A spot whose missing heat goes nowhere

The reason a sunspot is dark has been understood since Ludwig Biermann in 1941. A strong magnetic field in the column suppresses the convection that would have carried heat up. The spot’s surface radiates about a fifth of what the surrounding photosphere radiates, for its area. So the energy from below does not reach the surface there. That is where the usual account stops, with the field’s pressure in the picture, and there is a further problem which is obvious once it is stated. A spot 20,000 kilometres across is blocking something like $10^{29}$ erg a second, more than the whole output of a small red dwarf. Energy is conserved. So where does it go?

The obvious answer is that it flows round the plug. Hot gas that cannot rise through the spot would go sideways, and come out in a bright ring around it, so a spot’s dark centre should have a slightly brighter surrounding, with the same total energy. In the first half of the twentieth century people looked closely for this. The result, after several generations of careful photometry, is that there is hardly anything there, certainly not enough. In 1982 Hendrik Spruit gave an answer. It relies on a simple property of the convection zone, and it means that the flux missing from the spot does not come out anywhere on any timescale that matters. It is stored.

The Sun’s light does dip #

The obvious test is whether the Sun gets fainter when a spot crosses its disc, and this took until the first space radiometers to do. In 1980, the Active Cavity Radiometer on the Solar Maximum Mission made an astonishing discovery within a few months of launch: the total light of the Sun drops by up to about a quarter of a per cent for a week or so when a large spot group crosses the disc.

That dip is exactly what a dark patch of known area and contrast would make. So the Sun as a whole really is fainter, with no matching increase anywhere else. A simple measurement of solar irradiance, from a satellite that has no spatial resolution, refutes the most direct idea of what a blocked flux does. The luminosity of a star can change in the space of a few days, by amounts that exceed a thousandth.

It is worth emphasising how surprising this was at the time, since the Sun’s energy output was widely known as “the solar constant”. It had been measured by rockets and from mountaintops by Abbot for decades, and no one had been able to detect a change. It was taken for granted that the deep source of the Sun’s power, whose time constant is millions of years, could not change its surface output in less than that.

What actually happens is this. Luminosity is a surface property: what is radiated is decided in the top few hundred kilometres. In a stationary star it must equal what the core is producing on long time scales, but not on short ones. A star has a thick reservoir of stored heat. It is entirely possible for it to radiate less than it produces for quite a while, and the difference simply accumulates.

Now add the rest of the picture.

So there is a brightening, on either side of the dip. It is tempting to think of that as the missing heat, but it is not. Those bright regions, the faculae, are not spread around the spot in a way that conserves any quantity. They are a separate magnetic feature, made by thin flux tubes through which the surrounding hot walls can be seen, and they remain around for weeks after the spot itself has decayed. Over a cycle, facular brightening wins, and the Sun at sunspot maximum is about 0.1 per cent brighter than at sunspot minimum, in spite of carrying more spots. This counter-intuitive result is a result of a surface effect. So the fact that an active Sun is brighter says nothing at all about whether heat blocked by the spots is getting out.

Why there is no ring #

Think of the problem as steady heat flow. Below the surface the heat travels by turbulent convection, with a very large effective conductivity, of the kind a mixing length is used to describe, and at the surface it is radiated into space, by a sort of boundary condition. Now block part of the surface. How far from the blocked area does the extra flux emerge?

The answer depends on a comparison of two rates: how quickly heat spreads sideways through the interior, and how readily it escapes at the top. Consider a slab of depth $d$ with conductivity $K$, and a surface that responds to a temperature excess $\delta T$ with an excess flux $h\,\delta T$. Solve the steady problem, and an excess falls away from the plug exponentially, with a spreading length

$$L=\sqrt{\frac{K\,d}{h}} .$$

A low-conductivity medium would give a short length and a strong ring. For the solar convection zone it is huge. The turbulent conductivity is so great that a large part of the convection zone is practically isothermal on surfaces of constant pressure. A small excess of temperature in the interior is enough to carry the whole of the Sun’s flux. The surface, by contrast, responds only very weakly: raising the local radiated flux by one per cent requires a rise of a quarter per cent in surface temperature, and in the deep interior that much temperature excess would drive enormous flows.

The observation behind this figure is a set of high-precision photometric measurements around spots. Fowler, Foukal, Rast and others have measured bright rings of order a few tenths of a per cent in some cases and less in others, close around the spot. The most careful analyses, by Rast and collaborators around 2001, find a warm ring carrying only about a tenth of the spot’s missing flux. There is thus a small local ring, as one might expect from the shallowest layers. But most of the deficit does not reappear nearby.

It is the same as saying that the blocked flux is smeared out across the whole surface, and it is too small to detect at each point. If a spot group covers a thousandth of the disc and blocks 70 per cent of the flux there, then the whole remainder of the Sun would have to brighten by 0.07 per cent to compensate. That is well below what a satellite can resolve in a whole-Sun measurement separately from anything else. And that’s assuming it comes out at all.

The reservoir #

Spruit’s central point is that it doesn’t have to come out, not for a very long time. The energy flows into the convection zone and it heats it. The convection zone is enormous. It has a mass of about two and a half per cent of the Sun’s, with a heat content of about $7\times10^{45}$ erg.

So the energy blocked by spots through a whole solar cycle raises the temperature of the convection zone by less than a part in a million. That is the reason nothing happens. The change in surface temperature that results from this small rise is correspondingly small, and the Sun’s surface does not need to radiate the extra amount for a very long time. Its luminosity will come back into balance when the whole envelope has adjusted, over the thermal time of around a hundred thousand years; that’s far longer than any spot, and longer than the magnetic cycle that made it.

There’s another way to see this. The Sun’s luminosity at the surface is not something the interior dictates moment by moment. It’s more like the flow out of a tank with a large capacity. A partial blockage at the outlet briefly reduces the flow, and the level in the tank rises imperceptibly. There’s no reason why water would quickly appear at some other point. The only relevant timescale is how long it takes for the level to rise enough to change the pressure at the outlet, and for the Sun, that’s a very long time.

That has an important and surprising consequence for the energy balance of the Sun during the cycle. Between sunspot maximum and minimum the amount of energy stored changes by something like $10^{39}$ erg, a small fraction of the convection zone’s content, and on these time scales the Sun’s luminosity is not really a constant at all. Rather it’s the surface’s emissivity that changes, at fixed interior conditions. The whole 0.1 per cent solar cycle variation can be understood as a modulation of the surface, not any change in the production of energy.

Long enough for a star to answer #

What happens if spots stay for longer than the thermal time? On the Sun they do not, and the cover comes and goes every few years. But many active stars have coverage of tens of per cent, persistently. On a time scale long compared with the envelope’s thermal time, the blocked luminosity has to come out. A star cannot be out of energy balance forever. So its unspotted surface must rise, and this can only come about with the star adjusting its structure.

Then its response is a simple statement of the requirement. The star’s total emission is proportional to its area and to the mean surface flux. With a fraction $f$ blocked at contrast $c$, the flux falls by a factor $(1-fc)$. Since the surface temperature is regulated by the atmosphere’s opacity, at nearly the same value as before, the area must increase so as to restore the luminosity. So the star’s radius grows, on the thermal time of the envelope. The fact was worked out by Spruit and Weiss in 1986. It has an application to the radius inflation of active red dwarfs, and there has been a debate over whether spots are the cause.

Here the essay’s central question has a neat two-sided answer. On a short time scale, the missing flux goes nowhere and the star is fainter. On a long time scale the missing flux comes out everywhere, and the star is larger. Between them there’s no time at which it comes out as a bright ring. There is also an intermediate regime, in which the star’s luminosity changes on the time scale of its thermal relaxation. An observable effect might arise in stars whose magnetic activity changes on such time scales, for example if they are tidally spun up in close binaries; yet the envelope’s thermal time is a hundred thousand years for the Sun and a much longer time for a fully convective red dwarf, tens of millions of years. So an active red dwarf that has only recently spun up should not yet have finished responding.

What the model cannot distinguish #

A slab is not a spherical shell with rotation. Convection in the Sun’s rotating envelope is not an isotropic conductor. A proper treatment of the transport would use three-dimensional simulations. Those now exist for regions around spots, by Rempel and others, and they show the plug’s effect is more complicated: the Evershed flow and penumbra carry part of the flux, and a strong downward flow is found beneath some spots. Their conclusion on the overall energy balance agrees with the simple picture but with far more structure.

The spot’s depth is unknown. How far below the surface the strong field persists in a coherent column affects how much the flow beneath it can come back. A shallow spot blocks energy only in a thin layer, and there is more lateral flow beneath. Helioseismology has given conflicting results. The argument above is insensitive to this, since the reservoir is so large whichever depth applies, but the details of the ring depend on it.

The ring’s size is inferred from a one-dimensional strip. Real spots are round, and the strip gives larger lengths for a given amplitude. The labelled numbers are only order-of-magnitude guides, and in the figure the conductivity is not derived, just a chosen spreading length.

A faint Sun is not necessarily an energy deficit in every sense. Irradiance measures the flux radiated at the Earth’s direction. The distribution of a spot’s dimming around the Sun is not uniform. Part of what the dip implies for the whole-Sun luminosity is an estimate, and the same goes for the facular part, whose brightness has a strong angular dependence. Hence the actual luminosity variation is less well known than the observed irradiance, by a factor of about two.

A question from the climate #

The dip has an unusual place outside astronomy. Solar irradiance is the input to the Earth’s climate. The measured total solar irradiance has varied by about 0.1 per cent over the cycle since 1978. What it did before then, particularly in the Maunder Minimum, when there were almost no spots for about seventy years, is an important question and hard to answer.

One might have reasoned, before 1980, that a Sun with no spots must be brighter, since there is nothing blocking it. The opposite was found to be the case for the cycle, and it follows from faculae. Yet a long-term reconstruction needs the facular brightening in the absence of spots, which cannot be measured directly. Reconstructions have disagreed by a factor of several about how much fainter the Sun was in 1700, and the latest values are near the small end, around 0.1 per cent. The point of connection here is that all the variations under discussion are thought to be at the surface. Neither spots nor faculae change what is being produced in the core. If so, they change the Sun’s output only temporarily, in the sense that the energy is stored and then later released.

The same interpretation gives a way of thinking about the Sun’s long-term magnetic history as seen through isotopes. A reduction of a tenth of a per cent for a few decades is a few times $10^{39}$ erg, a tiny fraction of what the convection zone holds. If spots disappeared for a long time, the stored heat would not be perceptible in the Sun’s interior.

The same arithmetic on bigger spots #

Other stars extend the range. Young stars and those in tidally locked binaries, such as RS CVn systems, can vary by tens of per cent in brightness through rotation, implying large spots. Some have polar spots that are permanent and so cause no modulation. As a rotational light curve hides most of an even scatter of spots, the coverage has to be estimated in other ways. Molecular bands and line-depth ratios indicate coverage for active red giants up to about fifty per cent.

Such stars are likely to have reached the second regime, as their coverage has persisted for longer than their envelope’s thermal time, and this shows up as an offset on the Hertzsprung–Russell diagram. Very young stars in clusters, spotted heavily, appear at a position in the diagram that gives them an incorrect age and mass, because a spot-inflated cool star looks like a younger one. Measurements of the Pleiades and younger clusters suggest coverage in the tens of per cent, and models that include this effect, by Somers and others, change inferred ages by a factor of up to two. Here the missing flux has an unexpected consequence: an age error for an entire young cluster.

And the transit community has its own reason to care about this. A heavily spotted star has a larger radius than otherwise expected, so a planet’s inferred size from its transit depth inherits an error. It’s an indirect effect through a chain of inferences, starting from the fate of heat underneath a spot, and ending with a planet’s radius.

Still open: how far the ring can be pinned down #

The lack of a strong bright ring has been explained, but the small ring that does exist is still not fully accounted for. How much of the flux goes into it, as opposed to deep storage, depends on the depth of the spot and the structure of the convection below it. In simulations, the surrounding moat flow carries some energy outward at the surface, and the amount of brightening depends strongly on the assumed surface boundary. Observations have not settled how the ring’s flux depends on spot size, and the existing values vary among authors, from nearly nothing to several per cent of the blocked flux, in part because the contrast of the magnetic network near spots must be subtracted first.

Better photometry from space, together with the combination of disc-resolved images and irradiance, could tell how much of the blocked energy reappears nearby, as a function of spot size and age. The answer is unlikely to alter the energy budget of the Sun. It would, however, give a direct measurement of the effective conductivity of the top megametres of the convection zone, which is something otherwise known only through theory: a measurement, in other words, of how the Sun transports heat, made by observing where its heat fails to appear.