Concept

Plasma beta — where it appears

The ratio of gas pressure to magnetic pressure in a plasma, which decides which of the two governs the motion. Where it exceeds one the gas carries the field about; where it is below one the field dictates where the gas may go.

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

A spot is dark because it is squeezed. Pressure against depth below the quiet photosphere's optical surface, drawn as a ratio to the pressure there. The rising curve is the surrounding gas, which grows exponentially with a 140-kilometre scale height because that is what hydrostatic equilibrium in an ideal gas produces. The flat pair of bands is the spot's own budget: a magnetic pressure of 3.58·10⁵ dyn/cm² from a 3000-gauss field, which is 25 per cent of the total, plus the gas pressure left over. Horizontal balance requires the two columns to reach the same total at the same geometric level, and the level at which they do is 348 kilometres below the quiet surface — so the spot's own optical surface sits in a hollow. Measured Wilson depressions, obtained from the foreshortening of a spot near the limb, are four to six hundred kilometres. Nothing about the darkness was assumed: a field strength read off a Zeeman splitting fixes the magnetic share, the share fixes the depression, and the depression fixes the temperature the deeper layer must have to carry the reduced flux.

The darkness a field pays for

A sunspot is not cool because something is missing. It is cool because a three-thousand-gauss field supplies part of the pressure that holds the column up, the gas therefore supplies less, and the level at which that gas becomes opaque sits some hundreds of kilometres deeper than the surface around it.

stars · Plasma beta
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
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
Plasma beta from the corona's base outward. The ratio β of gas pressure to magnetic pressure against distance from the Sun's centre, one to ten solar radii on logarithmic scales, for a static isothermal corona in a dipole field. quiet Sun, 5 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.41 and crosses one again at 3.46 solar radii; quiet Sun, 10 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.1 and crosses one again at 5.00 solar radii; active region, 30 G at the base, with 3·10⁹ electrons per cubic centimetre at 2.5 million kelvin: β starts at 0.058 and crosses one again at 2.97 solar radii. The gas pressure falls steeply at first and then levels off, because gravity weakens with distance; the field's pressure, falling as the sixth power, never levels. Every corona therefore has a second place where the gas takes over, and it sits at two to four solar radii — where the closed loops of the corona end and the streamers' tips begin.

A corona held by its field and opened by its gas

A thousand kilometres above the Sun's surface the gas pressure has fallen so far that the magnetic field takes charge, and the corona's loops are the shape the field gives it. But the gas pressure of a hot atmosphere stops falling a few solar radii out, while the field's keeps falling as the sixth power of distance. So there is a second place where the gas takes over — and it is where the loops end, the streamers begin, and the solar wind is let out.

stars · Plasma beta
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
Elements sorted by how easily they lose an electron. The abundance of each element in the solar wind relative to the photosphere, normalised to oxygen, against its first ionisation potential, logarithmic: in the slow wind (filled) and the fast wind from coronal holes (open). Representative values. Elements easy to ionise — K, Na, Ca, Mg, Fe, Si, all below about 9 eV — are about three times over-abundant in the slow wind and about one and a half in the fast; those harder to ionise sit near one. The step comes at about 10 eV, which is the energy of a photon from the Lyman-α line of hydrogen: the sorting happens where hydrogen's own light sets the ionisation.

A wind that carries the ledger of its loops

The slow solar wind is richer in iron, magnesium and silicon than the Sun's surface is — about three times richer, relative to oxygen — and the elements that are enriched are exactly the ones that lose an electron easily. That sorting happens low in the chromosphere, inside closed magnetic loops, and it takes days. The fast wind from coronal holes barely shows it. So the composition of the wind arriving at the Earth is a record of where it was held, and for how long, before the field let it go.

stars · Plasma beta

Named alongside it

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

Dead zoneMagnetorotational instabilityAngular momentum transportCoronaDisc windHelmet streamerMagnetic pressureProtoplanetary discSolar windAccretionAlfven radiusAlfven speed

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