The part of a star that boils

The furnace at the centre of a star runs cooler than a compost heap per unit volume and still has to get its output to the surface. A star makes its energy in the middle and radiates it from the surface, and the interesting question is what happens in between. There are only two mechanisms available. Photons can carry the energy outward, diffusing through the material; or the material itself can carry it, hot gas rising and cool gas sinking.

Which of the two occurs is not a matter of degree. It is decided by an inequality, and on either side of the inequality a star has a completely different structure.

The criterion is a comparison of two slopes #

Take a small parcel of gas somewhere inside a star and displace it upward a little. It arrives where the pressure is lower, so it expands; the expansion is fast compared with the time it takes to exchange heat with its surroundings, so it is adiabatic.

The parcel is now at some new density. If it is denser than the material around it, it sinks back and the layer is stable. If it is less dense, it keeps rising, and the displacement grows: the layer is unstable and convection sets in.

Written in gradients — logarithmic derivatives of temperature with respect to pressure, which is the form that removes the scale — the condition for instability is

$$\nabla_{\rm rad} > \nabla_{\rm ad},$$

which is the Schwarzschild criterion, published by Karl Schwarzschild in 1906.

The right-hand side is a thermodynamic property of the gas. For a fully ionised monatomic ideal gas it is $1 - 1/\gamma = 0.4$, and it does not depend on the star at all.

The left-hand side is where all the astrophysics lives:

$$\nabla_{\rm rad} \propto \frac{\kappa L_r P}{M_r T^4}.$$

That expression is large in two circumstances, and they are the two circumstances that produce convection zones. Either the opacity $\kappa$ is high — which happens in cool outer layers where hydrogen is only partly ionised and the H⁻ ion is an enormously effective absorber — or the ratio $L_r/M_r$ is large, which happens when the energy generation is concentrated into a small central region.

Why radiation is slow #

The reason radiative transport can fail to keep up is worth making concrete, because the numbers are outside ordinary experience.

The mean free path is $1/\kappa\rho$, and at the centre of the Sun — an opacity around $0.17\ \mathrm{m^2\,kg^{-1}}$ and a density of $1.5\times10^5\ \mathrm{kg\,m^{-3}}$ — it is about four hundredths of a millimetre. Radiation moves energy at the speed of light and gets nowhere, because it is not going in a straight line.

That is why a steep temperature gradient is needed to push a given luminosity out radiatively: diffusion is driven by the gradient, and a slow diffusion needs a steep one. If the gradient required exceeds the one convection would establish, convection takes over, because it is the faster of the two.

The Sun: radiative inside, convective outside #

For the Sun the outcome is that the outer 29 per cent of the radius convects and everything below it does not.

The reason is the opacity. Near the surface the temperature has fallen to a few thousand kelvin, hydrogen is only partly ionised, and the H⁻ ion — a hydrogen atom with a second, very loosely bound electron — provides an opacity that rises steeply as the gas cools. $\nabla_{\rm rad}$ goes up with it, crosses 0.4, and the layer boils.

There is a second effect working the same way and it is easy to miss. In a partly ionised gas the adiabatic gradient itself falls, because energy put into the parcel goes into ionising hydrogen rather than into raising its temperature. So the two curves approach each other from both sides at once.

Both of those are statements about where the boundary sits inside a star whose structure is taken as given. It is worth seeing that structure once, because everything above is a comparison of two gradients through it and neither gradient means anything without it.

The massive star does it the other way round #

Now hold the opacity story fixed and change where the energy is made.

A concentrated source means a large $L_r$ inside a small $M_r$, which makes $\nabla_{\rm rad}$ large at small radius. The core becomes unstable.

So stellar structure has three regimes and they are ordered by mass. Below about 0.35 solar masses a star is convective throughout, top to bottom, because the opacity is high everywhere. Between 0.35 and 1.3 it has a radiative core and a convective envelope, like the Sun. Above 1.3 it has a convective core and a radiative envelope. The Sun and an O star have opposite interiors, and the reason is which nuclear reaction each of them runs.

The consequence that matters most is not structural but evolutionary. A convective core is continually mixed, so it burns all of its hydrogen rather than only the hydrogen at the exact centre — which enlarges the fuel supply, extends the main-sequence lifetime, and leaves behind a helium core of a well-defined mass. Every prediction about how a massive star ends depends on how big that mixed region is.

What was actually measured #

The base of the Sun’s convection zone is at $0.713 \pm 0.001$ solar radii. That is a measurement, not a model output, and how it is obtained is one of the more remarkable things in observational astrophysics.

The Sun rings. Millions of acoustic modes — standing sound waves — are excited by the convection itself, and each mode’s frequency depends on the sound speed along the path it samples. Doppler measurements of the solar surface, taken continuously for decades by instruments such as GONG and SOHO’s MDI, measure those frequencies to parts in $10^5$, and inverting the whole set gives the sound speed as a function of depth.

At the base of the convection zone the sound-speed gradient changes abruptly, because the temperature gradient does — a convective region follows the adiabatic gradient and a radiative one does not. The discontinuity is sharp, and locating it is what gives 0.713 to four significant figures.

The same inversions give the helium abundance of the envelope, from the signature of the helium ionisation zone, and they are what turned the solar oxygen abundance revision into a decade-long problem: the new abundances change the opacity in the radiative interior, which moves the predicted boundary away from 0.713 by several times the measurement error.

The other measurement is direct. Convection at the surface is visible.

What the boundary does to everything above it #

Three consequences that are usually presented as separate subjects and are all this one inequality.

The Hayashi track. A fully convective star cannot be arbitrarily cool: convection is so efficient at moving energy that a star’s surface temperature is pinned to within a narrow range, around 4,000 kelvin, almost independently of its luminosity. That produces a nearly vertical line on the Hertzsprung–Russell diagram, and a contracting protostar descends it before turning left onto the main sequence. The line is a forbidden boundary — no star in hydrostatic equilibrium exists to the right of it — and it is a statement about convective efficiency rather than about nuclear burning, which is why it applies to objects that have not started burning at all.

Lithium. Lithium is destroyed by protons at about 2.5 million kelvin, which is far below the temperature at the centre of any star and well above the temperature at its surface. So a star’s surface lithium survives only if the convection zone does not reach down to that depth. Measuring lithium in a cluster’s stars therefore measures how deep their convection zones are, star by star and mass by mass, and the observed depletion pattern is one of the few direct tests of the boundary in stars other than the Sun. It fails: the models under-predict the depletion, which is generally read as evidence for extra mixing that the Schwarzschild criterion does not describe.

And the dynamo. A convection zone is an electrically conducting fluid in rotation, which is the recipe for a magnetic dynamo. Stars with convective envelopes are magnetically active — spots, flares, X-ray emission, and a rotation that slows over time as the field carries angular momentum away in a wind. Stars without them are not. The boundary in the mass function at which the envelope becomes convective is visible directly in the rotation rates of stars in open clusters, as a sharp change in behaviour at about 1.3 solar masses. That is the same 1.3 the CNO cycle takes over at, and it is not a coincidence: both are the same structural transition.

How far past adiabatic a convecting layer actually sits #

The criterion is an inequality between two gradients, and it says nothing about by how much the inequality is satisfied. That margin turns out to vary by six orders of magnitude between the bottom of a convection zone and the top, and it is the reason the free parameter of the previous section matters where it does.

Convection carries heat by moving material, so the flux it delivers is roughly the density times the heat capacity times the velocity times the temperature excess of a rising parcel over its surroundings. In the deep interior the density is enormous — hundreds of kilograms per cubic metre at the base of the Sun’s convection zone — so an extremely small temperature excess suffices to carry the entire luminosity.

Putting the solar numbers in gives a superadiabaticity of about $10^{-6}$: the actual gradient exceeds the adiabatic one by a part in a million, and everywhere below the top few hundred kilometres the two are indistinguishable for any purpose. The gas is convecting vigorously and its stratification is adiabatic to six decimal places.

That is what makes the deep interior easy. A model does not need to know how convection works down there, only that it happens, because the gradient is pinned to a thermodynamic quantity that follows from the equation of state alone. The mixing length could be wrong by a factor of three and the structure would barely move.

Near the surface everything inverts. The density falls by orders of magnitude, so the same flux now requires a large temperature excess, and the superadiabaticity climbs to order unity in the last few hundred kilometres. The whole of the uncertainty in a stellar model’s convection is concentrated in a layer a thousandth of the star thick — and that layer is the one that sets the star’s radius, produces its granulation, and forms its spectral lines.

So the mixing length is fitted to the Sun’s radius because the radius is exactly the quantity the superadiabatic layer controls, and the fit is a statement about the top of the convection zone rather than about the bulk of it. A model that reproduces the Sun and fails for a red giant is failing in that thin layer, under conditions the calibration never covered.

What the picture cannot show #

Mixing-length theory, which is what the figures quietly assume away. The Schwarzschild criterion says whether a layer convects. It says nothing about how much energy the convection carries, how fast the gas moves, or how far a parcel travels before dissolving. The standard treatment invents a length — the mixing length, usually written as a multiple $\alpha$ of the pressure scale height — and $\alpha$ is a free parameter, fitted by requiring that a solar model reproduce the Sun’s radius and luminosity at its known age. Every stellar model in use contains that fitted number, and it is then applied to stars quite unlike the Sun.

Overshoot. A parcel arriving at the boundary of the unstable region is moving, and it does not stop there. It overshoots into the stable layer, mixing material across a boundary the criterion says is sharp. How far it goes is another parameter, and for a convective core it changes the amount of fuel available and therefore the star’s lifetime by tens of per cent.

And the model here is a polytrope. The interior used throughout is an $n = 3$ polytrope with a single Kramers opacity, integrated from the Lane–Emden equation. It puts the outer boundary at 89 per cent of the radius against the measured 71.3, and it produces a small convective core in the proton–proton case where a real solar model has essentially none. Those gaps are the price of a model with two ingredients, and they are stated here rather than hidden because the figure’s job is to show which way the inequality goes and where, not to be a solar model.

The other criterion, and why there are two #

There is a second version of the instability condition, and the difference between the two is a real physical question rather than a matter of taste.

The Schwarzschild criterion compares the two gradients assuming the gas has a uniform composition. Inside a star that has been burning for a while, it does not: the core is enriched in helium, which is heavier per particle, so there is a composition gradient as well as a temperature gradient. A parcel rising out of the core is then heavier than its surroundings for a reason that has nothing to do with temperature, and the extra weight can stabilise a layer the temperature gradient alone would destabilise.

Including that term gives the Ledoux criterion, which is more restrictive. Between the two lies a regime — Schwarzschild-unstable and Ledoux-stable — in which the layer neither convects properly nor sits still: it undergoes a slow, oscillatory mixing usually called semiconvection, which is not well described by either criterion and is treated in models by yet another free parameter.

Which criterion to use is an open question with consequences. It changes the size of a massive star’s convective core, therefore the amount of fuel it burns, therefore its lifetime and the mass of the remnant it leaves — and the two choices differ by enough that the same initial mass can produce a neutron star under one and a black hole under the other. The relation between a star’s mass and its fate has this ambiguity built into it.

The shear layer at the bottom #

The inversions that located the boundary at 0.713 measured something else at the same depth, and it was not predicted.

Sound waves travelling with the rotation and against it have slightly different frequencies, so the same set of oscillation modes that gives the sound speed also gives the rotation rate as a function of both radius and latitude. The result was a surprise in both regions it covered.

The convection zone does not rotate as a solid body, and it does not rotate on cylinders either, which is what a rotating convecting fluid was expected to do. Its rotation rate depends mainly on latitude and hardly at all on depth: the equator turns in about 25 days and the poles in about 34, at every level from the surface down to 0.71 of the radius.

The radiative interior beneath rotates almost rigidly, at a rate close to the convection zone’s mid-latitude value.

Between them, therefore, is a layer across which the rotation rate changes by twenty per cent, and the inversions put its thickness at under five per cent of the solar radius. That layer is the tachocline, and its discovery in the early 1990s changed where the dynamo was thought to operate.

The reason is that a shear layer is a place where a magnetic field can be wound up. A poloidal field threading a region whose upper and lower halves rotate at different rates is stretched into a toroidal one, and the stretching is efficient in proportion to the shear — so a thin layer with a large velocity difference across it is the natural site for the amplification the solar cycle requires. It sits at the bottom of the convection zone, where the field can be stored in a stably stratified region rather than being carried to the surface and shed by buoyancy as soon as it forms.

The measurement therefore located a structure nobody had asked for, and the theory of the solar cycle rearranged itself around it. Whether the whole dynamo lives there is now argued about again — distributed dynamo models put much of the action in the convection zone proper, and simulations do not reproduce the tachocline’s thinness — which is the usual condition of a subject where one number is measured to four figures and the mechanism around it is not.

The criterion is a comparison of two gradients, and both the gradient and the surface it produces are worth reading at a second setting.

Where the ladder goes next #

The rung above is three-dimensional convection: simulations that resolve the granulation directly and dispense with the mixing length, which reproduce line profiles and limb darkening without free parameters and are now the standard against which one-dimensional models are calibrated. The rung beside it is what the churning does other than carry heat — it generates a magnetic field, and the interface at the base of the convection zone is where the solar dynamo is thought to live.