The darkness a field pays for

A sunspot is about two thousand kelvin cooler than the surface it sits in, and the usual way of putting that — the field suppresses convection, so less heat gets out — is true and explains nothing. Suppressing convection would make a spot cool only if the heat had nowhere else to go, and it does: a spot occupies a thousandth of the Sun’s area, and the flux it fails to carry emerges elsewhere with no measurable change anywhere.

The useful question is not why a spot is cool. It is why a spot’s surface is where it is.

A field is a pressure, and that is the whole entry point #

A magnetic field exerts no force on neutral gas and does no work on anything moving along it. What it does do, in a conducting fluid, is push sideways: the field lines resist being squeezed together, with a pressure equal to the square of the field strength divided by eight pi in the units astronomy uses. At three thousand gauss — an ordinary umbral value — that is about three and a half times ten to the fifth dyne per square centimetre.

The number to set it against is the gas pressure at the level where the Sun becomes opaque, which is about one and a fifth times ten to the fifth. So the magnetic pressure inside a spot is roughly three times the gas pressure outside it at the same height.

That is an unusual state of affairs and it is worth registering how unusual. In almost every other part of the Sun the field is a spectator: the gas pressure exceeds the magnetic pressure by factors of a thousand or more, the gas goes where its own buoyancy sends it, and the field is carried along like dye in water. A spot is the exception, and it is an exception produced by concentration rather than by generation — the same total flux, gathered by convection into a few per cent of the area, raises the local field strength by the square root of the concentration factor in pressure terms and can therefore cross the threshold without any new field being made.

That comparison is the quantity called plasma beta, and it is the ratio of gas pressure to magnetic pressure. The whole of solar physics divides at beta equal to one.

Below the crossing the gas moves the field: convective cells shove flux tubes around, sweep them into the lanes between granules, and concentrate them there. Above it the field moves the gas, which is why the corona is structured into loops that follow no density gradient and why a flare can release in minutes an energy the gas at that height could not store.

A sunspot is the case where beta is below one at the photosphere, which is to say where the field has won at a depth the eye can see.

There is a second reading of the same ratio that is worth carrying, because it is the one that explains why spots exist as discrete objects rather than as a smooth magnetic haze. A flux tube in a high-beta medium is at the mercy of the flows around it; a flux tube whose internal magnetic pressure exceeds the external gas pressure is stiff, and stiff structures survive. So the same threshold that makes a spot dark is what allows it to hold together for weeks in a medium whose convective cells turn over in ten minutes. Darkness and longevity are the same fact seen twice.

An evacuated column has to sit lower #

Consider the spot and the quiet Sun side by side, at the same geometric height. Horizontal pressure balance is not optional: an imbalance of ten per cent would be erased at the sound speed, which crosses a spot in minutes, and spots last for weeks. So the total pressure — gas plus magnetic — must match across the boundary.

Inside the spot the field is supplying three units of pressure where outside there are four of gas. The gas inside therefore supplies one. It is evacuated, by a large factor, and evacuated gas is transparent.

Opacity in the solar photosphere comes almost entirely from the negative hydrogen ion, which needs both a neutral hydrogen atom and a free electron. Its abundance rises very steeply with temperature, because the electrons come from the ionisation of metals; the standard rule of thumb has the opacity varying as something like the ninth power of temperature in this range. So a spot is doubly transparent: it has less gas, and its gas is cooler and therefore far less absorbing per gram.

The consequence is geometric. Optical depth unity — the level a photon last scatters from, which is what a telescope sees as “the surface” — has to be reached somewhere, and in the spot it is reached deeper. That depth is the Wilson depression, and the figure at the head of this essay computes it: with a three-kilogauss field and an opacity contrast of about nine, some three hundred and fifty kilometres.

The estimate is sensitive to the opacity contrast in a way that is worth stating rather than hiding, because it is the weakest link. The magnetic pressure is known to a few per cent from the splitting; the scale height is known well; the contrast is a ratio of two opacities at two temperatures and two pressures, and getting it from a model atmosphere rather than from a measurement is what makes the computed depression uncertain at the level of a hundred kilometres or more. The sign of the effect and its order of magnitude are secure; the third significant figure is not, and no version of this calculation should be quoted as though it were.

The same reasoning about where a surface sits applies to every star and to every wavelength: an optical surface is not a place but a condition, and any change to the opacity moves it. What is unusual about a spot is that the thing moving it is a term in the pressure balance rather than a change in composition or temperature alone.

What was actually measured #

The depression is not inferred from the argument above; it was measured, and measured before anybody could compute it.

Alexander Wilson noticed in 1769 that a large spot near the solar limb looks asymmetric — the penumbra on the side nearer the limb appears narrowed or vanishes altogether, while the far side stays wide. That is what a saucer-shaped depression looks like when viewed obliquely: the near wall hides the floor. Fitting the foreshortening as a spot rotates across the disc gives a depth, and the modern versions of that measurement return four to six hundred kilometres.

A second and quite independent determination comes from spectroscopy. The magnetic field is measured by the splitting of a spectral line, and if the field is that of a roughly potential structure above a flux tube, its rate of change with height constrains where the observed level sits within the tube. A third comes from the wings of pressure-sensitive lines, which report the gas pressure at the level they form in directly. The three agree at the level of a hundred kilometres or so, which is about the accuracy each of them claims. What matters for the argument is that the depression is a real, measured geometric fact rather than a consequence of assuming the pressure balance — so the pressure balance is being tested rather than used.

Wilson’s own observation deserves a moment, because it is a good example of what the discipline can do with geometry alone. He had no idea what a sunspot was; the leading account at the time was that spots were holes through a luminous shell revealing a dark body beneath, and Wilson’s measurement was taken as support for it. The inference that survives is not his interpretation but his geometry: something about a spot is lower than the surrounding surface, by a distance small compared with the solar radius and large compared with nothing else in the photosphere. Two hundred years later that distance became the observable a pressure balance had to reproduce, which is the usual fate of a good measurement made for a bad reason.

The temperature follows, and so does the colour #

Having fixed where the surface is, the temperature follows from what the flux is. The spot is radiating from a level where the surrounding gas is hotter and denser, but the spot’s own gas has been evacuated and cannot carry the flux convectively; the field has made it stiff against the transverse motions that convection requires. The umbra ends up at about four thousand kelvin against the photosphere’s fifty-eight hundred, which is a factor of four in the emitted flux and rather more than that at short wavelengths. Nothing in that chain was fitted. A field strength read from a splitting fixes a magnetic pressure; the magnetic pressure fixes the depression; the depression and the suppression of convection fix the temperature; the temperature fixes the colour. Each step is a comparison of two numbers, and the numbers are measured independently at three of the four steps.

One number in it is genuinely strange and is worth pausing on. The energy a spot fails to radiate has to go somewhere, and the searches for it have come up empty: measurements of the total solar irradiance show the expected dip when a large spot crosses the disc, and no compensating excess around the spot at the level the deficit would require. The energy appears to be absorbed into the convection zone’s enormous heat capacity and released over a timescale far longer than the spot’s life, which is a satisfying answer with an uncomfortable feature — it means the spot’s deficit is not conserved locally in any way an observation can follow, and the bookkeeping is closed by an argument rather than by a measurement.

The same argument on other stars #

The Sun is one star, and a mechanism that works only there is not a mechanism. The check is that other stars have spots, and their spots are inferred from a light curve rather than seen. There is a second route on other stars that is worth naming because it does resolve the degeneracy. Molecular bands that form only below about four thousand kelvin — titanium oxide, in particular — appear in the spectrum of a spotted star and not in that of an unspotted one of the same type, and their strength gives the covered fraction directly. Combining a band strength with a light-curve amplitude separates coverage from contrast, and the answers are startling.

Young, rapidly rotating stars can be covered by spots over tens of per cent of their surface, against the Sun’s few thousandths at maximum, and the fields inferred from Zeeman broadening on such stars run to kilogauss averaged over the whole disc. That is a strong test of the pressure argument in an unexpected direction: with beta below one over much of the surface, such a star’s photosphere is magnetically structured everywhere rather than in isolated spots, and the resulting radius and temperature both shift measurably from what an unspotted model predicts. The discrepancy between the radii of magnetically active low-mass stars and the models of them is one of the standing problems in stellar structure, and it is this pressure term entering where nothing accounted for it.

What holds the tube together sideways #

One feature of the balance has been assumed and deserves its own paragraph, because it is what makes a spot an object rather than a smear.

A vertical flux tube in a stratified atmosphere is not in equilibrium along its length and does not need to be; what it needs is equilibrium across it, and that is the condition already used. But the same condition, applied at every height, fixes how the tube’s cross-section changes with depth. Deeper down the external gas pressure rises steeply, so a tube carrying fixed flux must be squeezed into a smaller area, and the field strength inside it must rise. Higher up the external pressure collapses and the tube fans out until neighbouring tubes meet and the field becomes a canopy covering everything.

That expansion is why a spot has a sharp edge at the surface and a much less definite one below it, why the field above an active region can be modelled as nearly potential a few thousand kilometres up, and why the transport of energy through the layer below is disturbed over a volume considerably wider than the dark area anybody photographs.

What the ratio does not explain #

Two things are worth separating out, because the beta argument is often asked to carry them and cannot.

It does not explain why the field is there. A three-kilogauss concentration is not what a field of a few gauss would do if left alone; it has been amplified and bundled by something, and that something is a dynamo working on the shear between the Sun’s interior and its convection zone. The spot is the outcome of that process, and its schedule is the eleven-year cycle. It also does not explain the penumbra. The umbra is the simple case, a nearly vertical field of a few kilogauss; the penumbra around it is a filamentary structure with an inclined field, systematic outflows and a heat flux most of the way back to the quiet value, and it is not a pressure balance at all but a convective mode that operates in a strongly inclined field. The penumbra is where the simple story stops, and it covers most of the spot’s area.

Nor does it explain why the field emerges where it does. Spots appear in pairs of opposite polarity, tilted systematically with respect to the equator, at latitudes that march toward the equator through a cycle — a set of regularities that has nothing to do with pressure balance and everything to do with how a subsurface toroidal field is made and how it becomes buoyant. The pressure argument takes the flux tube as given and works out what happens when it reaches the surface. Everything about how it got there is a different subject, and it is the one the polar field predicts a cycle ahead of time.

Where the boundary really is #

The ratio has one more use, and it is the reason it is worth naming rather than just computing.

Beta equal to one is not a place; it is a surface, and where that surface sits varies enormously across the Sun. Above a spot it is below the photosphere. Above the quiet network it is a thousand kilometres up. In the centre of a supergranule it may be higher still. So the boundary between the two regimes is corrugated, and the corrugation is what makes the chromosphere structurally complicated in a way the photosphere is not. That corrugation has consequences the essay can only gesture at. Waves generated by convection propagate freely where beta exceeds one and refract sharply where it does not, so the beta surface acts as a partial mirror for the acoustic energy that heats the chromosphere. Magnetic energy stored above the surface can only be released where the field dominates. And the field’s own footpoints are dragged about by convection below the surface and shaken free of it above, which is precisely the arrangement that winds a corona up until it fails.

The same surface, followed outward instead of upward, is where the solar wind’s own structure is decided. Field lines that close back to the Sun trap the gas on them; field lines that open carry it away, and the geometry those open lines take once they leave is fixed by nothing more than the wind’s speed and the star’s rotation. The beta surface is the boundary between the two behaviours, so it is also the boundary between the part of the atmosphere that stays and the part that goes.

The picture to take away is not that a sunspot is dark.There is a last consequence of the pressure argument that is easy to state and hard to accept, and it is the reason spots are useful beyond themselves. A spot’s field strength, its darkness, its depression and its lifetime are all one measurement seen four ways, and three of the four can be checked against the first. Nothing else in stellar physics offers that: a stellar radius is inferred from a flux and a distance, a stellar mass from an orbit, a stellar age from a model, and none of them can be re-derived from an independent observable of the same star. A sunspot can, because the field is measured directly and everything else follows from a balance of two pressures.

That chain is also, incidentally, the reason a sunspot is a good test object for radiative-transfer codes. An atmosphere with a known field, a known pressure deficit and an observed temperature is a considerably harder thing to reproduce than a quiet photosphere, and the codes that get the quiet Sun right do not automatically get an umbra right — the molecular opacities matter at four thousand kelvin and not at six. That is a small point with a large consequence: the umbral temperature scale rests on molecular physics that the rest of solar spectroscopy never has to get right.

The reason it can is that the Sun is resolved. The whole apparatus of this essay — a field strength at a point, a depression measured by foreshortening, a temperature from a spatially resolved spectrum — requires an image, and the Sun is the only star that has one at the necessary scale. Every statement about spots on other stars is a statement about a disc-integrated quantity with the geometry marginalised out, which is why coverage fractions there are lower bounds and why the radius discrepancy in active low-mass stars has resisted explanation for two decades.

It is that one dimensionless ratio, computed from a field strength that a spectral line reports and a gas pressure that a model atmosphere supplies, fixes a geometry that a telescope can check independently — and having been checked once at the surface of one star, it is the term that has to be carried into every other magnetised atmosphere. The next rung takes it outward, to the place where the same balance is struck against a wind instead of against a weight.