A corona held by its field and opened by its gas

The ratio of a plasma’s gas pressure to its magnetic pressure is called beta, and it decides which of the two is in charge. Where beta is large, the gas pushes the field around: field lines are carried by the flow, bent by convection, pressed into bundles. Where beta is small, the field pushes the gas around: the gas can move freely along field lines but not across them, and whatever structure the field has, the gas takes on. At the Sun’s visible surface beta is large almost everywhere, except inside the concentrated fields of sunspots, where the field supplies enough of the pressure to make the gas thin and cool and dark. A thousand kilometres higher, in the chromosphere, the gas pressure has fallen by a factor of a hundred thousand and the field’s by far less, and beta drops below one. From there upward the corona is the field’s.

That is why the corona has shapes. An eclipse photograph shows loops, arches and long tapering streamers, and a picture of the corona’s density alone could not have predicted any of them: the gas lies along field lines because it cannot cross them, and it glows where it is trapped. The loops are the field made visible.

But followed outward, the same comparison turns round a second time.

The second crossing is not a detail of this model. It follows from two facts about how the two pressures fall, and it says where the corona ends.

Why a hot atmosphere stops thinning #

A gas in hydrostatic equilibrium — the same balance of weight against pressure that holds up the star beneath it — in a uniform gravitational field thins exponentially with height, with a scale height set by its temperature and weight: $H = kT/\mu m_p g$. At the photosphere, at six thousand kelvin, that is about 150 kilometres, which is why the photosphere’s gas pressure falls so steeply that a thousand kilometres takes it through five orders of magnitude. In the corona, at one and a half million kelvin, the scale height is some two hundred and fifty times larger, about seventy-five thousand kilometres — a tenth of a solar radius.

A tenth of a solar radius is not small compared with the distances in question, and over distances comparable to the Sun’s radius gravity is not uniform. It falls as the inverse square of the distance from the centre. The hydrostatic solution in that field is

$$p(r) = p_0 \exp\!\left[-\frac{GM\mu m_p}{kTR}\left(1 - \frac{R}{r}\right)\right],$$

and its behaviour at large $r$ is the whole point. The term in brackets tends to one, not to infinity, so the pressure tends to a finite value: $p_0$ times the exponential of minus the ratio of the gravitational binding energy of a particle at the surface to its thermal energy. For the corona that ratio is about nine. It is the same comparison that decides whether a projectile escapes, kinetic energy against the depth of the well, made here for the thermal energy of a particle rather than a launch speed. The atmosphere thins by a factor of $e^9$, about eight thousand, and then stops.

The field has no such floor. A dipole field falls as the cube of distance and its pressure as the sixth power, and nothing about the field levels it off. The corona’s large-scale field is roughly a dipole at solar minimum, with a strength of a few gauss at its base. Its pressure starts larger than the gas’s by a factor of a few and falls a thousandfold between one and three solar radii. The gas pressure, over the same distance, falls by less, and then not at all. The two curves have to cross.

Where the crossing sits #

The crossing distance depends on the field strength and on the gas’s density and temperature at the base, and those vary over the Sun and through the cycle.

The range the figure gives, two to four solar radii, is where something visible happens. In eclipse photographs, the bright streamers over the Sun’s quieter latitudes — helmet streamers, named for their resemblance to spiked Prussian helmets — are closed loops at their base, nested arches with dense gas trapped inside, and they narrow to a cusp at two to three solar radii. Beyond the cusp each streamer continues as a thin ray, and the loops are gone. The cusp is where the gas trapped on the loops, heated to a million degrees, pushes hard enough to pull the outermost field lines open. The field holds the gas inside the cusp and the gas opens the field outside it, and the cusp marks the surface where the two change places.

The same surface is built into every model of the coronal field used for space weather. Those models compute the field above the photosphere from its measured values at the surface, assuming there are no currents in the corona, and they need an outer boundary where the field lines are forced to become radial — where the gas is assumed to drag them open. The boundary is called the source surface, and it is placed by convention at 2.5 solar radii because that is the radius that best reproduces the coronal holes and streamers seen on the Sun. The convention was fixed by comparison with observation, not from theory, and the crossing of the two pressure curves is the physics behind it.

A cooler corona, a lower cusp #

The exponent in the hydrostatic solution is inversely proportional to the temperature, and that makes the crossing sensitive to how hot the corona is. A corona at one million kelvin rather than one and a half has an exponent of about fourteen instead of nine, its pressure levels off lower by a factor of more than a hundred, and the field keeps control further out — unless the field is weaker as well.

That sensitivity turns the observed cusps into a thermometer. The gas trapped in streamers is measured, from the ratios of its emission lines, at about one and a half million degrees, and at that temperature the crossing falls at the two to three solar radii where the cusps are seen. A streamer at one million degrees would have its cusp at several solar radii, and none is observed there. The position of the cusp is a check on the temperature that is independent of any spectral line.

The pattern also runs through the solar cycle. At minimum the large-scale field is a clean dipole concentrated at the poles, and the streamers gather in a belt around the equator. At maximum, when the sunspot belts are full, the corona is hotter and denser over active regions, the large-scale field is a tangle of higher-order components that fall off faster than a dipole, and streamers appear at all latitudes with cusps that vary from place to place. The shape of the eclipse corona has been used since the nineteenth century as a marker of where the cycle stands, long before anyone could have said that it was a map of where two pressures cross.

A corona that cannot sit still #

The pressure the static corona levels off at raises a question that turned out to be one of the more important in solar physics. A hot atmosphere that keeps a finite pressure at infinity has to be held in by something at infinity.

That was Eugene Parker’s argument in 1958. A static corona at a million degrees has an asymptotic pressure millions of times larger than anything in interstellar space could supply, so it cannot be static. It must expand, and Parker showed that the expansion accelerates through a critical point at a few solar radii to become a supersonic outflow — a wind. Observers at the time had seen comet tails pointing away from the Sun more consistently than radiation pressure could explain, and proposed a steady stream of particles; Parker’s calculation said the stream was not optional. It was detected directly by the Mariner 2 spacecraft in 1962, at about four hundred kilometres per second, very close to his prediction. The same critical-point structure fixes the mass-loss rate of a hot star’s radiatively driven wind, where the push comes from line absorption rather than gas pressure.

The two arguments of this essay are therefore one. The coronal gas pressure that levels off is too large to be contained by the interstellar medium, and it is also large enough, at two to four solar radii, to overcome the field that contains it near the Sun. Where the field is strong enough to hold the gas — in the closed loops at the base of streamers, and over active regions — the gas stays. Where the gas wins, it opens the field and leaves. Coronal holes, the dark regions in X-ray images of the Sun, are where the field was open from the start, and they are the sources of the fast wind.

Once the gas leaves, beta falls again #

The static curves stop meaning anything once the gas starts to flow. A flowing gas is no longer in hydrostatic balance; its density is set by conservation of mass along an expanding flux tube, and once the wind is moving at a roughly constant speed, the density falls as the inverse square of distance. That is faster than the levelled-off static pressure, and the field, now open and radial, falls only as the inverse square as well, its pressure as the fourth power. So beta in the wind is not the large number the static curve predicts beyond its crossing. The gas that took over at three solar radii thins itself out by leaving.

The field in the wind is carried out by the gas, frozen into it, while its footpoints rotate with the Sun. The result is the Parker spiral: radial close in, increasingly wound with distance, its azimuthal part falling only as the inverse first power. At four hundred kilometres per second the winding angle reaches forty-five degrees near the Earth’s orbit, which is where the field’s pressure stops falling faster than the gas’s and beta flattens. Measured at the Earth, beta scatters around one — sometimes the gas dominates, sometimes the field — and the solar wind there is a plasma in which neither is in charge. It carries sound waves and magnetic waves at comparable speeds and is structured by both.

So beta traces a full circuit through the Sun’s atmosphere. It is large at the surface, small through the chromosphere and low corona, climbs through one at the streamer cusps, drops low again in the accelerating wind and climbs back to about one at the Earth. The first crossing makes the corona’s loops; the second opens them; the wind’s own beta sets what the Earth’s magnetosphere is pressed by.

Where the field lets go of the spin #

The same surface controls something the Sun has been losing for four and a half billion years: its rotation. Gas moving along a field line that is anchored in the rotating Sun is forced to corotate with its footpoint for as long as the field is stiff enough to force it — roughly, until the wind’s speed exceeds the Alfvén speed, at which point the field can no longer keep the gas turning. The gas therefore leaves the Sun with the angular momentum of a body rotating rigidly out to that radius, not the angular momentum of the surface. That lever arm is why a wind that carries almost no mass carries away almost all of a star’s spin.

The Alfvén radius and the beta-equals-one radius are not the same surface — one compares the flow’s kinetic energy with the field’s, the other the gas’s thermal energy — but in the corona they are close relatives and lie at similar distances, ten to twenty solar radii for the Alfvén surface. The Parker Solar Probe crossed below it for the first time in 2021, at about nineteen solar radii, and found the plasma there magnetically dominated, as the geometry predicts. Where these surfaces sit is set by the same comparison of a power-law field with a gas that has stopped thinning, and it is what the spin-down of Sun-like stars is ultimately measuring.

What the model leaves out #

The corona drawn here is isothermal and static below the crossing. The real corona is neither: its temperature rises through the lowest tenth of a solar radius from the chromosphere’s ten thousand degrees to its million, and the heating that keeps it there is not understood. The gas in closed loops is close to static, but the gas in coronal holes is already flowing at a few solar radii, and there the crossing is replaced by a gradual transition.

The field is a pure dipole. The real field near the surface is dominated by the small-scale fields of active regions and the network, which fall off much faster than a dipole and do not reach high into the corona; only the dipole and a few low-order components survive to two solar radii, which is why a dipole is a fair description of the field at the crossing and a poor one below it. During solar maximum the large-scale field is not a dipole at all, the streamers appear at every latitude, and the crossing radius varies over the Sun.

The density and temperature at the corona’s base, and the base field, are inputs, chosen to match the quiet Sun. Different choices move the crossing within the two-to-five radius range shown; none removes it, because the existence of the crossing depends only on the gas levelling off and the field not.

And beta in the solar wind is drawn from average scalings. The measured wind is a mixture of fast streams from coronal holes and slow wind from the edges of streamers, with transient ejections of magnetised gas passing through both, and beta at the Earth ranges from a tenth to ten from hour to hour. The value of about one is a typical value, not a steady one.

What was actually measured #

Coronal field strengths are among the hardest quantities in solar physics to measure, because the corona is faint and the Zeeman splitting of its hot, broad lines is tiny. Most of what is known comes from extrapolating the photospheric field upward, which assumes the answer to part of the question. Direct measurements have been made from the polarisation of infrared coronal lines, from radio emission at the gyrofrequency above active regions, and from the oscillations of coronal loops, whose periods give the Alfvén speed and hence the field. They agree with the extrapolations at the level of a factor of two and find fields of a few gauss at a solar radius above the quiet Sun.

The gas pressure is better known. The density comes from the brightness of white light scattered by electrons in eclipse and coronagraph images, which is proportional to the number of electrons along the line of sight, and the temperature from the ratio of emission lines of different ionisation stages. The streamer cusps are seen directly. So the crossing is known best by its consequence — the place where the loops stop — and the pressure argument is what says why they stop there rather than somewhere else.

Still open: whether beta decides where the slow wind comes from #

The fast wind, from coronal holes, is well understood as gas escaping along field lines that were always open. The slow wind is not. Its composition resembles the gas trapped in closed loops, not the gas in coronal holes, which suggests it began on closed field lines and was released. The most direct way to release it is at the streamer cusps and along the boundaries of coronal holes, where closed and open field lines meet and reconnect — that is, at exactly the surface where beta returns to one. Whether the slow wind is gas that leaked from closed loops at the second crossing, or gas from the edges of coronal holes that only resembles it, is a question the Parker Solar Probe and Solar Orbiter were designed to answer by sampling the wind close to where it forms, and their early results show the slow wind full of structures that look like loops pinched off at the cusps.