation radiusCosmic rays

All concepts