The circulation that should have stirred every fast rotator

Gravity darkening is usually introduced as a fact about the surface of a rapidly rotating star: its poles are hotter than its equator, by an amount von Zeipel’s law predicts from the local gravity. The measurement that tested the law found it only partly right. The temperature contrasts of the rapid rotators imaged by interferometry are smaller than the law’s exponent gives, and that discrepancy is a measurement of something the law leaves out: heat being carried sideways, from pole to equator, by motion inside the star.

The motion is not optional. In 1924 von Zeipel himself pointed out that his law, applied to the interior, leads to a contradiction. A star turning uniformly and transporting its energy by radiation cannot be in hydrostatic balance and in thermal balance at the same time. Resolving the contradiction requires the gas to move, and the resulting large-scale flow — rising at the poles, sinking at the equator, or the reverse, depending on depth — was worked out by Arthur Eddington and later by Peter Sweet, whose names it carries.

A flow that crosses the whole radiative interior does more than carry heat. It carries material. And the material at the bottom of a massive star has been through hydrogen burning.

Why a rotating star cannot sit still #

In a non-rotating star the surfaces of constant pressure, constant density and constant temperature are all concentric spheres, and radiation diffuses outward across them with a flux that the local energy generation exactly supplies. Rotation distorts the surfaces of constant pressure into flattened spheroids, closer together at the poles, where the effective gravity is larger. Radiation diffuses faster where those surfaces are packed together, so more flux leaves through the poles than the equator. That is the gravity darkening seen at the surface.

The trouble is underneath. The flux flowing through a thin shell must be replaced by energy released in the shell, and for uniform rotation the flux law requires the energy release to vary with latitude in a way that nuclear burning and gravitational contraction cannot supply. The shell heats up at some latitudes and cools at others. Buoyancy turns the imbalance into motion, and the motion carries heat from where there is too much to where there is too little until the imbalance is fed as fast as it is created.

The difference between those two contrasts, a factor of 1.39 against 1.29, is several hundred kelvin of heat that the star is not letting accumulate at its poles. Detailed models that include the circulation and the rotation’s own departures from uniformity reproduce exponents below 0.25 that fall further as rotation approaches critical. The surface temperature map and the interior flow are the same phenomenon seen from two sides.

How fast the circulation turns #

The speed of the flow can be estimated from how far out of balance the star is. The imbalance is proportional to the ratio of centrifugal to gravitational acceleration at the surface,

$$\chi = \frac{\Omega^2 R^3}{GM},$$

which for a star at a fraction ω of its critical rate is $(8/27)\,\omega^2$. The time for the imbalance to drive material across the star is the star’s thermal adjustment time, the Kelvin–Helmholtz time — how long it would shine on its stored heat alone — divided by that ratio:

$$t_{\rm ES} \approx \frac{t_{\rm KH}}{\chi}, \qquad t_{\rm KH} = \frac{GM^2}{RL}.$$

For the Sun the Kelvin–Helmholtz time is about thirty million years, and the Sun turns at less than one per cent of its critical rate, so χ is two parts in a hundred thousand and the circulation time is over a trillion years. The Sun’s radiative interior is not stirred by it. Its interior turns as one piece for other reasons.

The figure shows why the estimate is alarming. Massive stars live short lives but have short thermal times too — a 15-solar-mass star burns through its hydrogen in about twenty million years and would radiate away its stored heat in under a hundred thousand — and the ratio of the two is nearly the same across a wide range of mass. So the dividing line falls at about the same rotation rate for every star: a tenth of critical. Stars hotter than the convective boundary routinely turn at a third or half of critical, and some near break-up. By the classical estimate, almost all of them should be mixed from their cores to their surfaces several times over before they leave the main sequence.

The rotation rate at which that happens is far below the rate at which gravity darkening becomes conspicuous.

A star in this condition gives no sign at its surface that it rotates fast enough to matter. Its shape is indistinguishable from a sphere, and its pole is only about one per cent hotter than its equator, a difference of about a hundred kelvin on a star of 10,000. Yet the imbalance that drives the circulation scales with the square of the rotation rate just as the distortion does, and against a stellar lifetime the classical estimate says that small a push is enough.

What stops it #

Real stars are plainly not all fully mixed. A fully mixed star would burn all its hydrogen rather than only its core’s, would evolve along a completely different track, and would show hydrogen-burning products at its surface from early in its life. Most rapidly rotating B stars show nothing of the kind. Something slows the circulation far below the classical estimate.

The main brake is the star’s own composition. Hydrogen burning in the core converts hydrogen to helium and raises the mean molecular weight there. A parcel of core material carried upward by the circulation is heavier than the gas around it at the same pressure, and it resists being lifted, in the same way that salt water resists mixing upward into fresh water. Leon Mestel showed in 1953 that the gradient of molecular weight the core builds can throttle the circulation almost completely across the boundary of the burning region. The circulation then stirs the envelope but has difficulty reaching the material it would need to bring up.

A second brake, and a second source of mixing, is shear. Circulation carries angular momentum as well as heat, so a rotating star does not stay in uniform rotation; different layers come to turn at different rates, and the resulting shear drives small-scale turbulence that mixes on its own terms. How that turbulence behaves in a stably stratified, heat-conducting layer is only partly understood. The same difficulty appears in the thin layer below the Sun’s convection zone, which should have spread and has not, and in the edge of a convective core, where a parameter stands in for a theory.

Stellar models that include rotation therefore carry adjustable efficiencies for how strongly the circulation and the shear mix material, and those efficiencies are calibrated against observed stars. The calibration needs a tracer.

The quiet stars that show what stirring erases #

The clearest evidence that the circulation does act, in the envelopes it can reach, comes from the stars in which it is too weak to.

In a perfectly still radiative atmosphere, the elements do not stay evenly mixed. Heavy atoms sink under gravity, while atoms that absorb strongly in the outgoing radiation are pushed upward by it, and over millions of years the surface layers separate into an abundance pattern unlike the star’s bulk composition. A-type stars with exactly such patterns — some metals enhanced tenfold, calcium and scandium depleted — are common, and they are known as metallic-line stars.

They are also nearly all slow rotators. Among A stars, the chemically peculiar ones are concentrated at projected speeds below about a hundred kilometres a second, and above that the peculiarities largely vanish. The interpretation is direct: a star turning fast enough drives a circulation in its outer envelope that stirs faster than the separation can proceed, and the elements stay mixed. A star turning slowly, often because a close companion has locked its rotation to the orbit, lets them separate. The boundary in rotation between the two kinds of star is a measurement of where the circulation’s speed in the outer envelope overtakes the diffusion’s, and it falls where the classical estimate, applied to those layers, places it.

So the circulation is real and it mixes where it reaches. What the nitrogen question asks is how deep it reaches.

Nitrogen as the tracer #

Stars more massive than about one and a half solar masses burn hydrogen mainly through the CNO cycle, in which carbon, nitrogen and oxygen act as catalysts. The cycle does not change their total number, but it does change their proportions: its slowest step is the one that destroys nitrogen, so nitrogen piles up at the expense of carbon and eventually oxygen. In the burning core of a massive star, material at equilibrium has several times its original nitrogen.

Nitrogen at the surface is therefore a measurement of mixing. A star that brings even a little core material to its surface on the main sequence shows a nitrogen excess while its surface temperature, radius and luminosity barely change. A star that mixes more shows more, and a star that mixes more because it rotates faster should show a nitrogen excess that grows with rotation.

The prediction can be drawn for a population with stated rules.

The rules are a caricature — real models do not enrich as a simple function of speed and age — but they encode the one property every rotational-mixing model shares: nothing enriches a slowly rotating star. The shaded corner of slow, nitrogen-rich stars is empty except for measurement scatter.

Seen through a projected speed #

A spectrum does not measure a star’s equatorial speed. It measures the projected speed $v\sin i$, and a fast rotator seen close to its pole shows a small projected speed and a spectrum like a slow rotator’s. Plotted the way it would be observed, the toy population changes.

Projection can populate the corner, then, and the question is by how much. That depends only on geometry and on how many fast, enriched rotators there are to be tilted into it.

The numbers make the argument sharp. For a star turning at 250 km/s the chance of appearing slower than 50 km/s is one in fifty, and selecting stars by brightness barely changes it. To make a group of apparently slow, enriched stars entirely out of such rotators seen pole-on requires about fifty fast enriched rotators in the sample for every star in the group. A slow enriched group that is a sizeable fraction of a survey cannot come from projection.

Two groups that do not fit #

Large spectroscopic surveys of massive B stars in the Milky Way and in the Magellanic Clouds, whose lower abundances of heavy elements make nitrogen enrichment easier to see, have measured nitrogen and projected rotation for hundreds of stars. Many follow the predicted trend. Two groups do not.

The first is a group of fast rotators with little or no nitrogen excess. Some of these can be explained without new physics: stars early in their main-sequence lives have not had time to mix, and the toy population above has a few of them. Whether youth accounts for all of them depends on how quickly the mixing efficiency calibrated from other stars acts, and that is the quantity being calibrated.

The second is a group of slowly rotating stars with a substantial nitrogen excess. Line profiles put their projected speeds low, and projection, by the argument above, can supply only a small fraction of them. They are either genuinely slow rotators that were mixed by something other than rotation, or stars that rotated fast, mixed, and then lost their spin.

Both routes have physical candidates. A strong magnetic field can mix a star and, through a magnetised wind, remove its angular momentum far faster than an unmagnetised star loses it, leaving a slow rotator with an enriched surface; a few percent of massive stars are measured to have such fields. Binary interaction offers another route: a star that gains mass from a companion is spun up and mixed, and a star whose companion strips its envelope, or two stars that merge, can end up with processed material at the surface and a rotation rate that bears no relation to its history. Since most massive stars are born with a close companion, the second route is not exotic.

A longer life, and a younger-looking cluster #

Mixing changes more than the surface. Any hydrogen the circulation and the shear carry down into the burning core is extra fuel, and a core that is continually topped up lasts longer and grows larger than one that is not. Models of massive stars that include rotation at typical rates live something like ten to twenty-five per cent longer on the main sequence than non-rotating models of the same mass, and they leave it brighter.

That matters wherever an age is read from a cluster’s turnoff, the point where its most massive remaining stars are leaving the main sequence. How long a star of a given mass lasts is the clock, and rotational mixing slows the clock by an amount that depends on how fast the stars turn and on the very efficiencies the nitrogen is meant to calibrate. A cluster of rapid rotators looks younger than its true age if it is read with non-rotating models, and a cluster whose stars have a spread of rotation rates shows a spread of turnoffs even though its stars were born together — the same broadened turnoffs that gravity darkening smears from the outside, produced here from the inside.

The two effects act in the same direction on the diagram and are hard to separate: a rapid rotator is displaced by the angle at which its darkened surface is seen, and it is also a genuinely different star, larger and more luminous, because it has mixed. Disentangling the two in real clusters requires rotation speeds for individual stars, not only their positions, and it is one of the places where the calibration of rotational mixing feeds directly into ages quoted for star clusters.

What the darkening and the nitrogen measure together #

The temperature contrast across a rapid rotator’s surface and the nitrogen in its atmosphere look like unrelated measurements, one made with an interferometer and one with a spectrograph. They are two readings of the same interior flow. The first measures how much heat the circulation carries sideways near the surface; the second measures how much material the circulation and its shear carry up from the core. A star whose surface contrast is well below von Zeipel’s value is a star with an active circulation, and if its nitrogen is normal, something between the circulation and the core is stopping it.

That makes the few rapid rotators whose shapes and temperature maps have been imaged unusually valuable. They are nearby A stars rather than the B stars of the nitrogen surveys, burn hydrogen through a mixture of the two cycles, and have much smaller expected enrichments, so the direct comparison has not yet been made on the same objects. The chain from a measured exponent to a circulation speed to a mixing efficiency passes through a stellar model at every step, and it is the model — with its parameters for the edges of convective regions and its treatment of shear — that the two measurements are jointly testing.

Still open: why are some slowly rotating massive stars nitrogen-rich? #

The slow, enriched group is too large to be fast rotators seen pole-on, and its two leading explanations make different predictions. If magnetic braking made those stars slow, they should carry measurable surface magnetic fields, and the fraction of massive stars with detected fields is smaller than the fraction of stars in the group, though fields can decay and surveys are not sensitive to weak ones. If binary interaction made them, they should be preferentially found with companions, or show the signatures of having gained or lost mass, and should be commoner in populations rich in close binaries. The group’s size depends on where the lines between slow and fast and between enriched and normal are drawn, and on how the surveys were selected. Which mechanism dominates, and whether rotational mixing alone describes the stars that do follow the trend, is not settled — and until it is, the mixing efficiencies calibrated from those stars carry an uncertainty that propagates into every model of how a massive star evolves.