Series

Plasma beta — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · stars
  2. 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.

    part 2 · stars
  3. 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.

    part 3 · stars

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